geometry of b†cklund transformations
TRANSCRIPT
Geometry of Backlund Transformations
by
Yuhao Hu
Department of MathematicsDuke University
Date:Approved:
Robert Bryant, Supervisor
Hubert Bray
Lenhard L. Ng
Leslie Saper
Dissertation submitted in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in the Department of Mathematics
in the Graduate School of Duke University2018
Abstract
Geometry of Backlund Transformations
by
Yuhao Hu
Department of MathematicsDuke University
Date:Approved:
Robert Bryant, Supervisor
Hubert Bray
Lenhard L. Ng
Leslie Saper
An abstract of a dissertation submitted in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in the Department of Mathematics
in the Graduate School of Duke University2018
Copyright c© 2018 by Yuhao HuAll rights reserved except the rights granted by the
Creative Commons Attribution-Noncommercial Licence
Abstract
This thesis is a study of Backlund transformations using geometric methods. A
Backlund transformation is a way to relate solutions of two PDE systems. If such a
relation exists for a pair of PDE systems, then, using a given solution of one system,
one can generate solutions of the other system by solving only ODEs.
My contribution through this thesis is in three aspects.
First, using Cartan’s Method of Equivalence, I prove the generality result: a
generic rank-1 Backlund transformation relating a pair of hyperbolic Monge-Ampere
systems can be uniquely determined by specifying at most 6 functions of 3 variables.
In my classification of a more restricted case, I obtain new examples of Backlund
transformations, which satisfy various isotropy conditions.
Second, by formulating the existence problem of Backlund transformations as
the integration problem of a Pfaffian system, I propose a method to study how a
Backlund transformation relates the invariants of the underlying hyperbolic Monge-
Ampere systems. This leads to several general results.
Third, I apply the method of equivalence to study rank-2 Backlund transforma-
tions relating two hyperbolic Monge-Ampere systems and partially classify those that
are homogeneous. My classification so far suggests that those homogeneous Backlund
transformations (relating two hyperbolic Monge-Ampere systems) that are genuinely
rank-2 are quite few.
iv
To my parents.
v
Contents
Abstract iv
List of Abbreviations and Symbols viii
Acknowledgements ix
1 Introduction 1
2 Background 7
2.1 Exterior Differential Systems . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Hyperbolic Monge-Ampere Systems . . . . . . . . . . . . . . . . . . . 10
2.3 Equivalence of G-structures . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Backlund Transformations . . . . . . . . . . . . . . . . . . . . . . . . 16
3 The Problem of Generality 22
3.1 G-structure Equations for Backlund Transformations . . . . . . . . . 23
3.2 An Estimate of Generality . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Examples of Higher Cohomogeneity . . . . . . . . . . . . . . . . . . . 31
4 Backlund Transformations and Monge-Ampere Invariants 47
4.1 First Monge-Ampere Invariants . . . . . . . . . . . . . . . . . . . . . 47
4.2 The Backlund-Pfaff System . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Obstructions to Integrability . . . . . . . . . . . . . . . . . . . . . . 55
4.4 A Special Class of Backlund Transformations . . . . . . . . . . . . . 61
4.4.1 Type IIa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
vi
4.4.2 Type IIb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Homogeneous Rank-2 Backlund Transformations 69
5.1 Genericity Conditions and Structure Reduction . . . . . . . . . . . . 69
5.1.1 Case 1: pB1, B2q “ pB3, B4q “ p1, 0q . . . . . . . . . . . . . . 76
5.1.2 Case 2: Bi “ 0 pi “ 1, ..., 4q . . . . . . . . . . . . . . . . . . . 85
5.2 Assuming Genericity Conditions 1, 2 . . . . . . . . . . . . . . . . . . 86
5.2.1 Case: pB1, B2q “ pB3, B4q “ p1, 0q . . . . . . . . . . . . . . . . 88
5.2.2 Integration of the structure equations . . . . . . . . . . . . . . 102
5.3 Assuming Genericity Condition 1 . . . . . . . . . . . . . . . . . . . . 108
5.3.1 Case: pB1, B2q “ pB3, B4q “ p1, 0q . . . . . . . . . . . . . . . . 112
5.3.2 Integration of the Structure Equations . . . . . . . . . . . . . 123
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6 Conclusion 129
A Calculations for Theorem 3.3 131
B Invariants of an Euler-Lagrange System 134
Bibliography 138
Biography 140
vii
List of Abbreviations and Symbols
Symbols
pM, Iq An exterior differential system with manifold M and differentialideal I.
Ik Ă ΛkpT ˚Mq The subbundle corresponding to the degree-k piece of I, whenpM, Iq is an exterior differential system.
xθ1, ..., θ`y The ideal of Ω˚pUq generated by differential forms θ1, ..., θ` (de-fined on U) and their exterior derivatives.
xθ1, ..., θ`yalg The ideal of Ω˚pUq algebraically generated by differential formsθ1, ..., θ` defined on U .
rrθ1, ..., θ`ss The vector subbundle of ΛkpT ˚Uq generated by differential formsθ1, ..., θ` (defined on U) of the same degree k.
CpIq The Cartan system associated to the differential system pM, Iq,i.e., the Frobenius system whose integral curves are precisely theCauchy characteristics of pM, Iq.
X ω The interior product of a smooth vector fieldX with a differentialform ω.
SK Ă TM The distribution spanned by all vector fields X that satisfyX ω “ 0 for any ω P S Ă Ω˚pMq.
viii
Acknowledgements
I’d like to express my deepest gratitude to my advisor, Robert Bryant, for teaching
me how to do mathematics and for his encouragement and guidance during my
research. His influence on me goes beyond mathematics.
In the past years, I’ve benefited a great deal from attending lectures of and talking
with professors in or outside Duke. Special thanks go to
Hubert Bray, Lenny Ng and Leslie Saper, for being on my committee.
Jeanne Clelland, for her inspiring work and her interest in my research.
Clark Bray, for all the training he has given me in mathematical teaching.
David Schaeffer. The memory of your first-year course I’ll always cherish.
I’d like to acknowledge all my friends and fellow students who stood beside me
over the years. Especially, I’d like to thank
Ma Luo and Zhiyong Zhao, for our brotherly 5+ years.
Mendel Nguyen, for mathematics, physics, Chopin and Driade.
Gavin Ball, Ryan Gunderson and Mike Bell, for kindly sharing their research.
Zhennan Zhou, for taking me on hikes and teaching me how to cook.
Sean Lawley, for all our conversations during 2012-14.
Rosa Zhou, for moon, star, and rose.
Thank you, my dear parents. Your thoughts are constantly on me, my well-being.
How many times have you reminded me that I shall never give up. How deeply have
you, by your love, passed on to me the value of a simple and unassuming life.
ix
1
Introduction
In 1882, the Swedish mathematician A.V. Backlund proved the result (see [Bac83],
[BGG03] or [CT80]): Given a surface with a constant Gauss curvature K ă 0 in E3,
one can construct, by solving ODEs, a 1-parameter family of new surfaces in E3 with
the Gauss curvature K. This is the origin of the term “Backlund transformation”.
Classically, a Backlund transformation is a PDE system B that relates solutions
of two other PDE systems E1 and E2. More precisely, such a relation must satisfy
the property: given a solution u of E1 (resp. E2), substituting it in B, one obtains a
PDE system whose solutions can be found by ODE methods and produce solutions
of E2 (resp. E1).
For example, the Cauchy-Riemann system
#
ux ´ vy “ 0,
uy ` vx “ 0(1.1)
is a Backlund transformation; it relates solutions of the Laplace equation ∆z “ 0 for
zpx, yq in the following way: If u satisfies ∆u “ 0, then, substituting it in (1.1), we
obtain a compatible first order system for v, whose solutions can be found by ODE
1
methods and satisfy ∆v “ 0, and vice versa.
As another example, consider the system of nonlinear equations
#
zx ´ zx “ λ sinpz ` zq,
zy ` zy “1λ
sinpz ´ zq,(1.2)
where λ is a nonzero constant. One can show that (1.2) is a Backlund transformation
relating solutions of the sine-Gordon equation
uxy “12
sinp2uq. (1.3)
The system (1.2) is closely connected with the classical Backlund transformation
relating surfaces in E3 with a negative constant Gauss curvature. For details, see
[CT80].
In addition, a Backlund transformation may relate solutions of a parabolic equa-
tion (see [NC82]) or two equations that are nonequivalent (see [CI`09]). Numerous
other examples of Backlund transformations are discussed in [RS02]. Through these
examples, Backlund transformations are found to have rich connection with surface
theory in differential geometry and solitons in mathematical physics.
Among the examples discussed in [RS02], a Backlund transformation relating so-
lutions of the hyperbolic Tzitzeica equation is particularly interesting. The hyperbolic
Tzitzeica equation is the second-order equation for hpx, yq:
plnhqxy “ h´ h´2. (1.4)
This equation was discovered by Tzitzeica in his study of hyperbolic affine spheres
in the affine 3-space A3 (see [Tzi08] and [Tzi09]). He found that the system in α, β,
and h$
’
’
&
’
’
%
αx “ phxα ` λβqh´1 ´ α2,
αy “ βx “ h´ αβ,
βy “ phyβ ` λ´1αqh´1 ´ β2,
(1.5)
2
where λ is an arbitrary nonzero constant, is a Backlund transformation relating
solutions of (1.4). More explicitly, if h solves the hyperbolic Tzitzeica equation
(1.4), then, substituting it in the system (1.5), one obtains a compatible first-order
PDE system for α and β, whose solutions can be found by solving ODEs; for each
solution pα, βq, the function
h “ ´h` 2αβ
also satisfies the hyperbolic Tzitzeica equation (1.4). Furthermore, one can show
that, unlike the systems (1.1) and (1.2), substituting a solution h of (1.4) into (1.5)
yields a system whose solutions depend on 2 parameters instead of 1. Using our
terminology (Definition 2.13 of Chapter 2), one can verify that the system (1.5)
corresponds to a rank -2 Backlund transformation.
An ultimate goal of studying Backlund transformations is solving the Backlund
problem, which was considered by Goursat in [Gou25]. The statement of the Backlund
problem is: Find all pairs of systems of PDEs whose solutions are related by a
Backlund transformation. This problem remains unsolved.
However, recent works of Jeanne Clelland have shed new light on the classification
of Backlund transformations. Her paper [Cle01], in particular, focuses on Backlund
transformations relating solutions of two hyperbolic Monge-Ampere systems, which
are second-order PDEs in the plane arising frequently in differential geometry.
Clelland’s approach in [Cle01] to Backlund transformations involves several key
steps. First, a Backlund transformation relating two hyperbolic Monge-Ampere sys-
tems is formulated as an exterior differential system ([BCG`13]). This allows one to
study a Backlund transformation geometrically as a manifold N with a structure B.
As a result, concepts such as equivalence and symmetry can be easily defined. Sec-
ond, she applies Cartan’s method of equivalence to derive local invariants of such a
Backlund transformation. Third, by assuming all local invariants to be constants, she
3
obtains a complete classification of homogeneous rank-1 Backlund transformations
relating two hyperbolic Monge-Ampere systems, where homogeneous means that the
symmetry group of a Backlund transformation acts transitively on the underlying
manifold.
The aim of the present work is to address the following questions:
i. Up to equivalence, what is the generality of Backlund transformations
relating two hyperbolic Monge-Ampere systems?
ii. Given two hyperbolic Monge-Ampere systems, how to tell whether
they are related by a Backlund transformation?
iii. What can be concluded about the existence of Backlund transforma-
tions in higher ranks?
Of course, similar questions can be asked for Backlund transformations relating two
PDE systems in broader classes (elliptic, hyperbolic, parabolic), but, for clarity, in
this thesis, we focus on Backlund transformations relating two hyperbolic Monge-
Ampere systems.
This thesis is organized as follows.
In Chapter 2, we develop the basic concepts and techniques that are used in this
work. These include the notion of an exterior differential system, in particular, that
of a hyperbolic Monge-Ampere system; the notion of a G-structure, which is at the
heart of an equivalence problem; and the notion of a Backlund transformation.
In the first two sections of Chapter 3, we prove a generality result:
Theorem 3.3. A generic rank-1 Backlund transformation relating two hyperbolic
Monge-Ampere systems can be specified uniquely, up to equivalence, by initial data
consisting of at most 6 functions of 3 variables.
This theorem implies the following
4
Corollary 3.4. There exist hyperbolic Monge-Ampere systems that are not related
to any other hyperbolic Monge-Ampere system by a generic rank-1 Backlund trans-
formation.
Before summarizing the main ideas of proving Theorem 3.3, we briefly explain
what the term “generic” means. Given a rank-1 Backlund transformation pN,Bq
relating two hyperbolic Monge-Ampere systems, there is an intrinsic way to define
a tensor field T on N (Section 3.1). A rank-1 Backlund transformation is said to be
generic if the tensor T takes generic values.
To prove Theorem 3.3, we first show that, given a generic rank-1 Backlund trans-
formation pN,Bq relating two hyperbolic Monge-Ampere systems, where N is a 6-
manifold, there is a canonical way to define a local coframing on N . One can show
that two such coframings are locally equivalent up to diffeomorphism if and only if
the corresponding Backlund transformations are locally equivalent. Then, we apply
a theorem of Cartan ([Bry14]) to show that such local coframings, up to diffeomor-
phism, depend on at most 6 functions of 3 variables.
In the last section of Chapter 3, we focus on the case when two invariants of a
generic rank-1 Backlund transformation are assumed to be specific constants. We
can classify such Backlund transformations. In particular, we find new examples of
Backlund transformations with cohomogeneity 1, 2 and 3.
In Chapter 4, we study how obstructions to the existence of Backlund transfor-
mations may be expressed in terms of the invariants of the underlying hyperbolic
Monge-Ampere systems. Such obstructions can be found by applying techniques
of exterior differential systems ([BCG`13]) to a rank-4 Pfaffian system. We obtain
several results that inform us which pairs of hyperbolic Monge-Ampere systems may
be related by a rank-1 Backlund transformation of a particular type.
In Chapter 5, motivated by the example (1.5), we study rank-2 Backlund trans-
5
formations relating two hyperbolic Monge-Ampere systems, and partially classify
those that are homogeneous1. The approach to classification is analogous to that in
[Cle01]. The results of classification, as well as the cases that are work in progress,
are summarized in Section 5.4.
Most calculations in Chapters 3,4, and 5 are performed using MapleTM.
1 It turns out that the rank-2 Backlund transformation corresponding to (1.5) is nonhomogeneous.
6
2
Background
2.1 Exterior Differential Systems
Definition 2.1. Let M be a smooth manifold, I Ă Ω˚pMq a graded ideal that
is closed under exterior differentiation. The pair pM, Iq is said to be an exterior
differential system with space M and differential ideal I.
Given an exterior differential system pM, Iq, we use Ik to denote the degree-k
piece of I, namely, Ik “ I X ΩkpMq, where ΩkpMq stands for the C8pMq-module
of differential k-forms on M . If the rank of Ik, restricted to each point, is locally a
constant, then the elements of Ik are precisely smooth sections of a vector bundle
denoted by Ik.
Definition 2.2. An integral manifold of an exterior differential system pM, Iq is a
submanifold i : N ãÑM satisfying i˚φ “ 0 for any φ P I.
By this definition, the requirement of I being closed under exterior differentiation
is natural, because of the identity i˚ ˝ d “ d ˝ i˚.
Intuitively, an exterior differential system is a coordinate-independent way to
express a PDE system; an integral manifold, usually with a certain independence
7
condition satisfied, corresponds to a solution of the PDE system. We illustrate this
point by the following example.
Example 1. A single k-th order PDE in one unknown function u of n independent
variables x “ px1, ..., xnq can be expressed in the form
F px, u, Bα1upxq, ..., Bαkupxqq “ 0, (2.1)
where αi ranges over all multiindices of length i from 1, 2, ..., n. For simplicity, we
regard two multiindices as equivalent if and only if they are reorderings of each
other. Furthermore, if α is a multiindex of length i, then jα, where j P t1, ..., nu, is
a multiindex of length i` 1.
The equation (2.1) corresponds to an exterior differential system pM, Iq induced
from the canonical contact system on the k-jet bundle JkpRn,Rq. To be explicit,
JkpRn,Rq has the standard coordinates px, u, pα1 , ..., pαkq, where αi are multiindices
in the sense above. The canonical contact system C on JkpRn,Rq is algebraically
generated by the 1-forms
θ0 “ du´ pidxi,
θαi “ dpαi ´ pjαidxjpi “ 1, 2, ..., k ´ 1q
and their exterior derivatives.
Let w : Rn Ñ R be a Ck-function. Then w : Rn Ñ JkpRn,Rq, defined by
wpxq “ px,wpxq, Bα1wpxq, ..., Bαkwpxqq
is an n-dimensional integral manifold of the exterior differential system pJkpRn,Rq, Cq.
This w is called the lifting of w to JkpRn,Rq. Conversely, suppose that v : Rn Ñ
JkpRn,Rq is an n-dimensional integral manifold of pJkpRn,Rq, Cq on which x1, x2, ..., xn
are independent functions, in other words, on which dx1 ^ dx2 ^ ¨ ¨ ¨ ^ dxn ‰ 0 ev-
erywhere. We then have
pi ˝ v “B
Bxipu ˝ vq, pij ˝ v “
B2
BxiBxjpu ˝ vq, ...
8
Therefore, locally, a Ck-function w : Rn Ñ R is in one-to-one correspondence with
an integral manifold of pJkpRn,Rq, Cq on which dx1 ^ dx2 ^ ¨ ¨ ¨ ^ dxn ‰ 0.
Now let M Ă JkpRn,Rq be defined by the equation
F px, u, pα1 , ..., pαkq “ 0,
where F is as in (2.1). If ∇F is nonzero at a point, then, by shrinking to a neigh-
borhood of that point, we can assume M to be a smooth manifold. Under this
assmuption, let CM be the restriction of the contact system C to M . Then, by an
argument similar to the above, one can show that an integral manifold of the exte-
rior differential system pM, CMq on which dx1 ^ dx2 ^ ¨ ¨ ¨ ^ dxn ‰ 0 is in one-to-one
correspondence with a solution of (2.1) whose lifting to JkpRn,Rq is contained in M .
Regarding two exterior differential systems, the following notion of equivalence is
natural.
Definition 2.3. Two exterior differential systems pM, Iq and pN,J q are said to be
equivalent up to diffeomorphism, or equivalent, for brevity, if there exists a diffeo-
morphism φ : M Ñ N such that φ˚J “ I. Such a φ is called an equivalence between
both systems. An equivalence between pM, Iq and itself is called a symmetry of
pM, Iq.
Example 2. When J1pRn,Rq with coordinates pxi, u, piq is regarded as a contact
manifold with the contact form
θ “ du´ pidxi,
a symmetry of the exterior differential system pJ1pRn,Rq, Cq is just a contact trans-
formation of the space J1pRn,Rq to itself.
We end this section by presenting the Frobenius Theorem, formulated in terms of
exterior differential systems. This theorem is at the heart of our characterization of
9
a Backlund transformation. For a proof of the Frobenius theorem, see Chapter II of
[BCG`13].
Theorem 2.1. (Frobenius) Let pMn, Iq be an exterior differential system. If, on an
open U Ă M , the differential ideal I is generated by k linearly independent 1-forms
θ1, ..., θk satisfying
dθi ” 0 mod θ1, θ2, ..., θk,
then any p P U has an open neighborhood V Ă U on which there exists a coordinate
system px1, ..., xn´k, y1, ..., ykq such that the ideal I is generated by dy1, dy2, ..., dyk.
Remark 1. One can show ([BCG`13]) that a coordinate system px1, ..., xn´k, y1, ..., ykq
in the conclusion of Theorem 2.1 can be found by solving systems of ODEs. Once
such a coordinate system is obtained, setting the yi pi “ 1, ..., kq to be constants de-
fines an pn´kq-dimensional integral manifold of pM, Iq. Each such integral manifold
is called a leaf associated to the system pM, Iq.
Definition 2.4. Given an exterior differential system pM, Iq. If the assumption in
Theorem 2.1 holds on a neighborhood of every p P M for a constant k, then pM, Iq
is called a rank-k Frobenius system.
2.2 Hyperbolic Monge-Ampere Systems
Among second order PDEs for 1 unknown function of 2 independent variables, the
Monge-Ampere equations are of special interest, as they frequently arise in differential
geometry (see [Bry02]) and the calculus of variations (see [BGG03]).
The general form of a Monge-Ampere equation for zpx, yq is
Apzxxzyy ´ z2xyq `Bzxx ` 2Czxy `Dzyy ` E “ 0, (2.2)
where A,B,C,D,E are functions of x, y, z, zx, zy. A Monge-Ampere equation (2.2)
is said to be elliptic (resp., hyperbolic, parabolic) if AE´BD`C2 is negative (resp.,
10
positive, zero).
For example, the sine-Gordon equation (1.3) and the Tziteica equation (1.4) are
hyperbolic Monge-Ampere equations; the Cauchy-Riemann equation (1.1) and the
equation1 zxxzyy ´ z2xy “ 1 are elliptic Monge-Ampere equations; in the classical
calculus of variations, the Euler-Lagrange equation for a first-order functional
ż
Ω
Lpx, zpxq,∇zpxqqdx, Ω Ă R2, L : J1pΩ,Rq Ñ R
is always Monge-Ampere (see [BGG03]2).
A Monge-Ampere equation can be formulated as an exterior differential system
on a contact manifold. In the hyperbolic case, we follow [BGH95] to give the
Definition 2.5. A hyperbolic Monge-Ampere system pM, Iq is an exterior differential
system, where M is a 5-manifold, I being locally algebraically generated by θ P I1
and dθ,Ω P I2 satisfying
p1q θ ^ pdθq2 ‰ 0;
p2q rrdθ,Ωss, modulo θ, has rank 2;
p3q pλdθ ` µΩq2 ” 0 mod θ has two distinct solutions rλi : µis P RP1 pi “ 1, 2q.
In these three conditions, the first says that θ is contact; the second says that the
corresponding PDE system is nonempty; the third characterizes hyperbolicity, that
is, each integral surface of pM, Iq is foliated by two distinct families of characteristics.
Example 3. A PDE of the form
zxy “ F px, y, z, zx, zyq
1 cf. Jørgen’s Theorem in [Bry02].
2 In [BGG03], a Monge-Ampere equation is a second order PDE for 1 unknown function of nindependent variables, and it is shown that an Euler-Lagrange equation can be formulated as anEuler-Lagrange exterior differential system, which can be intrinsically characterized. In this thesis,we only consider the case when n “ 2.
11
corresponds to a hyperbolic Monge-Ampere system pM, Iq, where M “ J1pR2,Rq
with the standard coordinates px, y, z, p, qq; I is algebraically generated by
θ “ dz ´ pdx´ qdy,
dθ “ dx^ dp` dy ^ dq,
Ω “ pdp´ F px, y, z, p, qqdyq ^ dx.
In particular, Ω and dθ ` Ω are decomposable 2-forms.
Example 4. The oriented orthonormal frame bundle O over the Euclidean space
E3 consists of elements of the form px; e1, e2, e3q, where x P R3, and pe1, e2, e3q is an
oriented orthonormal frame at x. On O, we have the canonical structure equations
dωi “ ´ωij ^ ωj,
dωij “ ´ωik ^ ω
kj ,
where ωij “ ´ωji . Consider the exterior differential system pO, Iq, where
I “ xω3, dω3, dω12 ´Kω
1^ ω2
yalg, Kconst .
An integral surface of pO, Iq on which ω1 ^ ω2 ‰ 0 corresponds to a surface S in E3
with constant Gauss curvature K and with an orthonormal frame field pe1, e2, e3q
attached to it, e3 being normal to S.
Let X be a nonzero vector field on O that annihilates ω1, ω2, ω3, ω13 and ω2
3. For
each φ in the three algebraic generators of I above, we have
X φ “ 0, LXφ “ 0.
This implies that on the quotient space M5 of O by the flow of X, letting π : O ÑM
be the quotient map, there exists a well-defined differential form α such that π˚α “ φ.
It follows that I descends to M to be an exterior differential system J , in the sense
that I is algebraically generated by the elements of π˚J . Moreover, pM,J q is a
12
hyperbolic Monge-Ampere system if and only if the constant K ă 0. This, in part,
follows from the equality
pλdω3` µpdω1
2 ´Kω1^ ω2
qq2” pλ2
`Kµ2qpdω3
q2 mod ω3.
By Definition 2.5, on each hyperbolic Monge-Ampere system pM, Iq, locally there
exist 1-forms θ, ω1, ω2, ω3, ω4, linearly independent everywhere, such that
I “ xθ, ω1^ ω2, ω3
^ ω4y.
One can show that the pair of Pfaffian systems I10 “ xθ, ω1, ω2y and I01 “ xθ, ω
3, ω4y
are well-defined up to ordering. Restricted to an integral surface of pM, Iq, each
of I10 and I01 becomes a rank-1 Frobenius system whose integral curves are the
characteristics of pM, Iq in the usual sense of hyperbolic PDEs. This motivates the
Definition 2.6. Given a hyperbolic Monge-Ampere system3 pM, Iq, where I “
xθ, ω1 ^ ω2, ω3 ^ ω4y, the systems I10 “ xθ, ω1, ω2y and I01 “ xθ, ω
3, ω4y are called
the characteristic systems associated to pM, Iq.
2.3 Equivalence of G-structures
Definition 2.7. Let M be an n-dimensional manifold. A coframe at p P M is a
linear isomorphism up : TpM Ñ Rn. The set of all coframes on M forms a principal
right GLpn,Rq-bundle with the group action u ¨g :“ g´1u. This is called the coframe
bundle over M , denoted as FpMq. A local section of FpMq defined on U Ă M is
said to be a coframing on U .
Many (local) differential geometric structures on a smooth manifold Mn can be
equivalently expressed by a notion of ‘admissible’ coframings, with the property that
3 The notion of a characteristic system can apply to hyperbolic exterior differential systems ingeneral. See [BGH95].
13
any two such coframings differ pointwise by the action of a Lie group G Ă GLpn,Rq.
For example, a Riemannian metric g on Mn can be viewed as all local coframings
pω1, ω2, ..., ωnq satisfying g “ pω1q2 ` ¨ ¨ ¨ ` pωnq2, any two such coframings (defined
on the same domain) differing by a pointwise action of the orthogonal group Opnq;
a symplectic form Ω on M2n can be viewed as all local coframings pω1, ω2, ..., ω2nq
satisfying Ω “ ω1 ^ ω2 ` ¨ ¨ ¨ ` ω2n´1 ^ ω2n, any two such coframings differing by a
pointwise action of the symplectic group Spp2nq; an almost complex structure J on
M2n is characterized by any basis of p1, 0q-forms on M , whose real and imaginary
parts comprising a local coframing, any two such coframings differing by a pointwise
action of GLpn,Cq Ă GLp2n,Rq; etc. This motivates the following definition.
Definition 2.8. Let G Ă GLpn,Rq be a Lie subgroup. A G-structure on a smooth
manifold Mn is a principal G-subbundle of the coframe bundle FpMq.
When dealing with equivalence between two local geometric structures defined in
terms of coframes, it is necessary to take into account the ambiguity in the choice
of admissible coframings. For example, Let pMi, giq pi “ 1, 2q be two Riemannian
n-manifolds, where gi “ pω1piqq
2 ` ¨ ¨ ¨ ` pωnpiqq
2. These Riemannian structures are
locally equivalent if and only if there exists a (local) diffeomorphism φ : U1 Ñ U2
(Ui Ă Mi open) such that φ˚ωp2q “ γωp1q for some γ : U1 Ñ Opnq, where ωpiq “
pω1piq, ω
2piq, ..., ω
npiqq pi “ 1, 2q. However, the ambiguity represented by γ can be removed
by passing to G-structures, as we will describe below.
Definition 2.9. Let G be aG-structure onM , with π : G ÑM being the submersion.
The tautological 1-form τ on G is the Rn-valued 1-form determined by the equation
τpvq “ upπ˚pvqq, for any u P G and v P TuG.
Definition 2.10. Two G-structures, G1 and G2, with tautological 1-forms τ1, τ2,
respectively, are said to be equivalent if there exists a diffeomorphism φ : G1 Ñ G2,
such that φ˚τ2 “ τ1.
14
The following proposition is proved in [Gar89].
Proposition 2.1. Let Mi pi “ 1, 2q be two smooth n-manifolds, each with a cofram-
ing ωi defined on some open neighborhood Ui Ă Mi. Let G Ă GLpn,Rq be a Lie
subgroup. Let Gi be the G-structure on Ui defined by Gi “ tωippq ¨ h| p P Ui, h P Gu.
There exists a diffeomorphism φ : U1 Ñ U2 and a map g : U1 Ñ G such that
φ˚ω2 “ gω1 if and only if there exists an equivalence of G-structures φ : G1 Ñ G2.
Example 5. Consider a hyperbolic Monge-Ampere system pM, Iq. A local coframing
Θ “ pθ0, θ1, ..., θ4q defined on an open neighborhood U Ă M is said to be 0-adapted
if, on U , the differential ideal I can be expressed as
I “ xθ0, θ1^ θ2, θ3
^ θ4y.
This is a pointwise condition on Θ. Moreover, given any two 0-adapted coframings,
on a common domain, they must relate by a pointwise action of the subgroup G0 Ă
GLp5,Rq generated by matrices of the form
g “
¨
˝
a 0 0b1 A 0b2 0 B
˛
‚, a ‰ 0; A,B P GLp2,Rq; b1,b2 P R2,
and
J “
¨
˝
1 0 00 0 I2
0 I2 0
˛
‚.
Consequently, two hyperbolic Monge-Ampere systems are equivalent if and only if the
corresponding G0-structures are equivalent. This treatment of a differential system
as a geometric structure is what we mean by “geometry of differential systems”.
To close this section, let us mention a useful property of the tautological 1-form
of a G-structure on M : the so-called reproducing property.
15
Proposition 2.2. Let G be a G-structure on M . Let τ be the tautological 1-form on
G. For any local section σ : U Ñ G pU ĂM openq, we have
σ˚τ “ σ,
where, on the left-hand-side of the equality, σ is regarded as a differentiable map,
whereas, on the right-hand-side, it is regarded as a coframing on M .
In particular, if certain differential conditions are satisfied by admissible cofram-
ings, then the reproducing property leads to corresponding restrictions on the equa-
tions satisfied by the tautological 1-form on a G-structure.
2.4 Backlund Transformations
We follow [AF15] to define Backlund transformations, though we will, in later chap-
ters, mostly be concerned with those relating hyperbolic Monge-Ampere systems.
For the latter, a definition can be found in Chapter 4 of [BGG03].
Definition 2.11. Let pM, Iq be an exterior differential system. A rank-k inte-
grable extension of pM, Iq is an exterior differential system pN,J q with a submer-
sion π : N Ñ M that satisfies the condition: for each p P N , there exists an open
neighborhood U Ă N pp P Uq such that
p1q on U , the differential ideal J is algebraically generated by elements of π˚I
together with 1-forms θ1, ..., θk P Ω1pUq, where k “ dimN ´ dimM ;
p2q for any p P U , let Fp denote the fiber π´1pπppqq; the 1-forms θ1, ..., θk restrict
to TpFp to be linearly independent.
In Definition 2.11, roughly speaking, π : N Ñ M is a bundle and J defines a
‘connection’ on this bundle that is flat over the integral manifolds of I. In more
detail, condition p1q implies that, if S Ă M is an integral manifold of pM, Iq, then
J restricts to π´1pSq to be a Frobenius system; hence, π´1pSq is foliated by integral
16
manifolds of pN,J q. Condition p2q implies that, restricting to any integral manifold
of pN,J q, π is an immersion, whose image is an integral manifold of pM, Iq.
Example 6. Let pM, Iq be an exterior differential system. The obvious submersion
π : MˆRÑM induces an integrable extension pMˆR, xπ´1I, dtyq of pM, Iq, where
t is a coordinate on the R-factor.
Proposition 2.3. The composition of two integrable extensions is an integrable ex-
tension.
Proof. Let π1 : pM1, I1q Ñ pM2, I2q and π2 : pM2, I2q Ñ pM3, I3q be integrable
extensions. By definition, I2 is algebraically generated by π˚2I3 and certain 1-forms
α1, ..., αp; I1 is thus algebraically generated by pπ2 ˝ π1q˚I3, π˚1α1, ..., π
˚1αp and q 1-
forms β1, ..., βq. Clearly, for π2 ˝ π1, the first condition in the definition of integrable
extensions is satisfied. To check the second condition, suppose that there exist con-
stants ci, fj, x P M1 such that´
řpi“1 ciπ
˚1αi `
řqj“1 fjβj
¯
pvq “ 0 for any v P TxM1
satisfying π2˚pπ1˚pvqq “ 0. Since π1 is an integrable extension, each TxM1 is a direct
sum of V1 :“ kerxpπ1q and V2 :“ kerxpβ1, ..., βqq. For the previous equality to be
satisfied on V1, all fj must be equal to zero. Since π1˚ restrict to V2 to be an linear
isomorphism, for the equality to hold on V2X kerppπ2 ˝π1q˚q, ci must all be zero.
Definition 2.12. A Backlund transformation relating two exterior differential sys-
tems, pM1, I1q and pM2, I2q, is a quadruple pN,B; π1, π2q where, for each i P t1, 2u,
πi : N Ñ Mi makes pN,Bq an integrable extension of pMi, Iiq. Such a Backlund
transformation is represented by the diagram
pN,Bq
pM1, I1q pM2, I2q
π1 π2
17
Definition 2.13. In Definition 2.12, if M1,M2 have the same dimension, which
is not required in general, then the rank of pN,B; π1, π2q is the fiber dimension of
either π1 or π2. If pMi, Iiq pi “ 1, 2q are equivalent exterior differential systems, then
pN,B; π1, π2q is called an auto-Backlund transformation of either pMi, Iiq.
By Definitions 2.11 and 2.12, it is clear that, given a Backlund transformation
pN,B; π1, π2q relating pMi, Iiq pi “ 1, 2q, we can start with an integral manifold S
of pM1, I1q; restrict B to π´11 S, which becomes a Frobenius system; then solve this
Frobenius system and project, by π2, each leaf associated to it into M2. As a result,
one produces a family of integral manifolds of pM2, I2q. By Remark 1, it is easy to
see that only ODE methods are used in this process. This is what we mean by “a
Backlund transformation allows one to use a known solution of a PDE system and
ODE methods to obtain solutions of a second PDE system.”
Example 7. Let π : pN,J q Ñ pM, Iq be an integrable extension. The quadruple
pN,J ; π, πq is an auto-Backlund transformation of pM, Iq. However, such a Backlund
transformation does not produce new integral manifolds of pM, Iq from a given one.
Example 8. In Example 4, the orthonormal frame bundle O can be viewed as a
subgroup of GLp4,Rq; an element px; e1, e2, e3q P O corresponds to the matrix
ˆ
1 0 0 0x e1 e2 e3
˙
P GLp4,Rq.
Let K be a negative constant. Let θ, r be constants satisfying K “ ´ sin2 θr2. The
element
g “
¨
˚
˚
˝
1 0 0 0r 1 0 00 0 cos θ sin θ0 0 ´ sin θ cos θ
˛
‹
‹
‚
P O
induces a map ψg : O Ñ O defined by
ψgpuq “ ug, @u P O Ă GLp4,Rq.
18
Let I Ă Ω˚pOq be the differential ideal generated by elements of I and ψ˚gI. One can
verify that the quadraple pO, I; π, π ˝ ψgq defines an auto-Backlund transformation
of the system pM,J q. This is the classical Backlund transformation relating surfaces
in E3 with a negative constant Gauss curvature ([Bac83]).
Definition 2.14. Given a fiber bundle π : E Ñ B, for any p P E, the vertical
tangent space of E at p is the kernel of π˚ : TpE Ñ TπppqB.
Definition 2.15. A Backlund transformation pN,B; π1, π2q is called nontrivial if the
two fibrations π1, π2 have distinct vertical tangent spaces at each point p P N .
If pN,B; π1, π2q is a nontrivial Backlund transformation of rank k, then one can
show that, given any p-dimensional integral manifold S of pM1, I1q, π2pπ´11 Sq has
a dimension strictly greater than p; thus, π2pπ´11 Sq must be foliated by a family of
p-dimensional integral manifolds of pM2, I2q.
Example 9. Let pN,B; π, πq be a rank-1 Backlund transformation relating two hy-
perbolic Monge-Ampere systems pM, Iq and pM, Iq. On some open subsets U ĂM
and U Ă M , we have
I “ xη0, η1^ η2, η3
^ η4y, I “ xη0, η1
^ η2, η3^ η4
y.
Let V “ π´1UX π´1U . It is easy to see that the Cauchy characteristics of the system
xπ˚η0y pdefined on V q are precisely the fibers of π|V ; similarly for π|V . Therefore,
if pN,B; π, πq is nontrivial, then π˚η0 and π˚η0 must be linearly independent when
restricted to each tangent space of N . In particular, it follows that, on V , the
differential ideal B is algebraically generated by elements of π˚I and π˚η0, which
needs to be the same as the system algebraically generated by elements of π˚I and
π˚η0.
Definition 2.16. A nontrivial rank-1 Backlund transformation relating two hyper-
bolic Monge-Ampere systems is said to be normal if, in the notations of Example 9,
19
on N , the system rrdη0, dη0ss has rank 2 modulo η0, η0.
Consider two Backlund transformations, one relating pM1, I1q and pM2, I2q, the
other relating pM2, I2q and pM3, I3q, as the following diagram shows.
pN1,B1q pN2,B2q
pM1, I1q pM2, I2q pM3, I3q
π1 π2 π3 π4
The Whitney sum of the fiber bundles π2 : N1 ÑM2 and π3 : N2 ÑM2, denoted as
N1 ‘N2, admits two submersions
p1 : N1 ‘N2 Ñ N1, p2 : N1 ‘N2 Ñ N2
such that π2˝p1 “ π3˝p2. Let B denote the differential ideal on N1‘N2 algebraically
generated by p˚1B1 and p˚2B2.
Proposition 2.4. pN1‘N2,B; π1 ˝ p1, π4 ˝ p2q is a Backlund transformation relating
pM1, I1q and pM3, I3q.
Proof. By taking into account Proposition 2.3, it suffices to show that p1 and
p2 are integrable extensions. Let B1 be algebraically generated by α1, ..., αk and
π˚2I2, B2 by β1, ..., βl and π˚3I2. Hence, B is algebraically generated by p˚1α1, ..., p˚1αk,
p˚2β1, ..., p˚2βl and p˚1π
˚2I2 (the latter being the same as p˚2π
˚3I2). Now suppose that
v is tangent to a fiber of p1. By construction, p2˚v is tangent to a fiber of π3.
If p˚2pβiqpvq “ 0 for all i “ 1, ..., l, it is necessary that p2˚v “ 0, because π3 is
an integrable extension. Since p2, restricting to each fiber of p1, is an immersion,
we have v “ 0. This prove that p1 is an integrable extension. The case for p2 is
similar.
The above discussion suggests that, for exterior differential systems, being Backlund-
related is an equivalence relation. However, it is unknown whether this remains true
if one restricts to the notion of being Backlund-related at a particular rank.
20
We close this section with two more definitions, which will be useful later.
Definition 2.17. Let π : N ÑM be a submersion. A p-form on N that takes value
in π˚pΛppT ˚Mqq is said to be a π-semi-basic p-form.
Definition 2.18. Let M be a smooth manifold. Let E Ă ΛkpT ˚Mq be a vector
subbundle, and X a smooth vector field defined on M . We say that E is invariant
under the flow of X if, for any psmoothq local section ω : U Ñ E, where U Ă M is
open, the Lie derivative LXω remains a section of E defined on U .
Remark 2. It is easy to see that, to verify the condition in Definition 2.18, it suffices
to make a choice of basis sections σ1, ..., σk : U Ñ E passuming that E has rank kq,
and verify that, for each i, LXσi is a linear combination of σ1, ..., σk.
21
3
The Problem of Generality
In this chapter, the objects of study are nontrivial rank-1 Backlund transformations
relating a pair of hyperbolic Monge-Ampere systems. Since many classical exam-
ples belong to this category, it is highly desirable to have a complete classification of
Backlund transformations of this kind. In [Cle01], by establishing a G-structure asso-
ciated to a Backlund transformation, Clelland approached the classification problem
using Cartan’s Method of Equivalence, restricting to the case when all local invariants
of the structure are constants (a.k.a the homogeneous case). Her classification found
15 types, among which 11 are analogues of the classical Backlund transformation
between surfaces in E3 with a negative constant Gauss curvature.
Since homogeneous structures, up to equivalence, only depend on constants, the
following question remains to be answered: What kind of initial data do we need
to specify in order to determine a Backlund transformation? In Section 3.2, in the
generic case, we prove an upper bound for the magnitude of such initial data. In
Section 3.3, we provide several examples of Backlund transformations with higher
cohomogeneity, which we found by specifying only two structure invariants.
22
3.1 G-structure Equations for Backlund Transformations
According to Definition 2.12, it may appear that pN,B; π1, π2q being a Backlund
transformation imposes conditions on all components in this quadruple. However,
when it is a nontrivial rank-1 Backlund transformation relating two hyperbolic
Monge-Ampere systems, one only needs to impose conditions on the exterior dif-
ferential system pN,Bq, as the following proposition shows.
Proposition 3.1. ([Cle01]) An exterior differential system pN6,Bq is a nontrivial
rank-1 Backlund transformation relating two hyperbolic Monge-Ampere systems if
and only if, for each p P N , there exists an open neighborhood V Ă N pp P V q and a
coframing pθ0, θ0, θ1, ..., θ4q, defined on V , satisfying the conditions:
p1q the differential ideal B “ xθ0, θ0, θ1 ^ θ2, θ3 ^ θ4yalg;
p2q the vector bundles E0 “ rrθ0ss, E1 “ rrθ0, θ1, θ2ss and E2 “ rrθ0, θ3, θ4ss are
invariant along the flow of X (see Definition 2.18), where X is a nonvanishing
vector field on V that annihilates θ0, θ1, ..., θ4;
p2q1 the vector bundles E0 “ rrθ0ss, E1 “ rrθ0, θ1, θ2ss and E2 “ rrθ0, θ3, θ4ss are
invariant along the flow of X, where X is a nonvanishing vector field on V that
annihilates θ0, θ1, ..., θ4;
p3q for some nonvanishing functions A1, ..., A4 defined on V ,
dθ0” A1θ
1^ θ2
` A2θ3^ θ4 mod θ0,
dθ0” A3θ
1^ θ2
` A4θ3^ θ4 mod θ0.
Proof. In one direction, assume that pN,B; π, πq is a nontrivial rank-1 Backlund
transformation relating two hyperbolic Monge-Ampere systems pM, Iq and pM, Iq.
Maintaining the notation in Example 9 of Chapter 2, and dropping the pull-back
symbol when there is no confusion, we have that each p P N has an open neighbor-
23
hood V such that, on V ,
B “ xη0, η0, η1^ η2, η3
^ η4yalg “ xη
0, η0, η1^ η2, η3
^ η4yalg.
Thus, there exist nonvanishing functions Ai pi “ 1, ..., 4q defined on V such that
dη0” A1η
1^ η2
` A2η3^ η4 mod η0,
dη0” A3η
1^ η2
` A4η3^ η4 mod η0, η0. (3.1)
By adding appropriate multiples of η0 to η1, ..., η4, we can put (3.1) in the form
dη0” A3η
1^ η2
` A4η3^ η4 mod η0. (3.2)
It is easy to see that the resulting coframing pη0, η0, η1, ..., η4q satisfies the conditions
p1q and p3q. Next, we show that it also satisfies the conditions p2q and p2q1.
Using the congruence (3.2), it is easy to see that the Cartan system1
Cpxη0yq “ xη0, η1, ..., η4
y Ă Ω˚pV q.
As a result, π˚I is included in the intersection
Cpxη0yq X B “ xη0, η1
^ η2, η3^ η4
yalg,
Consequently, by switching pη1, η2q and pη3, η4q if needed, we have the following
relations of rank-3 vector bundles over V :
rrη0, η1, η2ss “ rrη0, η1, η2
ss, (3.3)
rrη0, η3, η4ss “ rrη0, η3, η4
ss. (3.4)
Let X be a vector field on V annihilated by η0, η1, ..., η4. By construction, the right-
hand-sides of (3.3) and (3.4) are invariant under the flow of X; it follows that the
same holds for rrη0, η1, η2ss and rrη0, η3, η4ss. This proves that pη0, η0, η1, ..., η4q satisfies
Condition p2q1. The verification of Condition p2q is similar.
1 Let pM, Iq be an exterior differential system, the Cartan system of pM, Iq is the Frobeniussystem that annihilates the Cauchy characteristic distribution of pM, Iq. See [BCG`13] for details.
24
For the other direction, assume that pθ0, θ0, θ1, ..., θ4q is a coframing defined on
an open set V Ă N and satisfying the conditions p1q-p3q. The quotient of V by the
integral curves of xθ0, θ1, ..., θ4y is a 5-manifold U . Let f : V Ñ U be the projection.
By Condition p2q, there exists a system I “ xη0, η1 ^ η2, η3 ^ η4y defined on U
satisfying
rrf˚η0ss “ rrθ0
ss, rrf˚η0, f˚η1, f˚η2ss “ rrθ0, θ1, θ2
ss, rrf˚η0, f˚η3, f˚η4ss “ rrθ0, θ3, θ4
ss.
It is then easy to show, using Conditions p1q and p3q, that I “ xη0, η1^η2, η3^η4yalg
is a hyperbolic Monge-Ampere ideal. A similar argument applies to the quotient of
V by the integral curves of xθ0, θ1, ..., θ4y. It follows that pN,Bq is a nontrivial rank-1
Backlund transformation relating two hyperbolic Monge-Ampere systems.
Corollary 3.1. Let pN6,B; π1, π2q be a nontrivial rank-1 Backlund transformation
relating two hyperbolic Monge-Ampere systems. A coframing defined on an open
subset V Ă N that satisfies Conditions p1q-p3q in Proposition 3.1 can always be
arranged to satisfy the extra condition: A2 “ A3 “ 1.
Proof. This is obtained by scaling θ0 and θ0.
Definition 3.1. A coframing as concluded in Corollary 3.1 is said to be 0-adapted
to the Backlund transformation pN,Bq.
Given a nontrivial rank-1 Backlund transformation pN,B; π1, π2q, one may wonder
whether its 0-adapted coframings are precisely the local sections of a G-structure
over N . However, this is not true. For example, consider a 0-adapted coframing
pθ0, θ0, θ1, ..., θ4q defined on an open subset U Ă N with corresponding functions
A1, A4. Let T : U Ñ GLp6,Rq be the transformation
T : pθ0, θ0, θ1, θ2, θ3, θ4q ÞÑ pA´1
1 θ0, A´14 θ0, θ3, θ4, θ1, θ2
q. (3.5)
The coframing on the right-hand-side is clearly 0-adapted. However, the same trans-
formation, when applied to a 0-adapted coframing with corresponding functions
25
A11, A14 that are different from A1, A4, may not result in a 0-adapted coframing.
One simple strategy, as taken by [Cle01], to avoid this imperfection is by, in
addition to understanding the subbundles rrθ0ss and rrθ0ss as an ordered pair, fixing
an order for the pair of subbundles rrθ0, θ0, θ1, θ2ss and rrθ0, θ0, θ3, θ4ss. Once this is
considered, all local 0-adapted coframings respecting such an ordering are precisely
the local sections of a G-structure, where G Ă GLp6,Rq is the Lie subgroup consisting
of matrices of the form
g “
¨
˚
˚
˝
detpBq 0 0 00 detpAq 0 00 0 A 00 0 0 B
˛
‹
‹
‚
, A “ paijq, B “ pbijq P GLp2,Rq. (3.6)
Now let G denote this G-structure on N . Let ω “ pω1, ω2, ..., ω6q be the tautolog-
ical 1-form on G. Let g be the Lie algebra of G. Using the conditions in Proposition
3.1 and the reproducing property, one can show that ω satisfies the following struc-
ture equations, recorded from [Cle01] with a slight change of notation:
d
¨
˚
˚
˚
˚
˚
˚
˝
ω1
ω2
ω3
ω4
ω5
ω6
˛
‹
‹
‹
‹
‹
‹
‚
“ ´
¨
˚
˚
˚
˚
˚
˚
˝
β0 0 0 0 0 00 α0 0 0 0 00 0 α1 α2 0 00 0 α3 α0 ´ α1 0 00 0 0 0 β1 β2
0 0 0 0 β3 β0 ´ β1
˛
‹
‹
‹
‹
‹
‹
‚
^
¨
˚
˚
˚
˚
˚
˚
˝
ω1
ω2
ω3
ω4
ω5
ω6
˛
‹
‹
‹
‹
‹
‹
‚
(3.7)
`
¨
˚
˚
˚
˚
˚
˚
˝
A1pω3 ´ C1ω
1q ^ pω4 ´ C2ω1q ` ω5 ^ ω6
ω3 ^ ω4 ` A4pω5 ´ C3ω
2q ^ pω6 ´ C4ω2q
B1ω1 ^ ω2 ` C1ω
5 ^ ω6
B2ω1 ^ ω2 ` C2ω
5 ^ ω6
B3ω1 ^ ω2 ` C3ω
3 ^ ω4
B4ω1 ^ ω2 ` C4ω
3 ^ ω4
˛
‹
‹
‹
‹
‹
‹
‚
,
where the matrix in α and β is a g-valued 1-form, called a pseudo-connection of G;
the second term on the right-hand-side is called the intrinsic torsion of G.
It is easy to see that the intrinsic torsion above, as a map defined on G, takes
value in a 10-dimensional representation of G and is G-equivariant. It is proved in
26
[Cle01] that this representation decomposes into 6 irreducible components, as shown
by the following equations, where u P G, and g is as in (3.6):
A1pu ¨ gq “detpAq
detpBqA1puq, A4pu ¨ gq “
detpBq
detpAqA4puq, (3.8)
ˆ
B1
B2
˙
pu¨gq “ detpABqA´1
ˆ
B1
B2
˙
puq,
ˆ
B3
B4
˙
pu¨gq “ detpABqB´1
ˆ
B3
B4
˙
puq,
ˆ
C1
C2
˙
pu ¨ gq “ detpBqA´1
ˆ
C1
C2
˙
puq,
ˆ
C3
C4
˙
pu ¨ gq “ detpAqB´1
ˆ
C3
C4
˙
puq.
Definition 3.2. Let G and G be as above. The Backlund transformation2 corre-
sponding to G is said to be generic if, at each point u P G, the intrinsic torsion takes
value in a G-orbit with the largest possible dimension.
3.2 An Estimate of Generality
In this section, we address the problem of generality for generic rank-1 Backlund
transformations relating two hyperbolic Monge-Ampere systems. The main recipe
are a G-structure reduction procedure described in [Gar89] and a theorem of Cartan
described in [Bry14].
Lemma 3.2. Let pN,Bq be a nontrivial rank-1 Backlund transformation relating two
hyperbolic Monge-Ampere systems. Let G be the associated G-structure. If pN,Bq is
generic, then, at each u P G, the intrinsic torsion takes value in an 8-dimensional
G-orbit.
Proof. Let W1 :“ spanppB1, B2q, pC1, C2qq and W2 :“ spanppB3, B4q, pC3, C4qq at
each point u P G. By (3.8), the function A1A4 and the dimensions of W1 and W2
are all invariant under the G-action. Let T denote the intrinsic torsion of G. We
claim that, for each u P G, the G-orbit of T puq is at most 8-dimensional and that
2 To be precise, this is a Backlund transformation with an ordered pair of characteristic systems.
27
this occurs precisely when W1 and W2 are both 2-dimensional. To see why this is
true, first note that if one of Wi pi “ 1, 2q has dimension less than 2 at u P G, then
the dimension of u ¨G is at most 7-dimensional. If both W1,W2 have dimension 2 at
u P G, then it is easy to show that there exists a unique g P G such that, at u1 “ u ¨g,
ˆ
B1 C1
B2 C2
˙
“
ˆ
ε1 00 1
˙
,
ˆ
B3 C3
B4 C4
˙
“
ˆ
ε2 00 1
˙
, (3.9)
where εi “ ˘1 pi “ 1, 2q. This completes the proof.
Let pN,Bq be a generic rank-1 Backlund transformation relating two hyperbolic
Monge-Ampere systems. Following from Lemma 3.2, each point p P N has a con-
nected open neighborhood U Ă N on which a canonical coframing pω1, ω2, ..., ω6q can
be determined. Such a coframing satisfies the equation (3.7), where all differential
forms are defined on U instead of G, and the equations (3.9), where the sign of each
εi is determined. This motivates the following
Definition 3.3. Let N be a 6-manifold. A coframing pω1, ω2, ..., ω6q defined on an
open subset U Ă N is said to be 1-adapted pto a generic rank-1 Backlund transfor-
mation relating two hyperbolic Monge-Ampere systemsq if there exist 1-forms αi, βi
pi “ 0, ..., 3q and two functions A1, A4 defined on U such that the equations (3.7)
and (3.9) are satisfied.
Theorem 3.3. Let N be a 6-manifold. For each p P N , a 1-adapted coframing
(Definition 3.3) defined on a small open neighborhood U Ă N of p can be uniquely
determined, up to diffeomorphism, by specifying at most 6 functions of 3 variables.
Proof. Let U Ă N6 be a sufficiently small connected open subset. Suppose that
ω “ pω1, ω2, ..., ω6q is a 1-adapted coframing on U in the sense of Definition 3.3. It
follows that there exist functions Pij pi “ 0, ..., 7; j “ 1, ..., 6q defined on U such that
ω satisfies (3.7) and (3.9) with
αi “ Pijωj, βi “ Pi`4,jω
jpi “ 0, ..., 3; j “ 1, ..., 6q.
28
There is a standard method to determine the generality of such a coframing ω
up to diffeomorphism (see [Bry14]). Our application of such a method involves three
major steps.
Step 1. By applying d2 “ 0 to (3.7), we find that Pij are related among themselves
and with the coefficients of their exterior derivatives. Repeating this, at a point, no
new relations among the Pij arise.
More explicitly, we can choose s expressions aα pα “ 1, ..., sq from Pij, find r
expressions bρ pρ “ 1, ..., rq, real analytic functions Fαi : Rr`s Ñ R and Ci
jk : Rr Ñ R
satisfying Cijk ` C
ikj “ 0, such that
pAq the equation (3.7), in general, takes the form
dωi “ ´1
2Cijkpaqω
j^ ωk; (3.10)
pBq daα, in general, takes the form:
daα “ Fαi pa, bqω
i; (3.11)
moreover, applying d2 “ 0 to (3.10) yields identities when we take into account both
(3.10) and (3.11);
pCq there exist functions Gρj : Rr`s Ñ R such that applying d2 “ 0 to (3.11) yields
identities when we replace dbρ by Gρjω
j and take into account (3.10) and (3.11).
Step 2. For the tableau of free derivatives associated to pFαi q, which is a subspace of
HompR6,Rsq defined at each point of Rr`s, compute its Cartan characters (an array
of 6 integers ps1, s2, ..., s6q) and the dimension δ of its first prolongation. For details,
see [Bry14]. Moreover, in our case, we verify thatř6i“1 si “ r.
Step 3. Restricting to a domain V Ă Rr`s where the Cartan characters are con-
stants, compare s :“ř6j“1 jsj with δ. By Cartan’s inequality, there are two possibil-
ities: either s “ δ (called the involutive case) or s ą δ.
In the involutive case, one can conclude that (see Theorem 3 of [Bry14]):
29
For any pa0, b0q P Rs`r there exists a coframing ω and functions a “
paαq, b “ pbρq defined on an open neighborhood of 0 P R6 that sat-
isfy (3.10), (3.11) and pap0q, bp0qq “ pa0, b0q. Moreover, locally, such
a coframing can be uniquely determined up to diffeomorphism by spec-
ifying sk functions of k variables, where sk is the last nonzero Cartan
character.
In the non-involutive case, which is the case we encounter, a natual step to take
is to prolong (see [Bry14]) the system by introducing the derivatives of bρ, carry out
similar steps as the above, and obtain new tableaux of free derivatives with Cartan
characters pσ1, σ2, ..., σ6q.
In practice, however, we do not actually prolong, for it is easy to show that, if sk
is the last nonzero character in ps1, ..., s6q, then σj “ 0 pj ą kq and σk ď sk. Using
this and the Cartan-Kuranishi Theorem ([BCG`13]), one can already conclude that
the ‘generality’ of 1-adapted coframings is bounded from above by sk functions of k
variables. (In our case, k “ 3 and sk “ 6.)
For the details of carrying out the steps above, see Appendix A. Most of our
calculations are performed using MapleTM.
Remark 3. Clearly, two 1-adapted coframings that are equivalent under a diffeo-
morphism correspond to equivalent Backlund transformations. Because of this, the
upper bound for the ‘generality’ of 1-adapted coframings in Theorem 3.3 applies to
the ‘generality’ of generic rank-1 Backlund transformations relating two hyperbolic
Monge-Ampere systems.
Corollary 3.4. There exist hyperbolic Monge-Ampere systems that are not related
to any other hyperbolic Monge-Ampere system by a generic rank-1 Backlund trans-
formation.
Proof. One can apply the same method used the the proof of Theorem 3.3 to
30
find the generality of hyperbolic Euler-Lagrange systems (see [BGG03]), which are
hyperbolic Monge-Ampere systems. We can show that, locally, a hyperbolic Euler-
Lagrange system can be determined uniquely by specifying 1 function of 5 variables.3
Note that a generic rank-1 Backlund transformation in consideration completely
determines the two underlying hyperbolic Monge-Ampere systems (up to equiva-
lence). The conclusion follows.
3.3 Examples of Higher Cohomogeneity
Following the discussion in the previous section, let U Ă N6 be a sufficiently small
connected open subset. Let ω be a 1-adapted coframing defined on U in the sense
of Definition 3.3. One can ask, when we specify several structure invariants, can we
classify the corresponding Backlund transformations, if any?
In the rest of this section, we consider the case when ε1 “ ε2 “ 1 in (3.9), and A1
and A4 (or P81 and P84 in the new notation) in (3.7) are specified to be A1 “ 1 and
A4 “ ´1.
The following procedure is similar to that in Appendix A. All calculations below
are performed with MapleTM.
First, all coefficients in (3.7) are expressed in terms of the remaining 40 Pij.
Defining their derivatives Pijk and applying the identity d2 “ 0 to (3.7), we obtain a
system of 106 polynomial equations for Pij and Pijk, which implies
P01 “ P41, P02 “ P42, P04 “ P44, P06 “ P46, P11 “ 0, P12 “ 0,
P16 “ 2P46, P21 “ ´1, P22 “ ´1, P23 “ 0, P35 “ 0, P36 “ ´1,
P51 “ 0, P52 “ 0, P54 “ 2P44, P61 “ 1, P62 “ ´1, P65 “ 0,
P73 “ 0, P74 “ ´1.
Using these relations and repeating the steps above, we obtain a system of 88
3 This is not surprising, as, in our case, a Lagrangian is a function depending on 5 variables.
31
equations, which implies
P25 “ ´2pP41 ` P44 ´ P42q, P26 “ P64, P63 “ ´2pP41 ` P42 ` P46q.
Using these and repeating, we obtain a system of 86 equations for the 17 Pij
remaining and 80 of their 102 derivatives. This system implies
P31 “ P32, P41 “ ´P44 ´ P46, P42 “ P44 ´ P46, P71 “ ´P72.
Using these and repeating, we obtain a system of 85 equations for the 13 Pij
remaining and 64 of their derivatives. This system implies
P03 “ ´1, P05 “ 0, P15 “ 0, P32 “ P72, P34 “ ´1,
P43 “ 0, P45 “ 1, P53 “ 0, P76 “ 1.
Using these and repeating, we obtain a system of 61 equations for P72, P44, P46, P64
and 22 of their derivatives. Solving this system leads to the two cases below.
Case 1: P72 ‰ 0.
In this case, we have
P44 “ 0, P46 “ 0.
Using these and applying d2 “ 0 to the structure equations, we find
P64 “1
P72
.
Using this and applying d2 “ 0 to the structure equations again, we find
dpP72q “ 0.
It follows that the only primary invariant remaining, P72, is a nonzero constant. The
32
structure equations read
dω1“ ω1
^ pω3` ω5
q ` ω3^ ω4
` ω5^ ω6,
dω2“ ´ω2
^ pω3` ω5
q ` ω3^ ω4
´ ω5^ ω6,
dω3“ pω1
` ω2´ P´1
72 ω6q ^ ω4
` ω1^ ω2,
dω4“ ´pP72ω
1` P72ω
2´ ω6
q ^ ω3` ω5
^ ω6,
dω5“ ´pω1
´ ω2` P´1
72 ω4q ^ ω6
` ω1^ ω2,
dω6“ pP72ω
1´ P72ω
2` ω4
q ^ ω5` ω3
^ ω4.
This, after the transformation
pω1, ω2, ω3, ω4, ω5, ω6q ÞÑ p
a
|P72|ω1,a
|P72|ω2, ω3,
a
|P72|ω4, ω5,
a
|P72|ω6q,
can be readily seen to belong to the Case 3D in Clelland’s classification ([Cle01]).
According to [Cle01], if P72 ă 0, then pN,Bq is a homogeneous Backlund transfor-
mation relating time-like surfaces of constant mean curvature H “ ´P72pP272 ` 1q´
12
in H2,1; if P72 ą 0, then pN,Bq is a homogeneous Backlund transformation relating
certain surfaces in a 5-dimensional quotient space of SO˚p4q.
Case 2: P72 “ 0.
In this case, all coefficients in (3.7) are expressed in terms of P44, P46, and P64.
Applying d2 “ 0 to the structure equations, no new relations between P44, P46 and
P64 arise. Furthermore, 16 of the 18 derivatives of P44, P46 and P64 are expressed in
terms of these three invariants; two derivatives, P644 and P646, are free.
It is easy to check that Theorem 3 in [Bry14] applies to ω, the expressions a “
pP44, P46, P64q, b “ pP644, P646q, and the functions Cijk and Fα
i determined during the
calculation above. The corresponding tableaux of free derivatives is involutive with
Cartan characters p1, 1, 0, 0, 0, 0q. We have thus proved the
Proposition 3.2. Locally, a generic rank-1 Backlund transformation pN,Bq relating
two hyperbolic Monge-Ampere systems with its 1-adapted coframing satisfying ε1 “
33
ε2 “ 1 and A1 “ ´A4 “ 1 can be uniquely determined by specifying 1 function of 2
variables.
We can study Case 2 in greater detail. For convenience, we introduce the following
new notation:
R :“ P44 ` P46, S :“ P44 ´ P46, T :“ P64;
T4 :“ P644, T6 :“ P646.(3.12)
In the new notation, ω satisfies (3.7) and (3.9) where
α0 “ ´Rω1` Sω2
´ ω3` 1
2pR ` Sqω4
` 12pR ´ Sqω6,
β0 “ ´Rω1` Sω2
` 12pR ` Sqω4
` ω5` 1
2pR ´ Sqω6,
α1 “ pR ´ Sqω6, α2 “ pR ` Sqω
5` Tω6
´ ω1´ ω2, (3.13)
α3 “ ´ω4´ ω6, β1 “ pR ` Sqω
4,
β2 “ pR ´ Sqω3` Tω4
` ω1´ ω2, β3 “ ´ω
4` ω6,
and ε1 “ ε2 “ 1, A1 “ ´A4 “ 1.
Moreover, the exterior derivatives of R, S and T are
dR “ ´R2ω1` pRS ´ 1qω2
`Rω3` 1
2pR2
`RS ´ 1qω4
`Rω5` 1
2pR2
´RS ` 1qω6,
dS “ ´pRS ´ 1qω1` S2ω2
´ Sω3` 1
2pS2
` SR ´ 1qω4 (3.14)
´ Sω5` 1
2pSR ´ S2
´ 1qω6,
dT “ 2p´RT ` Sqω1` 2pST ´Rqω2
` 2pR2´ S2
qpω3` ω5
q
` T4ω4` T6ω
6.
Using these equations, we can study the symmetry of the corresponding Backlund
transformation pN,Bq. Let U Ă N be the domain of a 1-adapted coframing ω. Let
the map Φ : U Ñ R3 be defined by
Φppq “ pRppq, Sppq, T ppqq.
34
Lemma 3.5. The map Φ can never have rank 0. Moreover, it has
• rank 1 if and only if 2RS “ 1 and T “ R2 ` S2;
• rank 2 if and only if it does not have rank 1 and satisfies either p1q 2RS “ 1
or p2q T4 “ pR ` SqpT ´ 1q and T6 “ pR ´ SqpT ` 1q;
• rank 3 if and only if it does not have rank 1 or 2.
Proof. In dR ^ dS ^ dT , the coefficients of ωi ^ ωj ^ ωk are polynomials in
R, S, T, T4 and T6. These coefficients have the common factor 2RS ´ 1. Calculating
with MapleTM, we find that the coefficients of ωi^ωj^ωk in p2RS´1q´1dR^dS^dT
all vanish if and only if T4 “ pR` SqpT ´ 1q and T6 “ pR´ SqpT ` 1q. This justifies
the conditions for having rank 2. The condition for rank 1 can be obtained by setting
the coefficients of ωi^ωj in dR^dS, dR^dT and dS^dT to be all zero. By (3.14),
it is clear that dR ‰ 0 everywhere; hence, Φ cannot have rank 0.
Definition 3.4. In the current case, if the corresponding Backlund transformation
pU,Bq has a symmetry whose orbits are of dimension 6 ´ k, then it is said to have
cohomogeneity k.
Case of cohomogeneity-1. This occurs precisely when rankpΦq “ 1. In this case,
locally Φ is a submersion to either branch of the curve in R3 defined by 2RS “ 1
and T “ R2 ` S2. Expressing S and T in terms of R, we have, on U Ă N ,
dR “ ´R2ω1´
1
2ω2`Rpω3
` ω5q `
1
4p2R2
´ 1qω4`
1
4p2R2
` 1qω6. (3.15)
It is clear that dR is nowhere vanishing. Since R is the only invariant, each
constant value of R defines a 5-dimensional submanifold NR Ă N , which has a Lie
group structure. The Lie group structure can be determined by setting the right-
hand-side of the equation (3.15) to be zero, obtaining, say,
ω1“ ´
1
2R2ω2`
1
Rω3`
2R2 ´ 1
4R2ω4`
1
Rω5`
2R2 ` 1
4R2ω6.
35
Substituting this into the structure equations, we obtain equations of dωi pi “
2, ..., 6q, expressed in terms of ω2, ..., ω6 alone. These are the structure equations on
each NR. Let X1, X2, ..., X5 be the vector fields tangent to NR and dual to ω2, ..., ω6,
such that ωipXjq “ δij`1 pi´ 1, j “ 1, 2, ..., 5q. We obtain the Lie bracket relations:
rX1, X2s “ 2X1 `1
RpX2 `X4q, rX1, X3s “ ´
1
2RX1 ´
p2R2 ´ 1q
4R2pX2 ´X4q `
1
RX3,
rX1, X4s “ 2X1 `1
RpX2 `X4q, rX1, X5s “
1
2RX1 `
2R2 ` 1
4R2pX2 ´X4q `
1
RX5,
rX2, X3s “ ´X1 ´1
RX2 ´X3 ´X5, rX2, X4s “ 0,
rX2, X5s “ ´2R2 ´ 1
2RX2 `X3 `
2R2 ` 1
2RX4 ´X5,
rX3, X4s “ ´2R2 ´ 1
2RX2 `X3 `
2R2 ` 1
2RX4 ´X5,
rX3, X5s “
ˆ
´R2`
1
2
˙
X2 `RX3 `
ˆ
R2`
1
2
˙
X4 ´RX5,
rX4, X5s “ X1 ´X3 `1
RX4 ´X5.
Using these relations, it can be verified that Xi pi “ 1, ..., 5q generate a 5-
dimensional Lie algebra that is solvable but not nilpotent. The derived series has
dimensions p5, 3, 1, 0, ...q. In fact, after introducing the following new basis
e1 “ RX1, e2 “ ´1
2pX2 ´X4q, e5 “
1
RX1 `
1
2R2pX2 `X4q,
e3 “ ´1
2RX1 ´
2R2 ´ 1
4R2pX2 ´X4q `
1
RX3,
e4 “1
2RX1 `
2R2 ` 1
R2pX2 ´X4q `
1
RX5,
we obtain the Lie bracket relations:
re1, e3s “ e3, re1, e4s “ e4, re1, e5s “ 2e5,
re2, e3s “ e4, re2, e4s “ ´e3, re3, e4s “ e5,
36
where all rei, ejs not on this list are zero. An equivalent way of writing these relations
is:
re1 ` ie2, e3 ` ie4s “ 2pe3 ` ie4q, re1 ´ ie2, e3 ` ie4s “ 0, re1 ` ie2, e5s “ 2e5,
re3 ` ie4, e3 ´ ie4s “ ´2ie5, re1 ` ie2, e1 ´ ie2s “ 0, re3 ` ie4, e5s “ 0.
It is then not hard to see that the Lie algebraÀ5
i“1 Rei is isomorphic to the Lie
algebra generated by the real and imaginary parts of the vector fields
Bw, e2wpBz ` izBλq, e2pw`wq
Bλ
on R ˆ C2 with coordinates pλ; z, wq. In fact, an isomorphism is induced by the
correspondence
e1 ` ie2 ÞÑ Bw, e3 ` ie4 ÞÑ e2wpBz ` izBλq, e5 ÞÑ e2pw`wq
Bλ.
Next, we describe the hyperbolic Monge-Ampere systems related by the Backlund
transformation being considered. In particular, we prove the
Proposition 3.3. A Backlund transformation in the current (cohomogeneity-1) case
is an auto-Backlund transformation of a homogeneous Euler-Lagrange system.
Proof. This proof is in two parts. First, we show that the underlying two hy-
perbolic Monge-Ampere systems are equivalent and are homogeneous. Second, we
verify that they are hyperbolic Euler-Lagrange systems in the sense of [BGG03] by
computing their local invariants.
Using the structure equations on U Ă N , if we let pθ0, θ1, ..., θ4q be either
`
Sω1,´Rpω1´ ω4
q ` ω3, Sω4,´Rpω1´ ω6
q ` ω5, Sω6˘
(3.16)
or`
´Rω2, Spω2` ω4
q ´ ω3, Rω4, Spω2´ ω6
q ´ ω5,´Rω6˘
, (3.17)
37
then one can verify (with MapleTM) that θi pi “ 0, ..., 4q, in both cases, satisfy the
same structure equations
dθ0“ ´2pθ1
` θ3q ^ θ0
` θ1^ θ2
` θ3^ θ4,
dθ1“ ´θ1
^ θ4` θ2
^ θ4` θ2
^ θ3,
dθ2“ ´θ1
^ θ4` θ2
^ θ4` θ2
^ θ3´ θ1
^ θ2` θ3
^ θ4, (3.18)
dθ3“ θ1
^ θ4´ θ2
^ θ4´ θ2
^ θ3,
dθ4“ ´θ1
^ θ4` θ2
^ θ4` θ2
^ θ3` θ1
^ θ2´ θ3
^ θ4.
It is easy to verify that (3.18) are the structure equations on a 5-manifold M with
a hyperbolic Monge-Ampere ideal I “ xθ0, θ1 ^ θ2, θ3 ^ θ4y. It follows that the
expressions (3.16) and (3.17) correspond to the pull-back of θi under two distinct
submersions π1, π2 : U ÑM . It is easy to see that pU,B; π1, π2q is an auto-Backlund
transformation of the system pM, Iq. Moreover, pM, Iq is homogeneous, since all
coefficients in (3.18) are constants.
Next, we verify that pM, Iq is a hyperbolic Euler-Lagrange system. In [BGG03], a
necessary and sufficient condition for a hyperbolic Monge-Ampere system to be Euler-
Lagrange is proved, which is summarized in our Section 4.1 with a slight change of
notation. In particular, pM, Iq is Euler-Lagrange if and only if an invariant tensor
S2 vanishes (see Proposition 4.2).
To compute S2, we choose a new coframing pηiq and 1-forms pφαq below
pη0, η1, η2, η3, η4q “ p
?2θ0,
?2θ1, θ1
` θ2´ θ0, θ3
` θ4´ θ0,
?2pθ4
´ θ0qq,
φ0 “1?
2η1` η3
´1?
2η4, φ1 “ ´
?2η0
´ η3´
1?
2η4, φ2 “ η0
`?
2η3, (3.19)
φ3 “ ´2η0` η1
´1?
2η2´?
2η3´ η4, φ4 “ φ0 ´ φ1, φ5 “ ´η
3´
1?
2η4,
φ6 “1?
2η3, φ7 “ ´η
0` η1
´?
2η2´?
2η3, φ8 “ φ0 ´ φ5.
38
These ηi pi “ 0, 1, ..., 4q and φα pα “ 1, ..., 8q are chosen such that they satisfy the
structure equations (4.1), and such that S1 and S2 are as simple as possible. One
can verify that, under this choice,
S1 “
ˆ
1 00 1
˙
, S2 “ 0.
This completes the proof.
Now, one may wonder whether the homogeneous Monge-Ampere system pM, Iq
considered in Proposition 3.3 has a symmetry of dimension greater than 5. Using
the method of equivalence, we prove that it is not the case.
Proposition 3.4. The hyperbolic Euler-Lagrange system in Proposition 3.3 has a
symmetry of dimension 5. In addition, any such symmetry is induced from a sym-
metry of the Backlund transformation pN,Bq.
Proof. Let pM, Iq denote the Euler-Lagrange system being considered. To show
that pM, Iq has a 5-dimensional symmetry, it suffices to show that there is a canonical
way to determine a local coframing on M . This can be achieved by applying the
method of equivalence. For details, see Appendix B.
By (3.16) and (3.17), it is easy to see that the fibers of πi : N ÑM pi “ 1, 2q are
everywhere transversal to the level sets of the functions R. The second half of the
statement follows.
To end the discussion of the cohomogeneity-1 case, we integrate the structure
equations (3.18) to express the corresponding hyperbolic Euler-Lagrange system in
local coordinates.
Proposition 3.5. The hyperbolic Monge-Ampere system pM, Iq with the differential
ideal I “ xθ0, θ1 ^ θ2, θ3 ^ θ4y, where θi satisfy (3.18), corresponds to the following
hyperbolic Monge-Ampere PDE up to contact equivalence:
pA2´B2
qpzxx ´ zyyq ` 4ABzxy “ 0, (3.20)
39
where A “ 2zx ` y and B “ 2zy ´ x.
Proof. Let U ĂM be a domain on which θi are defined. One can verify, using the
structure equations (3.18), that the 1-forms θ1 ´ θ2 ´ θ3 ´ θ4 and θ1 ` θ3 are closed.
Hence, by shrinking U if needed, there exist functions P,Q defined on U such that
dP “ ´pθ1` θ3
q, dQ “ θ1´ θ2
´ θ3´ θ4.
Moreover, if we let Θ “ θ2 ` iθ4, then, by a straight-forward calculation, we obtain
dΘ “ dpP ` iQq ^Θ.
It follows that there exist functions X, Y on U such that
Θ “ eP`iQdpX ` iY q.
Equivalently, we have
θ2“ eP pcosQdX ´ sinQdY q, θ4
“ eP psinQdX ` cosQdY q.
Now we can express θ1, ..., θ4 completely in terms of the coordinates X, Y, P,Q. By
computing the exterior derivative
dpe´2P θ0q “ e´2P
pθ1^ θ2
` θ3^ θ4
q, (3.21)
we notice that the right-hand-side of (3.21) is a symplectic form on a 4-manifold
on which θ1, ..., θ4 are well-defined, by (3.18). Thus, by Darboux’ Theorem, there
locally exist functions x, y, p, q such that the right-hand-side of (3.21) is equal to
dx^ dp` dy ^ dq.
In fact, in XY PQ-coordinates, the right-hand-side of (3.21) is equal to
d
ˆ
e´P
2pcosQ` sinQq `
Y
2
˙
^ dX ` d
ˆ
e´P
2pcosQ´ sinQq ´
X
2
˙
^ dY.
40
As a result, we can set
x “ X, y “ Y, p “ ´e´P
2pcosQ` sinQq ´
Y
2, q “ ´
e´P
2pcosQ´ sinQq `
X
2,
and write
e´2P θ0“ dz ´ pdx´ qdy,
for some function z independent of x, y, p, q. From these expressions, it is clear that
2p` y and 2q ´ x cannot simultaneously vanish.
Now let A “ 2p`y and B “ 2q`x. We can express θ1^θ2 in terms of x, y, z, p, q:
pA2 ´B2qdp^ dy ` pA`Bq2dx^ dp´ pA2 ´B2qdx^ dq ` pA´Bq2dy ^ dq
pA2 `B2q2.
Multiplying this expression by pA2`B2q2 then subtracting pA2`B2qpdx^dp`dy^
dqq, we obtain
pA2´B2
qpdp^ dy ´ dx^ dqq ` 2ABpdx^ dp´ dq ^ dyq.
The vanishing of this 2-form on integral surfaces implies that z must satisfy the
equation (3.20).
Case of cohomogeneity-2. By Lemma 3.5, this case can only occur when 2RS “ 1
and R2`S2 “ T do not both hold and either p1q 2RS “ 1 or p2q T4 “ pR`SqpT´1q,
T6 “ pR ´ SqpT ` 1q holds. We will focus on the latter case.
Proposition 3.6. When Φ has rank 2, and when T4 “ pR ` SqpT ´ 1q and T6 “
pR ´ SqpT ` 1q, the map Φ : N Ñ R3 has its image contained in a surface that is
defined by eitherR2 ` S2 ´ T
2RS ´ 1or its reciprocal being a constant.
Proof. First note that R2 ` S2 ´ T and 2RS ´ 1 cannot be both zero, for this
would reduce to the cohomogeneity-1 case; hence the conclusion has meaning. To
41
see that this statement is true, note that, in our current case, the pull-back of dR, dS
and dT via Φ to N are linearly dependent. To be precise, the 1-form
θ “ ´2pR2S ´ S3` ST ´RqdR ` 2pR3
´RS2´RT ` 2SqdS ` p2RS ´ 1qdT
equals to zero when pulled back to N . Since the tangent map Φ˚ has rank 2, this
can only happen if θ ^ dθ “ 0. It follows that θ is integrable. The primitives of
p2RS ´ 1q´2θ and pR2`S2´ T q´2θ are, respectively, the functionR2 ` S2 ´ T
2RS ´ 1and
its reciprocal, if 2RS´ 1 and R2`S2´T are, respectively, nonzero. This completes
the proof.
We now study the symmetry of the Backlund transformation pN,Bq being con-
sidered. Let Xi pi “ 1, 2, ..., 6q be the dual vector fields of ωi pi “ 1, 2, ..., 6q, defined
on U Ă N . Using the expressions of dR, dS and dT , it is easy to see that the rank-4
distribution on U annihilated by dR, dS and dT is spanned by the vector fields
Y1 “ RX2 ` SX1 `X5, Y2 “1
2pX1 ´X2q `X4,
Y3 “ X3 ´X5, Y4 “1
2pX1 `X2q `X6.
These vector fields generate a 4-dimensional Lie algebra l with
rY1, Y2s “R ` S
2Y3 ` Y4, rY1, Y3s “ 0, rY1, Y4s “ ´Y2 `
R ´ S
2Y3,
rY2, Y3s “ pR ` SqY3 ` 2Y4, rY2, Y4s “ Y1 ` pR ´ SqY2 `
ˆ
1
2´ T
˙
Y3 ´ pR ` SqY4,
rY3, Y4s “ 2Y2 ´ pR ´ SqY3.
It is easy to verify that 2Y1 ` Y3 belongs to the center of l. The quotient algebra
q “ lRp2Y1 ` Y3q, with the basis e1 “ rY2s, e2 “ rY3s and e3 “ rY4s, satisfies
re1, e2s “ pR ` Sqe2 ` 2e3,
re1, e3s “ ´Te2 ´ pR ` Sqe3,
re2, e3s “ 2e1 ` pS ´Rqe2.
42
According to the classification of 3-dimensional Lie algebras, see Lecture 2 in [Bry95]
for example, to identify the Lie algebra q, it suffices to find a normal form of the
matrix (note that it is symmetric)
C “
¨
˝
2 S ´R 0S ´R T R ` S
0 R ` S 2
˛
‚
under the transformation C ÞÑ detpA´1qACAT , where A P GLp3,Rq. Note that
detpCq “ ´2pR2 ` S2 ´ T q. We have the following
Proposition 3.7. If R2`S2 ă T , then q is isomorphic to sop3,Rq. If R2`S2 ą T ,
then q is isomorphic to slp2,Rq. If R2 ` S2 “ T , then q is isomorphic to the
solvable Lie algebra with a basis x1, x2, x3 satisfying rx2, x3s “ x1, rx3, x1s “ x2 and
rx1, x2s “ 0.
Proof. After a transformation of the form above, C can be put in the form
C 1 “
¨
˝
2 0 00 T ´R2 ´ S2 00 0 2
˛
‚.
According to [Bry95], the conclusion follows immediately.
Now consider the case when q is solvable, i.e., when R2 ` S2 “ T . By the
cohomogeneity-2 assumption, we must have 2RS ‰ 1. We proceed to identify the
Monge-Ampere systems related by such a Backlund transformation.
If we let pθ0, θ1, ..., θ4q be
pSω1,´Rpω1´ ω4
q ` ω3, Sω4,´Rpω1´ ω6
q ` ω5q, Sω6
q
and let F be defined by
F “2RS ´ 1
S2,
43
for which to have meaning we need to restrict to a domain on which S ‰ 0, then the
structure equations on N would imply
dθ0“ θ0
^ p2θ1´ Fθ2
` 2θ3´ Fθ4
q ` θ1^ θ2
` θ3^ θ4,
dθ1“ ´Fθ0
^ pθ2` θ4
q ´ θ1^ θ4
` θ2^ θ3
` pF ` 1qθ2^ θ4,
dθ2“ ´2Fθ0
^ θ2´ θ1
^ θ2´ θ1
^ θ4` θ2
^ θ3` p1´ F qθ2
^ θ4` θ3
^ θ4, (3.22)
dθ3“ Fθ0
^ pθ2` θ4
q ` θ1^ θ4
´ θ2^ θ3
` pF ´ 1qθ2^ θ4,
dθ4“ ´2Fθ0
^ θ4` θ1
^ θ2´ θ1
^ θ4` θ2
^ θ3` pF ` 1qθ2
^ θ4´ θ3
^ θ4,
and
dF “ 2F 2p2θ0
´ θ2´ θ4
q ` 2F pθ1` θ3
q, pF ‰ 0q. (3.23)
It can be verified that, in the equations (3.22) and (3.23), the exterior derivative
of the right-hand-sides are zero, by taking into account these equations themselves.
By the construction of the θi pi “ 0, ..., 4q, it follows that (3.22) and (3.23) are
the structure equations of one of the Monge-Ampere systems being related by the
Backlund transformation pN,Bq.
Proposition 3.8. The hyperbolic Monge-Ampere system pM, Iq with the differential
ideal I “ xθ0, θ1^ θ2, θ3^ θ4y, where θi satisfy (3.22) and (3.23), corresponds to the
following hyperbolic Monge-Ampere PDE up to contact equivalence:
pA2´B2
qpzxx ´ zyyq ` 4ABzxy ` εpA2`B2
q2“ 0. (3.24)
where A “ zx ´ y, B “ zy ` x; and ε “ ˘1.
Proof. The proof is very similar to that of Proposition 3.5. First it is easy to
verify that the 1-forms F p´2θ0 ` θ2 ` θ4q ´ θ1 ´ θ3 and θ1 ´ θ2 ´ θ3 ´ θ4 are closed.
Consequently, there locally exist functions f, g such that
df “ F p´2θ0` θ2
` θ4q ´ θ1
´ θ3, dg “ θ1´ θ2
´ θ3´ θ4.
44
Now the expression of dF can be written as dF “ ´2Fdf . This implies that there
exists a nonzero constant ε such that F “ εe´2f . Using the ambiguity in f , we can
set ε “ ˘1. In addition, if we let Θ “ e´f pθ2 ` iθ4q, it is easy to verify that Θ is
integrable. To be explicit,
dΘ “ i dg ^Θ.
Thus there exist functions X, Y such that Θ “ eigpdX ` idY q. From this we have
θ2“ ef pcos g dX ´ sin g dY q, θ4
“ ef psin g dX ` cos g dY q.
Using these, differentiating θ1 ` θ3 gives
dpθ1` θ3
q “ 2ε dX ^ dY.
This implies that there exists a function Z, independent of X, Y, f, g, such that
θ1` θ3
“ dZ ´ εpXdY ´ Y dXq.
Now, θ0, θ1, ..., θ4 can be completely expressed in terms of the functions X, Y, Z, f, g.
In particular,
´2e´2fθ0“ εpdpZ ` fqq ` p´Y ´ e´f psin g` cos gqqdX ` pX ` e´f psin g´ cos gqqdY.
If we make the substitution
x “ X, y “ Y, z “ εpZ`fq, p “ e´f pcos g`sin gq`Y, q “ e´f pcos g´sin gq´X,
the contact form θ0 is then a nonzero multiple of dz´pdx´qdy. The 2-form θ3^θ4,
after replacing dz by pdx` qdy, can be expressed as
θ3^ θ4
”1
8e4f
“
pA2´B2
qpdp^ dy ´ dx^ dqq`
pA`Bq2dq ^ dy ´ pA´Bq2dx^ dp` εpA2`B2
q2dx^ dy
‰
mod θ0,
where A “ p´ y, B “ q ` x. Note that, by construction, A,B cannot be simultane-
ously zero. The equation (3.24) follows.
45
In the current case, there remain several obvious questions to investigate. What
is the Monge-Ampere system corresponding to xω2, ω3^ω4, ω5^ω6y? Are the Monge-
Ampere systems being Backlund-related Euler-Lagrange? Is the Backlund transfor-
mation an auto-Backlund transformation? Answers to these questions can be ob-
tained in a very similar way as in the cohomogeneity-1 case. We thus have them
summarized in the following remark, omitting the details of calculation.
Remark 4. A. Whenever R ‰ 0,
pRω2, Spω2` ω4
q ´ ω3,´Rω4, Spω2´ ω6
q ´ ω5, Rω6q
form a well-defined coframing on a 5-manifold. The system xω2, ω3 ^ ω4, ω5 ^ ω6y
descends to correspond to the same equation (3.24) up to contact equivalence, where
ε has the same sign as 2RS ´ 1. Since the function 2RS ´ 1 is defined on N , ε must
be the same for the two underlying Monge-Ampere systems. The system pN,Bq is
therefore an auto-Backlund transformation of the equation (3.24).
B. The hyperbolic Monge-Ampere system in Proposition 3.8 is Euler-Lagrange.
In fact, by a transformation of the θi pi “ 0, ..., 4q, the structure equations (3.22) can
be put in the form of (4.1) with
S1 “
ˆ
1 00 1
˙
, S2 “ 0.
Applying the procedure described in Appendix B, we find the corresponding Monge-
Ampere invariants
Q01Q04 ´Q02Q03 “ ´1
2p1` F 2
q ‰ 0, Q00 “ 0.
Following from this and the expression of dF , one can show pin a way that is similar
to the proof of Proposition 3.4q that the underlying Euler-Lagrange system has a
symmetry of dimension 4. Such a symmetry is induced from the symmetry of the
Backlund transformation pN,Bq.
46
4
Backlund Transformations and Monge-AmpereInvariants
Given two hyperbolic Monge-Ampere systems pMi, Iiq pi “ 1, 2q, suppose that we
have chosen for each of them a 0-adapted coframing in the sense of Example 5 in
Chapter 2. The problem of finding a nontrivial Backlund transformation relating
these two Monge-Ampere systems is a problem of integration, that is, to find a
6-manifold N Ă M1 ˆ M2 with the obvious submersions πi : N Ñ Mi such that
the differential ideal B generated by π˚1I1 and π˚2I2 makes pN,B; π1, π2q a Backlund
transformation. From this point of view, it is desirable to express the obstructions to
integrability in terms of the invariants of the two hyperbolic Monge-Ampere systems.
4.1 First Monge-Ampere Invariants
Let pM, Iq be a hyperbolic Monge-Ampere system. Let G0 be the G0-structure on
pM, Iq (Example 5, Chapter 2). In [BGG03], the reduction of G0 to a G1-structure
47
G1 is performed such that the following structure equations hold on G1:
d
¨
˚
˚
˚
˚
˝
ω0
ω1
ω2
ω3
ω4
˛
‹
‹
‹
‹
‚
“ ´
¨
˚
˚
˚
˚
˝
φ0 0 0 0 00 φ1 φ2 0 00 φ3 φ4 0 00 0 0 φ5 φ6
0 0 0 φ7 φ8
˛
‹
‹
‹
‹
‚
^
¨
˚
˚
˚
˚
˝
ω0
ω1
ω2
ω3
ω4
˛
‹
‹
‹
‹
‚
(4.1)
`
¨
˚
˚
˚
˚
˝
ω1 ^ ω2 ` ω3 ^ ω4
pV1 ` V5qω0 ^ ω3 ` pV2 ` V6qω
0 ^ ω4
pV3 ` V7qω0 ^ ω3 ` pV4 ` V8qω
0 ^ ω4
pV8 ´ V4qω0 ^ ω1 ` pV2 ´ V6qω
0 ^ ω2
pV3 ´ V7qω0 ^ ω1 ` pV5 ´ V1qω
0 ^ ω2
˛
‹
‹
‹
‹
‚
,
where φ0 “ φ1 ` φ4 “ φ5 ` φ8, and G1 Ă G0 is the subgroup generated by
g “
¨
˝
a 0 00 A 00 0 B
˛
‚, A,B P GLp2,Rq, a “ detpAq “ detpBq, (4.2)
and
J “
¨
˝
1 0 00 0 I2
0 I2 0
˛
‚P GLp5,Rq. (4.3)
Definition 4.1. Let pM, Iq be a hyperbolic Monge-Ampere system. A 1-adapted
coframing1 of pM, Iq with domain U ĂM is a section η : U Ñ G1.
Following [BGG03], we introduce the notation2
S1 :“
ˆ
V1 V2
V3 V4
˙
, S2 :“
ˆ
V5 V6
V7 V8
˙
. (4.4)
It is shown in [BGG03] that
Proposition 4.1. Along each fiber of G1,
Sipu ¨ gq “ aA´1SipuqB, pi “ 1, 2q (4.5)
1 This is not to be confused with a 1-adapted coframing in the sense of Definition 3.3.
2 Note that these Si are half of those defined in [BGG03] with the same notation.
48
for any g “ diagpa;A;Bq in the identity component of G1. Moreover,
S1pu ¨ Jq “
ˆ
´V4 V2
V3 ´V1
˙
, S2pu ¨ Jq “
ˆ
V8 ´V6
´V7 V5
˙
. (4.6)
Proposition 4.1 has a simple interpretation: the matrices S1 and S2 correspond
to two invariant tensors under the G1-action. More explicitly, one can verify that
the quadratic form
Σ1 :“ V3 ω1ω3
´ V1 ω1ω4
` V4 ω2ω3
´ V2 ω2ω4 (4.7)
and the 2-form
Σ2 :“ V7 ω1^ ω3
´ V5 ω2^ ω3
` V8 ω1^ ω4
´ V6 ω2^ ω4 (4.8)
are G1-invariant, which implies that Σ1,Σ2 are locally well-defined on pM, Iq.
An infinitesimal version of Proposition 4.1 will be useful: for i “ 1, 2,
dSi ”
ˆ
φ4 ´φ2
´φ3 φ1
˙
Si ` Si
ˆ
φ5 φ6
φ7 φ8
˙
mod ω0, ω1, ..., ω4. (4.9)
An Euler-Lagrange system, in the classical calculus of variations, is the system
of PDEs whose solutions correspond to the stationary points of a given first order
functional. In [BGG03], an Euler-Lagrange system is formulated as a Monge-Ampere
system3; moreover, it is shown:
Proposition 4.2. ([BGG03]) A hyperbolic Monge-Ampere system is locally equiva-
lent to an Euler-Lagrange system if and only if S2 vanishes.
Remark 5. Proposition 4.2 says that the property of being Euler-Lagrange is intrin-
sically defined.
Proposition 4.3. ([BGG03]) A hyperbolic Monge-Ampere system corresponds to the
wave equation zxy “ 0 (up to contact equivalence) if and only if S1 “ S2 “ 0.
3 See Definitions 1.3 and 1.4 of [BGG03]
49
Proposition 4.4. A hyperbolic Monge-Ampere system pM, Iq, where I is alge-
braically generated by θ, ω1 ^ ω2 and ω3 ^ ω4, locally corresponds to a PDE of
the form zxy “ F px, y, z, zx, zyq (up to contact equivalence) if and only if each of
the characteristic systems I10 “ xθ, ω1, ω2y and I01 “ xθ, ω3, ω4y admits a rank-1
integrable subsystem.
Proof. One direction is obvious, since dx, dy respectively belong to the two char-
acteristic systems and are integrable. For the other direction, assume that pM, Iq
has the property that each of I10 and I01 has a rank-1 integrable subsystem; and let
pθ, ω1, ..., ω4q be a coframing satisfying dθ ” ω1^ω2`ω3^ω4 mod θ. For I10, this
means that a certain linear combination Aθ ` Bω1 ` Cω2, where A,B,C (not all
zero) are functions on M , is closed; hence, it locally equals to dx for some function
x. Of course, B,C cannot both be zero, since θ is a contact form. Without loss of
generality, assume that B ‰ 0. Let ω1 “ Aθ ` Bω1 ` Cω2 “ dx, and ω2 “ 1Bω2.
Similarly, there locally exist functions A1, B1, C 1 (assuming B1 ‰ 0) and y such that
ω3 “ A1θ `B1ω3 ` C 1ω4 “ dy; let ω4 “ 1B1ω
4.
Now dθ ” dx ^ ω2 ` dy ^ ω4 mod θ; hence the system xθ, dx, dyy is Frobenius.
As a result, there locally exists a function z such that xθ, dx, dyy “ xdz, dx, dyy. In
other words, there exist functions g, p, q pg ‰ 0q such that 1gθ “ dz´pdx´qdy. This
implies that dx^dp`dy^dq ” 1gpdx^ ω2`dy^ ω4q mod θ. By Cartan’s Lemma,
there exists a function F such that
ω2” gdp´ gFdy mod dx, θ; ω4
” gdq ´ gFdx mod dy, θ.
The vanishing of θ and ω1 ^ ω2 on integral surfaces then implies that locally the
corresponding Monge-Ampere equation is equivalent to zxy “ F px, y, z, zx, zyq.
Now we focus on hyperbolic Euler-Lagrange systems.
By Proposition 4.1, the sign of detpS1q is independent of the choice of 1-adapted
coframings. Hence, each hyperbolic Euler-Lagrange system belongs to one of the
50
following three types.
Definition 4.2. Given a hyperbolic Euler-Lagrange system pM, Iq, it is said to be
i. positive if detpS1q ą 0;
ii. negative if detpS1q ă 0;
iii. degenerate if detpS1q “ 0.
Example 10. The hyperbolic Monge-Ampere systems in Proposition 3.5 and Propo-
sition 3.8 in Chapter 3 are Euler-Lagrange of the positive type.
Example 11. In Example 4 of Chapter 2, let K “ ´1 for convenience. Recall that
the differential ideal I on O is algebraically generated by
θ “ ω3, dθ “ ´pω31 ^ ω
1` ω3
2 ^ ω2q, Ψ “ ω1
^ ω2` ω1
3 ^ ω23.
Consider the change of basis
ω1“ ´η1
` η3, ω2“ η2
` η4, ω3“ 2η0, ω1
3 “ η2´ η4, ω2
3 “ η1` η3.
In terms of ηi pi “ 0, ..., 4q, we have I “ xη0, η1 ^ η2, η3 ^ η4y. It can be verified
that η “ pη0, η1, ..., η4q, pulled back by a loal section σ : U Ñ O (U ĂM open), is a
local coframing 1-adapted to pM,J q with V2 “ V3 “ 1 and all other Vi being zero.
It follows that pM,J q is a hyperbolic Euler-Lagrange system of the negative type.
Example 12. Consider an equation of the form zxy “ fpzq. (This is called an
f -Gordon equation.) It corresponds to a hyperbolic Euler-Lagrange system of the
degenerate type, for it is easy to verify that
η0“ dz ´ pdx´ qdy, η1
“ dx, η2“ dp´ fpzqdy, η3
“ dy, η4“ dq ´ fpzqdx
is a 1-adapted coframing for the corresponding hyperbolic Monge-Ampere system.
Under this coframing, V3 “ ´f1pzq and all other Vi are identically zero.
51
Remark 6. By Proposition 4.4, a hyperbolic Euler-Lagrange system of either the
positive or the negative type is non-equivalent to any PDE of the form zxy “
F px, y, z, zx, zyq. To see this, one can use (4.1) to show that, given a hyperbolic
Monge-Ampere system pM, Iq, if S2 “ 0 and S1 nonsingular, then the derived sys-
tems (see [BCG`13]) of each of the two characteristic systems of pM, Iq will terminate
at zero.
4.2 The Backlund-Pfaff System
In this section, we show that given two hyperbolic Monge-Ampere systems, the
existence of a normal rank-1 Backlund transformation (Definition 2.16) relating them
can be formulated as the integrability of a rank-4 Pfaffian system.
To start with, let us fix some notation.
Let pM, Iq and pM, Iq be two hyperbolic Monge-Ampere systems. Let G1 and
G1 be the respective G1-structures (see Section 4.1). Let α “ pα0, α1, ..., α4q and
β “ pβ0, β1, ..., β4q be the tautological 1-forms on G1 and G1, respectively. And let
P :“
pu, u, s, t, µ, εq| u P G1, u P G1; s, t P R4;µ ě 1; ε “ ˘1 pεµ2‰ 1q
(
Ă G1 ˆ G1 ˆ R9ˆ t˘1u. (4.10)
Proposition 4.5. An immersed 6-manifold φ : N Ñ M ˆ M is a normal rank-1
Backlund transformation if and only if N admits a lifting φ : N Ñ P that is an
integral manifold of a rank-4 Pfaffian system J with the independence condition
α1 ^ α2 ^ α3 ^ α4 ^ α0 ^ β0 ‰ 0, where J is generated by θi pi “ 1, ..., 4q:
θ1 “ β1 ´ s1α0 ´ t1β0 ´ µα1,
θ2 “ β2 ´ s2α0 ´ t2β0 ´ µα2,
θ3 “ β3 ´ s3α0 ´ t3β0 ´ µ´1α3,
θ4 “ β4 ´ s4α0 ´ t4β0 ´ εµ´1α4,
(4.11)
at each pu, u, s1, ..., s4, t1, ..., t4, µ, εq P P.
52
P
N6 M ˆ M
pM, Iq pM, Iq
τ
φ
φ
π π
Proof. By the construction of G1 and G1, on P , we have
pπ ˝ τq˚pIq “ xα0, α1 ^ α2, α3 ^ α4y,
rrpπ ˝ τq˚pθqss ” rrα0ss,
and similarly for I and the βi. Now assume that φ admits a lifting φ that integrates
the Pfaffian system J . It is easy to see that, on N ,
φ˚prrα1 ^ α2, α3 ^ α4ssq ” φ˚prrβ1 ^ β2, β3 ^ β4ssq
” φ˚prrdα0, dβ0ssq mod φ˚α0, φ˚β0.
In the last equality, we have used the assumption that µ2 ‰ ε, which guarantees that
the bundle rrα1^α2`α3^α4, β1^ β2` β3^ β4ss has rank two modulo α0, β0 when
pulled back to N . It follows that φ : N ÑM ˆ M defines a normal rank-1 Backlund
transformation.
Conversely, suppose that φ : N Ñ M ˆ M defines a normal rank-1 Backlund
transformation. Let η :“ pη0, η1, ..., η4q (resp. ξ :“ pξ0, ξ1, ..., ξ4q) be a local 1-adapted
coframing defined on a domain in M (resp. M). We have, by definition,
φ˚prrη1 ^ η2, η3 ^ η4ssq ” φ˚prrξ1 ^ ξ2, ξ3 ^ ξ4ssq mod φ˚η0, φ˚ξ0.
By switching the pairs pη1, η2q and pη3, η4q if needed, we can assume that, on N ,
η1 ^ η2 ” ξ1 ^ ξ2 mod η0, ξ0,
η3 ^ η4 ” ξ3 ^ ξ4 mod η0, ξ0,
53
where the pull-back symbol is dropped for convenience. Consequently, there exist 16
functions s1, ..., s4, t1, ..., t4, u1, ..., u8 defined on N such that, when restricted to N ,
ξ1 “ s1η0 ` t1ξ0 ` u1η1 ` u2η2,
ξ2 “ s2η0 ` t2ξ0 ` u3η1 ` u4η2,
ξ3 “ s3η0 ` t3ξ0 ` u5η3 ` u6η4,
ξ4 “ s4η0 ` t4η0 ` u7η3 ` u8η4,
(4.12)
where u1u4´u2u3 and u5u8´u7u6 are both nonvanishing. Moreover, since rrdη0, dξ0ss
has rank two modulo η0, ξ0, we have u1u4 ´ u2u3 ‰ u5u8 ´ u6u7.
Using the flexibility of choosing the 1-adapted coframings η and ξ, we can nor-
malize some of the ui . To be specific, we can apply SLp2,Rq-actions to pη1, η2q and
pη3, η4q to arrange u2, u3, u6, u7 to be zero, and to arrange that u4 “ ˘u1, u8 “ ˘u5,
pu1, u5 ą 0q. Then we transform pη0, η1, ..., η4q by a diagonal matrix in G1 to arrange
that u1 “ u4 ą 0 and u3 “ u´11 , u4 “ εu´1
1 , where ε “ ˘1. Meanwhile, this action
scales si pi “ 1, ..., 4q. Finally, if u1 “ u4 ă 1, then we switch the pairs of indices
p1, 2q and p3, 4q for ξ and η in (4.12), and multiply the new η0, η2, η4 by ε.
It is easy to see that the resulting pη, ξ, psiq, ptiq, u1, εq defines a map φ : N Ñ P
that is a lifting of φ and integrates J .
In light of Proposition 4.5, we make the
Definition 4.3. The system pP ,J q in Proposition 4.5 is called the 0-refined Backlund-
Pfaff system for normal rank-1 Backlund transformations relating two hyperbolic
Monge-Ampere systems.
Definition 4.4. An integral manifold of the 0-refined Backlund-Pfaff system pP ,J q
is called a 0-refined lifting of the underlying Backlund transformation.
In these terms, Proposition 4.5 says that each normal Backlund transformation
has a 0-refined lifting. Of course, given a normal Backlund transformation φ : N Ñ
54
M ˆ M , its 0-refined liftings are not unique.
Lemma 4.1. Let φ : N Ñ M ˆ M be a normal Backlund transformation. The
functions µ ˝ φ and ε ˝ φ are independent of the choice of 0-refined liftings φ of φ.
Proof. Clearly, for different choices of φ, the 1-forms φ˚α0 and φ˚β0 only change by
scaling. On N , the quotient λ1λ2 between the two solutions λ1, λ2 of the equation
pdβ0 ` λdα0q2” 0 mod α0, β0
is independent of the scaling of α0 and β0 but may depend on the order of the pair
pλ1, λ2q. By (4.11), on N , either λ1λ2 or λ2λ1 must be equal to εµ4.
Since we have arranged µ ě 1 for 0-refined liftings, the ambiguity of ordering
λ1 and λ2 can be removed by letting λ1λ2 “ εµ4. It follows that µ and ε are both
independent of the lifting.
Remark 7. pAq There is a simple geometric interpretation for the two possible values
of ε. Let pM, Iq and pM, Iq be as above. Suppose that φ : N Ñ M ˆ M defines a
normal rank-1 Backlund transformation. The 4-plane field D :“ rrα0, β0ssK on N is
independent of the choice of 0-refined liftings of φ. When restricted to D, the 4-forms
pdα0q2 and pdβ0q
2 define two orientations. If ε “ 1, these two orientations are the
same; if ε “ ´1, they are distinct.
pBq If a Backlund transformation is normal, which we will always assume for the
rest of this chapter, then ε “ 1 implies µ ą 1.
4.3 Obstructions to Integrability
In this section, we express the obstructions to integrability of pP ,J q in terms of the
invariants of the two hyperbolic Monge-Ampere systems.
For convenience, we introduce new notation below.
pAq Let η1 :“ α1, η2 :“ α2, η3 :“ α3, η4 :“ α4, η5 :“ α0, η6 :“ β0.
55
pBq The components of the pseudo-connection 1-form on G1 psee (4.1)q are de-
noted by ϕ0, ϕ1, ..., ϕ8; those on G1 are denoted by ψ0, ψ1, ..., ψ8.
On P , differentiating the θi and reducing modulo θ1, ..., θ4 yields the following
equations
dθi “ ´πiα ^ ηα ` τi mod θ1, ..., θ4, (4.13)
where πij are linear combinations of the 1-forms in
S :“ tdps1q, ..., dps4q; dpt1q, ..., dpt4q; dµ;ϕ0, ..., ϕ3, ϕ5, ϕ6, ϕ7;ψ0, ..., ψ3, ψ5, ψ6, ψ7u,
and τi, called the torsion, are of the form
τi “ Tijkηj ^ ηk, Tijk “ ´Tikj,
for some functions Tijk defined on P .
We can use a standard method (see [BCG`13]) to obtain from (4.13) obstructions
to integrability of pP ,J q. The key is to absorb the torsion terms (i.e., terms in τi)
as much as possible by adding linear combinations of η1, ..., ηk to the 1-forms in S.
In our case, the matrix pπiαq (called the tableau) takes the form
pπiαq “
¨
˚
˚
˝
π1 π2 0 0 π3 π4
π5 π6 0 0 π7 π8
0 0 π9 π10 π11 π12
0 0 π13 π14 π15 π16
˛
‹
‹
‚
,
where π1, ..., π16 are 1-forms independent of the θ1, ..., θ4, η1, ..., η6 and independent
among themselves. It is easy to see that all terms in τi can be absorbed except the
η3 ^ η4 terms in τ1 and τ2, and the η1 ^ η2 terms in τ3 and τ4.
Calculation yields
dθ1 ” ´µ2s1 ` εt1
µ2η3 ^ η4 mod θ1, ..., θ4, η1, η2, η5, η6,
dθ2 ” ´µ2s2 ` εt2
µ2η3 ^ η4 mod θ1, ..., θ4, η1, η2, η5, η6,
56
dθ3 ” ´pµ2t3 ´ s3qη1 ^ η2 mod θ1, ..., θ4, η3, η4, η5, η6,
dθ4 ” ´pµ2t4 ´ s4qη1 ^ η2 mod θ1, ..., θ4, η3, η4, η5, η6.
As a result, we have proved
Lemma 4.2. Integral manifolds of pP ,J q only exist in the locus L Ă P defined by
the equations
s3 “ ´t3µ2, s4 “ ´t4µ
2,
t1 “ ´εs1µ2, t2 “ ´εs2µ
2.
Definition 4.5. Let P1 Ă P be the locus defined by the equations
s1 “ t1 “ s3 “ t3 “ 0, s4 “ ´t4µ2, t2 “ ´εs2µ
2.
Let J1 be the restriction of J to P1. The rank-4 Pfaffian system pP1,J1q is called the
1-refined Backlund-Pfaff system for normal rank-1 Backlund transformations relating
two hyperbolic Monge-Ampere systems.
Definition 4.6. An integral manifold of the 1-refined Backlund-Pfaff system pP1,J1q
is called a 1-refined lifting of the underlying Backlund transformation.
Proposition 4.6. Let pM, Iq and pM, Iq be as above. Any normal rank-1 Backlund
transformation φ : N ÑM ˆ M admits a 1-refined lifting.
Proof. By previous discussion, there exists a 0-refined lifting φ of φ such that, when
pulled-back via φ to N ,
β1 “ s1α0 ´ εs1µ2β0 ` µα1,
β2 “ s2α0 ´ εs2µ2β0 ` µα2,
β3 “ ´t3µ2α0 ` t3β0 ` µ
´1α3,
β4 “ ´t4µ2α0 ` t4β0 ` εµ
´1α4.
57
If s2 ‰ 0, transform the pairs pα1, α2q and pβ1, β2q simultaneously using
h1 “
ˆ
1 ´s1s2
0 1
˙
If s2 “ 0 but s1 ‰ 0, transform the pairs pα1, α2q and pβ1, β2q simultaneously using
h2 “
ˆ
0 ´11 0
˙
such that the previou step applies. This will yield a 0-refined lifting with s1 “ 0.
Further, we can apply simultaneous SLp2,Rq-actions on pa3, a4q and pb3, b4q to arrange
t3 “ 0.
Now J1 is generated by
θ1 “ b1 ´ µa1,
θ2 “ b2 ´ s2a0 ` εs2µ2b0 ´ µa2,
θ3 “ b3 ´ µ´1a3,
θ4 “ b4 ` t4µ2a0 ´ t4b0 ´ εµ
´1a4.
Given a normal rank-1 Backlund transformation φ : N Ñ M ˆ M , it is easy to
see that whether the product s2t4 locally vanishes is independent of the choice of
1-refined liftings of φ. The following proposition says that the case of s2t4 “ 0 is
quite restricted when both pM, Iq and pM, Iq are Euler-Lagrange systems.
Proposition 4.7. Let pM, Iq and pM, Iq be two hyperbolic Euler-Lagrange systems.
If there exists a normal rank-1 Backlund transformation φ : N Ñ M ˆ M such that
s2t4 “ 0 on a 1-refined lifting of φ, then both pM, Iq and pM, Iq must be equivalent
to the system corresponding to the wave equation zxy “ 0.
Proof. By the Euler-Lagrange assumption and Proposition 4.2, S2 “ S2 “ 0. Let
S1 “
ˆ
V1 V2
V3 V4
˙
, S1 “
ˆ
W1 W2
W3 W4
˙
.
58
By Proposition 4.3, it suffices to show that, on any integral manifold of the 1-refined
Backlund-Pfaff system pP1,J1q, we have S1 “ S1 “ 0.
First, we assume s2 “ 0. Restricted to the locus defined by s2 “ 0 in P1, the
tableau pπiαq associated to pP1,J1q satisfies
πiα “ 0, pi “ 1, 2; α “ 3, ..., 6q.
As a result, the ηi ^ ηj terms in dθ1 and dθ2, i, j “ 3, ..., 6 and i ‰ j, cannot be
absorbed and the corresponding coefficients must vanish on integral manifolds of
pP1,J1q. Calculating with MapleTM, we find
dθ1 ” µpV1η3 ` V2η4q ^ η5 ´1
µpW1η3 ` εW2η4q ^ η6 `W2t4µ
2η5 ^ η6,
dθ2 ” µpV3η3 ` V4η4q ^ η5 ´1
µpW3η3 ` εW4η4q ^ η6 `W4t4µ
2η5 ^ η6,
both congruences being reduced modulo θ1, ..., θ4, η1, η2. It follows that S1 “ S1 “ 0.
The case of t4 “ 0 is similar.
Proposition 4.8. On any 1-refined lifting of a normal rank-1 Backlund transforma-
tion relating two hyperbolic Euler-Lagrange systems, the following expressions must
vanish
Φ1 :“ ´µ4V1 ` εW1, Φ2 :“ ´µ4V2 `W2,
Φ3 :“ µ4W4 ´ V4, Φ4 :“ µ4W2 ´ V2.
Proof. This is evident when such a Backlund transformations satisfies s2t4 “ 0. In
fact, by Proposition 4.7, the functions Vi and Wi vanish identically on P1. Otherwise,
restricting to the open subset of P1 defined by s2t4 ‰ 0, a calculation shows that
the torsion of the Pfaffian system J1 can be absorbed only if Φi pi “ 1, ..., 4q are all
zero.
59
Corollary 4.3. If two hyperbolic Euler-Lagrange systems are related by a normal
rank-1 Backlund transformation with ε “ 1, then they are either both degenerate or
both nondegenerate.
Proof. We have noted above (see Remark 7) that, if a normal Backlund transforma-
tion satisfies ε “ 1, then µ ‰ 1. The vanishing of Φ2 and Φ4 on a 1-refined lifting of
such a Backlund transformation then implies that, on such a lifting,
V2 “ W2 “ 0.
By the vanishing of Φ1 and Φ3, it is easy to see that the matrices S1 and S1 are either
both degenerate or both nondegenerate.
Note that, in the proof of Corollary 4.3, the condition V2 “ W2 “ 0 is meaningful
only if it is independent of the choice of 1-refined liftings of a Backlund transforma-
tion. To make this point explicit, we state the following proposition which shows how
1-refined liftings of a normal rank-1 Backlund transformation relate to each other.
Proposition 4.9. Let φ be a normal rank-1 Backlund transformation relating two
hyperbolic Monge-Ampere systems satisfying s2t4 ‰ 0 on its 1-refined liftings. There
exists a subgroup H Ă G1 ˆG1 such that any two 1-refined liftings of φ are related,
in the G1 ˆ G1 component, by an H-valued transformation. Moreover,
i. if µ ą 1, then H is the subgroup generated by elements of the form h “ pg, g1q
where
g “
¨
˚
˚
˚
˚
˝
h0 0 0 0 00 h1 0 0 00 h3 h0h
´11 0 0
0 0 0 h2 00 0 0 h4 h0h
´12
˛
‹
‹
‹
‹
‚
P G1,
and g1 is the result of replacing h4 in g by εh4.
ii. if µ “ 1, then H is generated by the subgroup in case i and the element
hJ “ pJ1, Jq,
60
where J is as in (4.3) and J 1 “ diagp´1, 1,´1, 1,´1qJ.
4.4 A Special Class of Backlund Transformations
In the previous section, we have seen that, to a normal rank-1 Backlund transfor-
mation φ : N Ñ M ˆ M relating two hyperbolic Monge-Ampere systems pM, Iq
and pM, Iq, we can associate a function µ : N Ñ r1,8q that is independent4 of the
0-refined liftings of φ.
Definition 4.7. A normal rank-1 Backlund transformation relating two hyperbolic
Monge-Ampere systems is said to be special if µ “ 1.
For the rest of this section, we will focus on special Backlund transformations
relating two hyperbolic Euler-Lagrange systems. A motivation to this is that many
classical Backlund transformations are of this type (cf. [Cle01], [CI13]).
By Proposition 4.8, given a special rank-1 Backlund transformation φ : N Ñ
M ˆ M relating two hyperbolic Euler-Lagrange systems, the following equalities
must hold on any 1-refined lifting of φ:
ε “ ´1, W1 “ ´V1, W2 “ V2, W4 “ V4. (4.14)
Now let Ps Ă P1 be defined by the equations
µ “ 1, ε “ ´1.
Proposition 4.10. Any 1-refined lifting of a special rank-1 Backlund transforma-
tion relating two hyperbolic Euler-Lagrange systems is completely contained in Ps.
Moreover, on such a lifting, in addition to (4.14), we have
V3 `W3 ` 2s2t4 “ 0. (4.15)
4 In fact, it is easy to see that µ is an invariant of the corresponding Backlund transformation.
61
Proof. Restricting to Ps, the generators θi of J1 satisfy equations of the form
dθi “ ´πiα ^ ηα ` τi.
The tableau pπiαq now takes the form
pπiαq “
¨
˚
˚
˝
´π1 ´π2 0 0 s2π8 s2π8
´π3 π1 ´ π7 0 0 ´s2π7 ´ π9 ´π9
0 0 ´π4 ´π5 ´t4π10 t4π10
0 0 ´π6 π7 ´ π4 t4π7 ` π11 ´π11
˛
‹
‹
‚
,
where, reduced modulo η0, η1, ..., η4,
π1 ” ϕ1 ´ ψ1, π4 ” ϕ5 ´ ψ5, π7 ” ϕ0 ´ ψ0, π10 ” ψ6
π2 ” ϕ2 ´ ψ2, π5 ” ϕ6 ` ψ6, π8 ” ψ2, π11 ” ´dpt4q ` t4ψ5.
π3 ” ϕ3 ´ ψ3, π6 ” ´ϕ7 ´ ψ7, π9 ” ´dps2q ` s2ψ1,
By a calculation using MapleTM, it is easy to see that the torsion τi can be absorbed
only if the equations (4.14) and (4.15) hold.
The equalities (4.14) and (4.15) tell us which Euler-Lagrange systems may be
Backlund-related. In particular, by Propositions 4.9 and 4.1, it is easy to see that
whether V2 (hence W2) vanishes is independent of the choice of 1-refined liftings.
Thus, we may locally5 classify special rank-1 Backlund transformations relating two
hyperbolic Euler-Lagrange systems into the following three types:
Type I. V2 “ 0, detpS1q “ V1V4 ‰ 0;
Type II. V2 “ 0, detpS1q “ V1V4 “ 0;
Type III. V2 ‰ 0.
Lemma 4.4. pAq A Type I special Backlund transformation cannot be an auto-
Backlund transformation of a nondegenerate hyperbolic Euler-Lagrange system.
5 Namely, the conditions below hold on an entire open subset of N .
62
pBq Two hyperbolic Euler-Lagrange systems related by a Type III special Backlund
transformation cannot be both degenerate.
Proof. Part pAq is immediate, because, in this case, on a 1-refined lifting,
detpS1q “ ´ detpS1q ‰ 0.
To prove Part pBq, first apply Proposition 4.9 to show that, in this case, one
can always find a 1-refined lifting on which V1 “ V4 “ 0. For such a 1-refined
lifting, by (4.15), it is easy to see that the two Euler-Lagrange systems can be both
degenerate only when s2t4 “ 0. By Proposition 4.7, both S1 and S1 must vanish,
which impossible since we have assumed V2 ‰ 0.
Remark 8. Lemma 4.4 allows us identify the types of various known Backlund trans-
formations. For example, any special Backlund transformation relating a pair of
f -Gordon equations must appear as Type II (the classical auto-Backlund transfor-
mation of the sine-Gordon equation is of this type); the classical auto-Backlund
transformation relating surfaces in E3 with a negative constant Guass curvature,
which can be verified to be special, must be of Type III.
Now we focus on Type II. The analysis of Type I and III will be included in a
future work.
Let φ : N ÑM ˆ M be a Type II special rank-1 Backlund transformation in the
sense above. It is easy to see using Propositions 4.1 and 4.9 that whether the pair
pV1, V4q vanishes is independent of the choice of 1-refined liftings of φ. It follows that
φ must be one of the following two types.
Type IIa: on any 1-refined lifting of φ, pV1, V4q “ 0;
Type IIb: on any 1-refined lifting of φ, pV1, V4q ‰ 0.
63
4.4.1 Type IIa
In this case, (4.14) implies that
W1 “ W2 “ W4 “ 0.
If locally either V3 or W3 is zero, which is independent of the choice of 1-refined
liftings, the underlying Backlund transformation must relate a hyperbolic Euler-
Lagrange system with the system corresponding to zxy “ 0. See [CI`09] for a
classification of all hyperbolic Monge-Ampere systems that are rank-1 Backlund-
related to the equation zxy “ 0.
Now suppose that V3 and W3 are both nonzero on any 1-refined lifting of φ.
Theorem 4.5. If two hyperbolic Euler-Lagrange systems are related by a Type IIa
special rank-1 Backlund transformation, then each of them corresponds (up to contact
equivalence) to a second order PDE of the form
zxy “ F px, y, z, zx, zyq.
Proof. By definition, any Type IIa special Backlund transformation admits a 1-
refined lifting that is completely contained in the locus PIIa Ă P1 defined by the
equations
µ “ 1, ε “ ´1, V1 “ V2 “ V4 “ W1 “ W2 “ W4 “ 0.
Let G2 Ă G1 be the subbundle defined by V1 “ V2 “ V4 “ 0; similarly, let G2 Ă G1 be
the subbundle defined by W1 “ W2 “ W4 “ 0. It is clear that PIIa is the product of
G2, G2, and a space of parameters with coordinates ps2, t4q.
By (4.9), on G2, there exist functions Pij and V3j such that
ϕ2 “ P20α0 ` P21α1 ` ¨ ¨ ¨ ` P24α4,
ϕ6 “ P60α0 ` P61α1 ` ¨ ¨ ¨ ` P64α4,
dpV3q “ pϕ1 ` ϕ5qV3 ` V30α0 ` V31α1 ` ¨ ¨ ¨ ` V34α4.
64
Similarly, on G2, there exist functions Qij and W3j such that
ψ2 “ Q20β0 `W21β1 ` ¨ ¨ ¨ `Q24β4,
ψ6 “ Q60β0 `W61β1 ` ¨ ¨ ¨ `Q64β4,
dpW3q “ pψ1 ` ψ5qW3 `W30β0 `W31β1 ` ¨ ¨ ¨ `W34β4.
There is freedom to add linear combinations of α0, ..., α4 (resp. β0, ..., β4) to ϕi
(resp. ψi) without changing the form of the corresponding Monge-Ampere structure
equations. Using this, we can arrange the following expressions to be zero:
P21, P22, P63, P64; Q21, Q22, Q63, Q64.
Moreover, applying d2 “ 0 to the Monge-Ampere structure equations yields
dpdα1q ” P24V3α0 ^ α3 ^ α4 mod α1, α2,
dpdα2q ” V34α0 ^ α3 ^ α4 mod α1, α2,
dpdα3q ” P62V3α0 ^ α1 ^ α2 mod α3, α4,
dpdα4q ” V32α0 ^ α1 ^ α2 mod α3, α4.
This implies that, on G2,
P24V3 “ V34 “ P62V3 “ V32 “ 0. (4.16)
By a similar argument, one can show that, on G2,
Q24W3 “ W34 “ Q62W3 “ W32 “ 0.
Restricted to PIIa, the generators θi pi “ 1, ..., 4q of J1 satisfy equations of the
form
dθi “ ´πiα ^ ηα ` τi.
The tableau takes the form
pπiαq “
¨
˚
˚
˝
´π1 0 0 0 0 0´π2 π1 ´ π5 0 0 ´s2π5 ` π6 π6
0 0 ´π3 0 0 00 0 π4 ´π3 ` π5 t4π5 ´ π7 π7
˛
‹
‹
‚
,
65
where, modulo ηi,
π1 ” ϕ1 ´ ψ1, π3 ” ϕ5 ´ ψ5, π5 ” ϕ0 ´ ψ0, π7 ” dpt4q ´ t4ψ5,
π2 ” ϕ3 ´ ψ3, π4 ” ϕ7 ` ψ7, π6 ” dps2q ´ s2ψ1.
Assuming s2, t4 to be both nonzero and calculating with MapleTM, we find that the
torsion can be absorbed only if the following expressions are zero
P20, P23, P24, P60, P61, P62; Q20, Q23, Q24, Q60, Q61, Q62.
One can verify that, on the subbundle of G2 defined by the vanishing of P20, P23, P24,
P60, P61 and P62, the following structure equations hold
dα0 “ ´ϕ0 ^ α0 ` α1 ^ α2 ` α3 ^ α4,
dα1 “ ´ϕ1 ^ α1,
dα2 “ ´ϕ3 ^ α1 ` pϕ1 ´ ϕ0q ^ α2 ` V3α0 ^ α3,
dα3 “ ´ϕ5 ^ α3,
dα4 “ ´ϕ7 ^ α3 ` pϕ5 ´ ϕ0q ^ α4 ` V3α0 ^ α1.
Clearly, the systems xα1y and xα3y are both integrable. It is a similar case for the
structure equations on G2. By Proposition 4.4, the proof is complete.
4.4.2 Type IIb
In this case, on a 1-refined lifting of φ, either V1 or V4 vanishes. By Proposition 4.9
(in particular, using hJ), we can arrange V1 ‰ 0 and V4 “ 0 on a 1-refined lifting
of φ. Such a 1-refined lifting can always be chosen to further satisfy V1 “ 1 and
V3 “ W3 “ 0 or 1.
In the next proposition we show that the case of V1 “ 1 and V3 “ W3 “ 0 is
impossible. Then we characterize the case when φ admits a 1-refined lifting for which
V1 “ 1, V3 “ W3 “ 1.
66
Proposition 4.11. Restricting to the locus in P1 defined by
µ “ 1, ε “ ´1, V1 “ ´W1 “ 1, V2 “ V3 “ V4 “ W2 “ W3 “ W4 “ 0,
J1 is not integrable.
Proof. If there exists a 1-refined lifting of a special Backlund transformation such
that V3 “ W3 “ 0, then the equality (4.15) enforces that s2t4 “ 0 on such a lifting.
By Proposition 4.7, both Monge-Ampere systems must be contact equivalent to the
wave equation zxy “ 0. In particular, Vi and Wi must all be zero on G1 and G1,
respectively. This is a contradiction.
Proposition 4.12. Let pM, Iq and pM, Iq be two hyperbolic Euler-Lagrange systems.
If φ : N ÑMˆM defines a Type IIb special rank-1 Backlund transformation relating
pM, Iq and pM, Iq, then each of pM, Iq and pM, Iq must have a characteristic system
that contains a rank-1 integrable subsystem.
Proof. The idea is similar to that of Proposition 4.5. We restrict the differential ideal
J1 to the locus PIIb Ă P1 defined by the equations
µ “ 1, ε “ ´1, V1 “ ´W1 “ V3 “ W3 “ 1, V2 “ V4 “ W2 “ W4 “ 0
and analyse the obstructions to integrability of the resulting rank-4 Pfaffian system.
By (4.9), on the subbundle of G1 defined by V1 “ V3 “ 1 and V2 “ V4 “ 0, there
exist functions Pij such that
ϕ2 “ P20α0 ` P21α1 ` ¨ ¨ ¨ ` P24α4 ` ϕ0 ´ ϕ1 ` ϕ5,
ϕ3 “ P30α0 ` P31α1 ` ¨ ¨ ¨ ` P34α4 ` ϕ1 ` ϕ5,
ϕ6 “ P60α0 ` P61α1 ` ¨ ¨ ¨ ` P64α4.
Using the freedom in the choice of ϕi, we can arrange P21, P22, P23, P31, P32, P34, P64
to be zero. By expanding dpdαiq “ 0, we find
P24 “ 0, P61 “ ´P62, P63 “ 0.
67
Similarly, on the subbundle of G1 defined by ´W1 “ W3 “ 1 and W2 “ W4 “ 0,
there exist functions Qij such that
ψ2 “ Q20β0 `Q21β1 ` ¨ ¨ ¨ `Q24β4 ´ ψ0 ` ψ1 ´ ψ5,
ψ3 “ Q30β0 `Q31β1 ` ¨ ¨ ¨ `Q34β4 ´ ψ1 ´ ψ5,
ψ6 “ Q60β0 `Q61β1 ` ¨ ¨ ¨ `Q64β4.
Using the freedom in the choice of ψi, we can arrange Q21, Q22, Q23, Q31, Q32, Q34, Q64
to be zero. By expanding dpdβiq “ 0, we find
Q24 “ 0, Q61 “ Q62, Q63 “ 0.
Denote the restriction of J1 to PIIb as JIIb. Calculating with MapleTM, it is easy
to see that the torsion of pPIIb,JIIbq can be absorbed only if the following expressions
are zero:
s2t4 ` 1, P20 ´Q20, P60, P62, Q60, Q62.
In particular, the vanishing of P60, P62, Q60 and Q62 implies that
dα3 “ ´ϕ5 ^ α3, dβ3 “ ´ψ5 ^ β3.
The conclusion of the proposition follows.
68
5
Homogeneous Rank-2 Backlund Transformations
Rank-2 Backlund transformations are interesting because they arise naturally in at
least the following three ways1: one, constructed from a single or a 1-parameter
family of rank-1 Backlund transformations; two, constructed as the composition (see
Proposition 2.4) of two rank-1 Backlund transformations; three, arise as geometric
examples such as the classical auto-Backlund transformation (1.5) of the hyperbolic
Tzitzeica equation (1.4). An ultimate goal of studying Backlund transformations of
higher ranks might be to answer the question: How many more differential systems
are Backlund-related when we allow Backlund tranformations of a higher rank?
In this chapter, for rank-2 Backlund transformations relating two hyperbolic
Monge-Ampere systems, we formulate the corresponding equivalence problem, and
provide a partial classification in the homogeneous case.
5.1 Genericity Conditions and Structure Reduction
Let pN,B; π, πq be a rank-2 Backlund transformation relating two hyperbolic Monge-
Ampere systems pM, Iq and pM, Iq. There exists a local coframing pθ, η1, ..., η4q
1 We are not claiming that these cases are disjoint.
69
defined on a domain U1 ĂM such that θ is a contact form and that the differential
ideal I on U1 can be written as I “ xθ, η1 ^ η2, η3 ^ η4yalg. Similarly, we have
I “ xθ, η1^ η2, η3^ η4yalg for a coframing pθ, η1, ..., η4q defined on a domain U2 Ă M .
Definition 5.1. Let π : N Ñ M be a submersion. A vector subbundle J Ă T ˚N is
said to be transversal to π if, at each point p P N , pJpqK X kerppπ˚q “ 0 Ă TpN .
By Definitions 2.11 and 2.12 of Chapter 2, there exists a rank-two subbundle
J Ă B1 that is transversal to π and satisfy B1 “ J ‘ π˚pI1q. Similarly, there exists
a rank-two subbundle J Ă B1, transversal to π, satisfying B1 “ J ‘ π˚pI1q. Clearly,
B1 Ă T ˚N has rank three.
Now we start to formulate the equivalence problem for rank-2 Backlund trans-
formations relating two hyperbolic Monge-Ampere systems. At several points of our
analysis, we define genericity conditions to help us distinguish between cases.
To start with, we define the
First genericity condition: π˚θ and π˚θ, as sections of B1, are linearly
independent at each point of their common domain in N .
This first genericity condition will always be assumed in this chapter. As a result
of this assumption, on an open subset U Ă N , there exists a 1-form γ P Ω1pUq such
that π˚θ, π˚θ, γ form local basis sections of the bundle B1 Ñ N . Moreover, by the
definition of an integrable extension, γ can be chosen in such a way that π˚θ, γ (resp.
π˚θ, γ) restrict to each fiber of π (resp. π) to be linearly independent.
We have thus obtained a coframing pπ˚θ, π˚θ, γ, π˚η1, π˚η2, π˚η3, π˚η4q on U Ă N .
Dropping the pull-back symbol for convenience, we have, by Definition 2.12,
B “ xθ, θ, γ, η1^ η2, η3
^ η4yalg “ xθ, θ, γ, η
1^ η2, η3
^ η4yalg.
Now we define the
70
Second genericity condition: dθ and dθ are everywhere linearly inde-
pendent modulo θ, θ, γ.
We now assume the second genericity condition2. Consequently, as rank-2 vector
bundles over U ,
rrdθ, dθss ” rrη1^ η2, η3
^ η4ss ” rrη1
^ η2, η3^ η4
ss mod θ, θ, γ.
In addition, we have
B “ xθ, θ, γ, dθ, dθyalg.
Such a choice of coframing on U Ă N can be normalized. In fact, one can start
with a 0-adapted coframing pθ, η1, ..., η4q on U that satisfies the additional condition
dθ ” η1^ η2
` η3^ η4 mod θ.
The same congruence then holds for the pull-backs of θ, η1, ..., η4 under π. After the
change of notations θ ÞÑ ω0, θ ÞÑ ω0, and ηi ÞÑ ωi, we obtain, on U ,
dω0” ω1
^ ω2` ω3
^ ω4 mod ω0, (5.1)
dω0” A1ω
1^ ω2
` A2ω3^ ω4 mod ω0, ω0, γ, (5.2)
dγ ” A3ω1^ ω2
` A4ω3^ ω4 mod ω0, ω0, γ, (5.3)
where A1, ..., A4 are functions defined on U (at this point, Ai are essentially functions
on πpUq). Here A1, A2 are non-vanishing because ω0^pdω0q2 ‰ 0. Moreover, A1 ‰ A2
by the second genericity assumption.
As a result of the above, one can successively perform the following steps without
violating the normalizations made in earlier steps:
1) add multiples of ω0, ω0 to γ to arrange A3 “ A4 “ 0;
2) add multiples of ω0 to ω1, ..., ω4 such that the congruence (5.2) holds modulo
only ω0 and γ;
2 This assumption is to be removed in Section 5.3.
71
3) scale ω0 to arrange A1 “ 1;
4) depending on the sign of A2, scale ω3 and ω0 to put (5.1) and (5.2) in the form
dω0” Aω1
^ ω2` ω3
^ ω4 mod ω0,
dω0” ω1
^ ω2` εAω3
^ ω4 mod ω0, γ,
where ε “ ˘1, A ą 0, and A2 ‰ ε.
5) if needed, switch pω1, ω2q with pω3, ω4q and scale ω0 and ω0 to arrange A ě 1;
6) add multiples of γ to ω3, ω4 such that
dω0” Aω1
^ ω2` ω3
^ ω4` pB3ω
3`B4ω
4q ^ γ mod ω0, (5.4)
dω0” ω1
^ ω2` εAω3
^ ω4` pB1ω
1`B2ω
2q ^ γ mod ω0, (5.5)
where B1, ..., B4 are functions defined on U . Note, in particular, that there cannot
be a pω0 ^ γq-term in dω0, because ω0 ^ pdω0q3 “ 0.
Finally, there exist functions C0, C1, ..., C4, D1, ..., D4 on U such that
dγ ” C0ω0^ ω0
` Ciωi^ ω0
`Diωi^ ω0 mod γ. (5.6)
Now we define the
Third genericity condition: C0 is locally non-vanishing; by multiply-
ing γ by 1C0
, we arrange C0 “ 1.
Remark 9. The parameter ε is intrinsic to a rank-2 Backlund transformation. It
determines whether pdω0q2 and pdω0q2 define the same or the opposite orientations
on the rank-4 distribution pB1qK Ă TU .
For the rest of this section, we will focus on the case when all three genericity
conditions are satisfied and ε “ 1. In particular, this enforces A ą 1 after the
normalization above.
72
Definition 5.2. Let pN,B; π, πq be a rank-2 Backlund transformation satisfying all
three genericity conditions and ε “ 1. A coframing pω0, ω0, γ, ω1, ..., ω4q defined on
a domain U Ă N is said to be 0-adapted if it satisfies
rrω0ss “ rrπ˚θss, rrω0
ss “ rrπ˚θss, rrω0, ω0, γss “ B1 (5.7)
and the equations
dω0” Aω1
^ ω2` ω3
^ ω4` pB3ω
3`B4ω
4q ^ γ mod ω0, (5.8)
dω0” ω1
^ ω2` Aω3
^ ω4` pB1ω
1`B2ω
2q ^ γ mod ω0, (5.9)
dγ ” ω0^ ω0
` Ciωi^ ω0
`Diωi^ ω0 mod γ, (5.10)
with A ą 1.
Lemma 5.1. For a rank-2 Backlund transformation pN,Bq satisfying all three gener-
icity conditions and ε “ 1, the 0-adapted coframings are local sections of a G-
structure G over N , where G Ă GLp7,Rq is generated by
g “
¨
˚
˚
˚
˚
˝
detpbq 0 0 0 00 detpaq 0 0 00 0 detpaq detpbq 0 00 0 0 a 00 0 0 0 b
˛
‹
‹
‹
‹
‚
,detpaq “ detpbq ‰ 0,
a, b P GLp2,Rq.(5.11)
Proof. Given a 0-adapted coframing ω define on U Ă N , it is easy to check that
ω ¨ g “ g´1ω remains a 0-adapted coframing for any g : U Ñ G.
Converserly, by (5.7) and A ą 1, changing from one 0-adapted coframing to an-
other does not change the order of the bundles rrω0, ω0, γ, ω1, ω2ss and rrω0, ω0, γ, ω3, ω4ss.
Consequently, if ω1 and ω2, both defined on U Ă N , are two 0-adapted coframings,
then there exist a function g : U Ñ GLp7,Rq of the form
g “
¨
˝
Ψ 0 0˚ a 0˚ 0 b
˛
‚,
73
where Ψ P GLp3,Rq is lower triangular, and a, b P GLp2,Rq, such that ω2 “ ω1 ¨ g.
It is then not hard to see that, in order for ω1, ω2 to both satisfy (5.8), (5.9), and
(5.10), g must be of the form (5.11).
Concerning 0-adapted coframings, we prove the following
Lemma 5.2. piq Let Ξ10 “ rrθ, η1, η2ss and Ξ01 “ rrθ, η
3, η4ss, associated to pM, Iq. As
vector bundles over U1 Ă M , their pull-backs via π are, up to ordering, rrω0, ω1, ω2ss
and rrω0, ω3 ´B4γ, ω4 `B3γss.
piiq Let Ξ10 “ rrθ, η1, η2ss and Ξ01 “ rrθ, η
3, η4ss, associated to pM, Iq. As vector
bundles over U2 Ă M , their pull-backs via π are, up to ordering, rrω0, ω3, ω4ss and
rrω0, ω1 ´B2γ, ω2 `B1γss.
Proof. We only prove piq; the proof of piiq is identical.
By (5.8), the vector bundle associated to the Cartan system Cpxω0yq has a basis
of sections ω0, ω1, ω2, ω3´B4γ and ω4`B3γ. Since π˚pIq Ă B, π˚prrθssq “ rrω0ss and
CpIq “ Cpxθyq, the ideal generated by π˚pIq must be contained in the intersection
of B and Cpxω0yq, which equals to
xω0, ω1^ ω2, pω3
´B4γq ^ pω4`B3γqyalg. (5.12)
It follows that the system generated by the sections of π˚pIq is equal to BXCpxω0yq.
Comparing the characteristic systems lead to piq.
By Lemma 5.2, the 1-forms ω0, ω1, ω2, ω3´B4γ, ω4`B3γ are π-semi-basic. More-
over, the exterior differential system (5.12) is invariant in the fiber directions of π
(see Definition 2.18).
To be explicit, letX0, X0, Xγ, X1, ..., X4 be the dual vector fields of ω0, ω0, γ, ω1, ω2,
ω3´B4γ, ω4`B3γ; and let Y0, Y0, Yγ, Y1, ..., Y4 be the dual vector fields of ω0, ω0, γ, ω1´
B2γ, ω2 ` B1γ, ω
3, ω4. One can show that any 0-adapted coframing defined on an
open subset U Ă N must satisfy the
74
Invariance Property:
(Let ΓpKq denote the set of sections of a bundle K Ñ U ; let σ P ΓpKq
be an arbitrary element; let LX denote the Lie derivative along a vector
field X.)
When K is either rrω0, ω1, ω2ss or rrω0, ω3 ´B4γ, ω4 `B3γss, we have
LX0σ, LXγσ P ΓpKq, @σ P ΓpKq; (5.13)
When K is either rrω0, ω3, ω4ss or rrω0, ω1 ´B2γ, ω2 `B1γss, we have
LY0σ, LYγσ P ΓpKq, @σ P ΓpKq. (5.14)
Proposition 5.1. Let N be a 7-manifold. Suppose that there exists a coframing
ω “ pω0, ω0, γ, ω1, ..., ω4q defined on a domain U Ă N satisfying (5.8), (5.9), and
(5.10) with A ą 1. Suppose, in addition, that ω satisfies the Invariance Property
described by (5.13) and (5.14). Such a coframing ω is then 0-adapted to a rank-2
Backlund transformation satisfying all three genericity conditions with ε “ 1.
Proof. The proof is in the same spirit as that of Proposition 3.1 of Chapter 3,
so we omit the most of it. However, we need to show, for instance, that the system
xω0, ω1, ω2, ω3´B4γ, ω4`B3γy is Frobenius, in order to justify the existence of rank-2
fibers. This amounts to showing that rX0, Xγs is annihilated by these 5 generating
1-forms. For example, we have
ω1prX0, Xγsq “ ´dω1
pX0, Xγq “ ´pX0 dω1qpXγq.
By (5.13), X0dω1 is a linear combination of ω0, ω1, ω2, hence annihilates Xγ. Other
cases are similar.
A simple calculation shows that, under the transformation u ÞÑ u ¨ g “ g´1u P G,
where g is as in (5.11), the coefficients B1, B2, B3, B4 in (5.8) and (5.9) transform by
ˆ
B1
B2
˙
pu ¨ gq “ detpaqaTˆ
B1
B2
˙
puq,
ˆ
B3
B4
˙
pu ¨ gq “ detpbqbTˆ
B3
B4
˙
puq.
75
Since SLp2,Rq acts transitively on R2zt0u, we can always, by choosing an appropriate
0-adapted coframing, reduce to one of the following 4 cases:
Case 1: B1 “ B3 “ 1, B2 “ B4 “ 0;
Case 2: Bi “ 0 pi “ 1, ..., 4q;
Case 3: B2, B3, B4 “ 0, B1 “ 1;
Case 31: B1, B2, B4 “ 0, B3 “ 1.
Cases 3 and 31 may be turned in to one another by switching the submersions π
and π. We thus consider them as essentially one case.
Geometrically, in Cases 1 and 3, the fibers of π and π are transversal; in Case 2,
the fibers of π and π are non-transversal.
We will mainly focus on Case 1. In particular, we will prove that this seemingly
most ‘generic’ case contains no homogeneous Backlund transformation. Case 2 turns
out to be less interesting, in that locally N admits a 6-dimensional quotient which is a
rank-1 Backlund transformation relating the same pair of hyperbolic Monge-Ampere
systems. Case 3 is work in progress.
5.1.1 Case 1: pB1, B2q “ pB3, B4q “ p1, 0q
This reduces to a G1-structure G1 Ă G. Here, G1 consists of elements g P G satisfying
a12 “ b12 “ 0 and a22pa11q2 “ b22pb11q
2 “ 1. In particular, G1 is 3-dimensional.
We can apply the reproducing property (Proposition 2.2 of Chapter 2) to the
equations (5.8)-(5.10) to obtain the structure equations on G1:
d
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
ω0
ω0
γω1
ω2
ω3
ω4
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
“ ´
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
α 0 0 0 0 0 00 α 0 0 0 0 00 0 2α 0 0 0 00 0 0 ´α 0 0 00 0 0 β1 2α 0 00 0 0 0 0 ´α 00 0 0 0 0 β2 2α
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
^
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
ω0
ω0
γω1
ω2
ω3
ω4
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
`
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
Ω0
Ω0
ΓΩ1
Ω2
Ω3
Ω4
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
, (5.15)
76
where rrα, β1, β2ss Ă T ˚G1 has rank 3 and is transversal to the fibers of G1, and
Ω0“ Aω1
^ ω2` ω3
^ pω4` γq ` pP0ω
0`Kγ ` Piω
iq ^ ω0,
Ω0“ ω1
^ pω2` γq ` Aω3
^ ω4` pQ0ω
0` Lγ `Qiω
iq ^ ω0, (5.16)
Γ “ ω0^ ω0
` Ciωi^ ω0
`Diωi^ ω0, pA ą 1q
for functions A,Ci, Di, P0, K, Pi, Q0, L,Qi pi “ 1, ..., 4q defined on G1. In particular, Γ
takes the form above because one can add any linear combination of ω0, ω0, γ, ω1, ..., ω4
to α without changing the form of (5.15) and (5.16).
Let U Ă N be a sufficiently small domain. Let σ : U Ñ G1 be any section. The
pull-back by σ of the tautological 1-form on G1 must then satisfy the Invariance
Property (see (5.13) and (5.14)).
Lemma 5.3. pAq If we express the 2-forms Ω1,Ω2 under the basis tω0^ ω0, ..., ω3^
ω4u, the coefficients of the following terms are zero:
γ ^ ω3, γ ^ ω4, γ ^ ω0, ω0^ ω3, ω0
^ ω4.
Proof. By construction, the vector fields X0, Xγ are annihilated by the 1-forms
ω0, ω1, ω2, ω3, ω4 ` γ and are dual to ω0 and γ. Using this and (5.15), we obtain
LX0ω0” 0 mod ω0,
LX0ωi ” X0 Ωi mod ω0, ω1, ω2
pi “ 1, 2q,
LXγω0” 0 mod ω0,
LXγωi ” Xγ Ωi mod ω0, ω1, ω2pi “ 1, 2q.
The conclusion then follows from the Invariance Property.
By similar arguments, we can prove
Lemma 5.3. pBq If we express the 2-forms Ω1 and Ω2´γ^C2ω0`pC3ω
3`C4ω4q^ω0
under the basis tω0^ ω0, ..., ω3^ω4u, the coefficients of the terms in each pair below
77
are the same:
pγ ^ ω0, ω2^ ω0
q, pγ ^ ω3, ω2^ ω3
q, pγ ^ ω4, ω2^ ω4
q;
the coefficients of the following terms are zero:
ω0^ ω3, ω0
^ ω4.
pCq In the expressions of Ω3 and Ω4, the coefficients of the following terms are zero:
γ ^ ω1, γ ^ ω2, γ ^ ω0, ω0^ ω1, ω0
^ ω2.
pDq In the expressions of Ω3 and Ω4´γ^D4ω0`pD1ω
1`D2ω2q^ ω0, the coefficients
of the terms in each pair below are the same:
pγ ^ ω1, ω4^ ω1
q, pγ ^ ω2, ω4^ ω2
q, pγ ^ ω0, ω4^ ω0
q;
the coefficients of the following terms are zero:
ω0^ ω1, ω0
^ ω2.
By Lemma 5.3, Ω1, ...,Ω4 must take the form
Ω1“ T 1
0ω0^ ω0
` ω0^ pT 1
1ω1` T 1
2 pω2` γqq ` ω0
^ pT 11ω
1` T 1
2ω2q
` γ ^ pR11ω
1`R1
2ω2q `
1
2T 1ijω
i^ ωj, (5.17)
Ω2“ T 2
0ω0^ ω0
` ω0^ pC3ω
3` C4ω
4` T 2
1ω1` T 2
2 pω2` γqq ` ω0
^ pT 21ω
1` T 2
2ω2q
` γ ^ pC2ω0`R2
1ω1`R2
2ω2q `
1
2T 2ijω
i^ ωj, (5.18)
Ω3“ T 3
0 ω0^ ω0
` ω0^ pT 3
3ω3` T 3
4 pω4` γqq ` ω0
^ pT 33ω
3` T 3
4ω4q
` γ ^ pR33ω
3`R3
4ω4q `
1
2T 3ijω
i^ ωj, (5.19)
Ω4“ T 4
0 ω0^ ω0
` ω0^ pD1ω
1`D2ω
2` T 4
3ω3` T 4
4 pω4` γqq ` ω0
^ pT 43ω
3` T 4
4ω4q
` γ ^ pD4ω0`R4
3ω3`R4
4ω4q `
1
2T 4ijω
i^ ωj, (5.20)
78
where T kij “ ´Tkji, and T 1
23, T124, T
223, T
224, T
314, T
324, T
414, T
424 are zero.
The coefficients in Ω1, ...,Ω4 are not all determined. In fact, by adding appropriate
linear combinations of ω0, ω0, γ, ω1, ..., ω4 to β1, we can arrange that
T 212 “ T 2
13 “ T 214 “ T 2
1 “ T 21 “ R2
1 “ 0;
in a similar manner, by adjusting β2, we can arrange that
T 43 “ T 4
3 “ R43 “ T 4
13 “ T 423 “ T 4
34 “ 0.
As a standard step in the method of equivalence, we apply d2 “ 0 to (5.15) and
reduce modulo appropriate differential forms. From this, we obtain relations among
the torsion functions (i.e., the coefficients of Ω0, Ω0,Γ, ...,Ω4).
In particular, expanding the expressions
dpdω0q mod ω0, ω1, dpdω0
q mod ω0, ω2,
dpdω0q mod ω0, ω3, dpdω0
q mod ω0, ω4,
and
dpdγq mod γ, ω0, ω0,
we find the following relations:
T 323 “ P2, T 1
34 “ 0, T 334 “ ´P4 `K ´R3
3 ´R44, D4 “ ´P0 ` T
33 ` T
44 ,
T 313 “ P1, T 2
34 “ 0, T 114 “ ´Q4, T 3
12 “ 0,
T 112 “ ´Q2 ` L´R
11 ´R
22, C2 “ ´Q0 ` T
11 ` T
22 , T 1
13 “ ´Q3, T 412 “ 0,
D2 “ ´1
AC2, C4 “ ´
1
AD4, D1 “ ´
1
AC1, C3 “ ´
1
AD3,
In the equations above, we replace the functions on the left-hand-side by expres-
sions on the right-hand-side. Then we compute
dpdω0q mod ω0, dpdω0
q mod ω0.
79
From this, we find that the expression of dA is determined:
dA
A“ pQ0 ´ T
33 ´ T
44 qω
0` pP0 ´ T
11 ´ T
22 qω
0´ pP1 ´Q1qω
1´ pP2 ´Q2qω
2
` pP3 ´Q3qω3` pP4 ´Q4qω
4` pL´R3
3 ´R44qγ,
with the extra condition: K ´R11´R
22 “ L´R3
3´R44. This shows that the function
A pA ą 1q is an invariant of the G1-structure.
Homogeneity Assumption: Now we assume that the underlying rank-2 Backlund
transformation is homogeneous, that is, its symmetry group acts locally transitively.
Making this assumption will imply that any local structure invariant is a constant.
Following from the homogeneity assumption, dA “ 0. This implies
P3 “ Q3, P4 “ Q4, Q1 “ P1, Q2 “ P2,
K “ R11 `R
22, P0 “ T 1
1 ` T22 , L “ R3
3 `R44, Q0 “ T 3
3 ` T44 .
Further differentiation of the structure equations yields
dpdω1q ” T 1
2ω3^ ω4
^ γ mod ω0, ω0, ω1, ω2,
dpdω1q ” pAT 1
2 ` T12 qω
3^ ω4
^ ω2 mod ω0, ω0, ω1, γ
dpdω2q ”
1
ArAp´T 3
3 ´ T44 ` T
11 q
´ T 33 ´ T
44 ` T
11 ` T
22 sω
3^ γ ^ ω4
` T 20ω
3^ pω4
` γq ^ ω0 mod ω0, ω1, ω2,
and
dpdω3q ” T 3
4ω1^ ω2
^ γ mod ω0, ω0, ω3, ω4,
dpdω3q ” pAT 3
4 ` T34 qω
1^ ω2
^ ω4 mod ω0, ω0, ω3, γ,
dpdω4q ”
1
ArAp´T 1
1 ´ T22 ` T
33 q
´ T 11 ´ T
22 ` T
33 ` T
44 sω
1^ γ ^ ω2
` T 40ω
1^ pω2
` γq ^ ω0 mod ω0, ω3, ω4.
80
These congruences imply
T 12 “ T 1
2 “ T 20 “ T 3
4 “ T 34 “ T 4
0 “ 0,
T 11 “ T 3
3 ` T44 `
1
A2 ´ 1pAT 4
4 ` T22 q, T 3
3 “ T 11 ` T
22 `
1
A2 ´ 1pAT 2
2 ` T44 q.
Using these, we compute
dpdω1q ” T 1
0ω3^ pω4
` γq ^ ω0 mod ω0, ω1, ω2,
dpdω3q ” T 3
0ω1^ pω2
` γq ^ ω0 mod ω0, ω3, ω4,
which implies
T 10 “ T 3
0 “ 0.
In addition, we have
dpdω1q ” pdR1
2 ´ 5R12αq ^ γ ^ ω
2 mod ω0, ω0, ω1, ω3, ω4, (5.21)
dpdω1q ”
R12
A2 ´ 1pAT 4
4 ` T22 qγ ^ ω
0^ ω4
`R1
2D3
Aγ ^ ω0
^ ω3 mod ω1, ω2, (5.22)
dpdω3q ” pdR3
4 ´ 5R34αq ^ γ ^ ω
4 mod ω0, ω0, ω1, ω2, ω3, (5.23)
dpdω3q ”
R34
A2 ´ 1pAT 2
2 ` T44 qγ ^ ω
0^ ω2
`R3
4
AC1γ ^ ω
0^ ω1 mod ω3, ω4. (5.24)
Lemma 5.4. R12 and R3
4 are both zero on an open subset of G1.
Proof. Suppose that locally R12 ‰ 0. The equation (5.22) then implies
D3 “ 0, T 22 “ ´AT
44 .
Following from this, we have
dpdγq ” ´Aω3^ ω4
^ ω0 mod γ, ω1, ω2, ω0,
which is impossible because A ą 1. Therefore, R12 “ 0. An analogous argument
leads to R34 “ 0.
81
Note that there remains freedom to add a multiple of γ to α without changing
the form of the structure equations. Using this, we can arrange
R11 `R
22 “ ´pR
33 `R
44q.
Recall that K “ R11`R
22. We now remind the reader that all torsion coefficients are
expressed in terms of the constant A and the functions
K, P1, P2, Q3, Q4, T 11 , T 2
2 , T 33 , T 4
4 ,
C1, T 22 , T 4
4 , D3; R11, R3
3.
The torsion cannot be absorbed further.
By applying d2 “ 0 to the structure equations, we can find how3 G1 acts on these
remaining torsion functions. For simplicity, we introduce the new notation:
F1 :“A
A2 ´ 1pAT 2
2 ` T44 q F3 :“
A
A2 ´ 1pAT 4
4 ` T22 q.
Infinitesimally, the G1-action on the torsion functions can be expressed as:
dD3 ” ´2αD3 ` F3β2, dF3 ” αF3,
dC1 ” ´2αC1 ` F1β1, dF1 ” αF1,
dP1 ” β1P2 ´ αP1, dP2 ” 2αP2,
dQ3 ” β2Q4 ´ αQ3, dQ4 ” 2αQ4,
dK ” 2αK, dR11 ”2αR1
1, dR33 ” 2αR3
3,
dT 11 ” αT 1
1 , dT 22 ” αT 2
2 , dT 33 ” αT 3
3 , dT 44 ” αT 4
4 ,
where all congruences are modulo the semi-basic 1-forms ω0, ω0, γ, ω1, ..., ω4.
3 What we can obtain by applying d2 “ 0 to the structure equations is essentially an infinitesimalversion of the G1-action on the torsion components. Hence, it only tells us how the identitycomponent of G1 acts. Performing a structure reduction using the action by the identity componentof G1 thus may not distinguish two equivalent coframings that differ by a discrete transformation.However, this would not pose a problem, as we can examine whether there is such an equivalenceat the end of our classification.
82
In the proof of Lemma 5.4, we have seen that, when pD3, F3q or pC1, F1q locally
vanishes, we have a contradiction. In particular, D3 cannot locally vanish, otherwise,
by the equation (5.22), F3 must vanish. For a similar reason, C1 cannot locally vanish.
Hence, for the pair of torsion functions pD3, F3q (and similarly for the pair pC1, F1q),
there are two possibilities:
I. Locally on G1, D3 ‰ 0, F3 “ 0. This implies that G1 scales D3;
II. Locally on G1, F3 ‰ 0. By the G1-action on F3, it is easy to see that
F3 ‰ 0 on the entire group fibers. In this case, it is possible to reduce to
a G12-structure G 12 Ă G1, defined by D3 “ 0. Restricting to G 12, the 1-form
β2 becomes semi-basic.
Lemma 5.5. Case I is empty.
Proof. If F3 “ 0, then D3 scales under the G1-action. We can then reduce to the
subbundle defined by either D3 “ 1 or D3 “ ´1. Now let G2 Ă G1 be defined by
D3 “ 1.
On G2, α is semi-basic. In other words, there exist functions H0, H0, Hγ, H1, ..., H4
on G2 such that
α “ H0ω0` H0ω
0`Hγγ `H1ω
1` ...`H4ω
4.
In addition, the following functions are constant along each fiber of G2
K, P2, Q4, R11, R3
3, T 11 , T 2
2 , T 33 , T 4
4 , F1.
By the homogeneity assumption, they must be constants on G2.
We now compute
dpdγq ”1
Ap2AH2 ` 2AP2 ´ F1H3 ` F1Q3qω
2^ ω3
^ ω0
` p1´ 2Q4 ` 2K ´ 2H4qω3^ ω4
^ ω0
`1
AF1pH4 ´Q4qω
4^ ω2
^ ω0 mod ω0, γ, ω1.
83
If F1 ‰ 0, we must have Q4 “ H4 and 2K “ 4H4 ´ 1. Using this, we compute
dpdγq ” ´1
ApA2
´ 1qω0^ ω3
^ ω4 mod ω0, γ, ω1, ω2,
which is impossible since A ą 1.
It follows that, F1 “ 0 on G2. With this, we compute
dpdγq ” 2pP2ω2`Q4ω
4q ^ ω0
^ ω0` ¨ ¨ ¨ mod γ, ω1.
Therefore, P2 “ Q4 “ 0. Using this, we have
dpdγq ” p1`2K´2H4qω3^ω4
^ω0´
1
ApA2`2K´2H4qω
0^ω3
^ω4 mod γ, ω1, ω0^ω0.
This implies A2 “ 1, which is impossible, by assumption.
By a similar argument, one can show that there is no homogeneous structure in
the case when C1 ‰ 0 and F1 “ 0. This, together with Lemma 5.5, implies that
the only case remaining is when both F3 and F1 are locally nonzero. In this case,
one can reduce to the subbundle on which D3 “ C1 “ 0. There, β1 and β2 both
become semi-basic. Further, we can reduce to an e-structure on which F3 “ 1. On
this e-structure, F1 is a nonzero constant, by homogeneity.
Lemma 5.6. The case when both F3 and F1 are locally nonzero is empty.
Proof. On the e-structure defined by D3 “ C1 “ 0 and F3 “ 1, the 1-forms α, β1
and β2 are semi-basic:
α “ H0ω0` H0ω
0`Hγγ `H1ω
1` ...`H4ω
4,
β1 “M0ω0` M0ω
0`Mγγ `M1ω
1` ...`M4ω
4,
β2 “ N0ω0` N0ω
0`Nγγ `N1ω
1` ...`N4ω
4.
Differentiation gives
dpdγq ” ´1
ApA2
´H3 `N4 `Q3qω3^ ω4
^ ω0` p1´H3 `N4 `Q3qω
3^ ω4
^ ω0
mod γ, ω1, ω2, ω0^ ω0.
84
Clearly, A2 “ 1, which is a contradiction.
Combining Lemmas 5.5 and 5.6, the following theorem is immediate.
Theorem 5.7. There exist no homogeneous rank-2 Backlund transformation satis-
fying all three genericity conditions, ε “ 1, pB1, B2q ‰ 0, and pB3, B4q ‰ 0.
Remark 10. pAq In Theorem 5.7, the condition ε “ 1 can be removed. The case
when ε “ ´1 has a proof that is only a slight modification of the arguments above.
pBq It still remains to be answered whether there exists a (non-homogeneous)
rank-2 Backlund transformation satisfying all three genericity conditions and pB1, B2q ‰
0, pB3, B4q ‰ 0.
5.1.2 Case 2: Bi “ 0 pi “ 1, ..., 4q
In this case, we prove the
Proposition 5.2. Suppose that pN,Bq is a rank-2 Backlund transformation (not
necessarily homogeneous) satisfying all three genericity conditions, ε “ 1 and Bi “ 0
pi “ 1, ..., 4q, then pN,Bq admits a 6-dimensional quotient that is a rank-1 Backlund
transformation relating the same pair of hyperbolic Monge-Ampere systems.
Proof. Using previous notation, the vector fields Xγ, Yγ coincide when Bi “ 0
pi “ 1, ..., 4q. Shrink N if needed, and let N 1 be the quotient of N by the flow
of Xγ. The Invariance Property ((5.13) and (5.14)) then implies that the vector
bundles rrω0, ω1, ω2ss, rrω0, ω3, ω4ss, rrω0, ω1, ω2ss and rrω0, ω3, ω4ss annihilate Xγ and
are invariant under the flow of Xγ. Hence, their intersections are the pull-backs of
vector subbundles of T ˚N 1. In particular, there locally exist 1-forms η0, η0, η1, ..., η4
on N 1 such that the following relations hold (dropping the pull-back symbol): rrω0ss “
rrη0ss, rrω0ss “ rrη0ss, rrω1, ω2ss “ rrη1, η2ss, rrω3, ω4ss “ rrη3, η4ss. This implies that pull-
back of the corresponding Monge-Ampere systems are respectively xη0, η1^η2, η3^η4y
85
and xη0, η1^η2, η3^η4y. It follows that pN 1,B1q, with B1 “ xη0, η0, η1^η2, η3^η4y and
the obvious submersions to M, M , defines a rank-1 Backlund transformation.
Remark 11. One can prove that the conclusion in Proposition 5.2 remains true when
ε “ ´1.
5.2 Assuming Genericity Conditions 1, 2
Without assuming the third genericity condition, we need a new definition of 0-
adapted coframings.
Definition 5.3. Let pN,B; π, πq be a rank-2 Backlund transformation (relating two
hyperbolic Monge-Ampere systems) satisfying only the first two genericity conditions
and ε “ 1. A coframing pω0, ω0, γ, ω1, ..., ω4q defined on an open subset U Ă N is
said to be 0-adapted if it satisfies
rrω0ss “ rrπ˚θss, rrω0
ss “ rrπ˚θss, rrω0, ω0, γss “ B1
and
dω0” Aω1
^ ω2` ω3
^ ω4` pB3ω
3`B4ω
4q ^ γ mod ω0, (5.25)
dω0” ω1
^ ω2` Aω3
^ ω4` pB1ω
1`B2ω
2q ^ γ mod ω0, (5.26)
dγ ” Ciωi^ ω0
`Diωi^ ω0 mod γ, (5.27)
with A ą 1.
Lemma 5.8. Given a rank-2 Backlund transformation pN,Bq (relating two hyper-
bolic Monge-Ampere systems) satisfying only the first two genericity conditions and
ε “ 1, its 0-adapted coframings are local sections of a G-structure G over N , where
G Ă GLp7,Rq is generated by
g “
¨
˚
˚
˚
˚
˝
detpbq 0 0 0 00 detpaq 0 0 00 0 c 0 00 0 0 a 00 0 0 0 b
˛
‹
‹
‹
‹
‚
,detpaq “ detpbq ‰ 0, c ‰ 0
a “ paijq, b “ pbijq P GLp2,Rq.(5.28)
86
Proof. We omit this proof since the arguments are similar to Lemma 5.1.
It is easy to show that, on G,
ˆ
B1
B2
˙
pu ¨ gq “c
detpaqaT
ˆ
B1
B2
˙
puq,
ˆ
B3
B4
˙
pu ¨ gq “c
detpbqbT
ˆ
B3
B4
˙
puq.
It follows that one can normalize Bi pi “ 1, ..., 4q to be one of the following:
Case 1: B1 “ B3 “ 1, B2 “ B4 “ 0;
Case 2: Bi “ 0 pi “ 1, ..., 4q;
Case 3: B2, B3, B4 “ 0, B1 “ 1;
Case 31: B1, B2, B4 “ 0, B3 “ 1.
Lemma 5.9. A rank-2 Backlund transformation in the current case arises as a
1-parameter family of rank-1 Backlund transformations relating the same pair of
hyperbolic Monge-Ampere systems if and only if γ is integrable.
Proof. Suppose that γ is integrable. It is easy to see that each leaf of γ is a
rank-1 Backlund transformation. Conversely, suppose that pN,Bq is constructed in
the natural way from a 1-parameter family of rank-1 Backlund transformations. Let
t be the parameter. Then we have that B1 is generated by the pull-back of θ, θ and
dt. It follows that, given a local 0-adapted coframing on N , there exists a linear
combination ψ “ γ ` λω0 ` µω0 that is integrable. Computing dψ and reducing
modulo ω0, ω0 and γ, it is easy to see, by (5.25)-(5.27), that λ and µ are zero. It
follows that γ must be integrable.
We will only be interested in the case when γ is not integrable. In the current
case, this is given by the condition: locally the functions Ci, Di pi “ 1, ..., 4q are not
all zero.
87
5.2.1 Case: pB1, B2q “ pB3, B4q “ p1, 0q
In this case, we reduce to a G1-structure G1 Ă G, where the subgroup G1 Ă G is
defined by (maintaining the notation in (5.28))
a22 “ b22 “ c, a12 “ b12 “ 0.
The structure equations on G1 can be written as
dω “ ´Φ^ ω `T, (5.29)
where
ω “ pω0, ω0, γ, ω1, ..., ω4qT , T “ pΩ0, Ω0,Γ,Ω1, ...,Ω4
qT ,
Φ “
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
α ` φ 0 0 0 0 0 00 α ` φ 0 0 0 0 00 0 φ 0 0 0 00 0 0 α 0 0 00 0 0 β1 φ 0 00 0 0 0 0 α 00 0 0 0 0 β2 φ
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
.
Here, by the reproducing property of the tautological 1-form ω and by adding an
appropriate multiple of γ to φ, we can arrange
Ω0“ Aω1
^ ω2` ω3
^ ω4` ω3
^ γ ` pP0ω0`Kγ ` Piω
iq ^ ω0,
Ω0“ ω1
^ ω2` Aω3
^ ω4` ω1
^ γ ` pQ0ω0` Lγ `Qiω
iq ^ ω0,
Γ “ Ciωi^ ω0
`Diωi^ ω0.
It is not hard to see that Lemma 5.2 and Lemma 5.3 apply to the current case.
Following from this, Ω1, ...,Ω4 have expressions (5.17)-(5.20) where T kij “ ´Tkji, and
T 123, T
124, T
223, T
224, T
314, T
324, T
414, T
424 are all zero.
In (5.29), the torsion T can be further absorbed by adjusting Φ. In fact, by
adding appropriate semi-basic 1-forms to α, we can arrange
T 11 “ ´T
33 , T 1
1 “ ´T33 , T 1
12 “ T 323, T 1
14 “ ´T334, R1
1 “ ´R33, T 1
13 “ T 313 “ 0.
88
There remains freedom of adding a multiple of γ to φ without changing the form of
the structure equations. Using this, we can arrange
K “ ´L.
By adding appropriate semi-basic 1-forms to β1 and β2, we can arrange
T 21 “ T 2
1 “ R21 “ T 2
12 “ T 213 “ T 2
14 “ T 43 “ T 4
3 “ R43 “ T 4
13 “ T 423 “ T 4
34 “ 0.
By expanding the expressions
dpdω0q mod ω0, ω1, γ, dpdω0
q mod ω0, ω2, γ,
dpdω0q mod ω0, ω3, γ, dpdω0
q mod ω0, ω4, γ
and
dpdγq mod γ, ω0, ω0,
we find the following relations
P2 “ AT 134 ` T
323, P0 “ T 3
3 ` T44 ´D4, P1 “ AT 2
34,
Q4 “ AT 312 ` T
334, Q0 “ T 2
2 ´ T33 ´ C2, Q3 “ ´AT
412,
D2 “ ´1
AC2, C3 “´
1
AD3, C4 “ ´
1
AD4, D1 “ ´
1
AC1.
Further calculation yields
dpdω0q ” pL` T 3
34 `R33 `R
44 ` P4qγ ^ ω
3^ ω4
` pAL` AR22 ´ AR
33 ` T
312qω
1^ ω2
^ γ
` p´AP4 ` AT334 ` T
312qω
1^ ω2
^ ω4´ AT 2
34ω1^ ω3
^ γ ´ 2AT 234ω
1^ ω3
^ ω4
´ AT 134ω
2^ ω3
^ γ ` p´AP3 ´ T412qω
3^ ω1
^ ω2
` ApD4 ` T22 ´ 2T 3
3 ´ T44 qω
0^ ω1
^ ω2` dA^ ω1
^ ω2 mod ω0,
dpdω0q ” pL´ T 3
23 `R33 ´R
22 ´Q2qω
2^ ω1
^ γ ` p´AL` AR33 ` AR
44 ` T
134qω
3^ ω4
^ γ
` p´AQ2 ` AT323 ` T
134qω
2^ ω3
^ ω4` p´AQ1 ´ T
234qω
1^ ω3
^ ω4
` AT 412ω
3^ ω1
^ γ ´ AT 312ω
4^ ω1
^ γ
` ApC2 ´ T22 ` 2T 3
3 ` T44 qω
0^ ω3
^ ω4` dA^ ω3
^ ω4 mod ω0.
89
This completely determines dA as a linear combination of the semi-basic 1-forms.
Hence, A is a structure invariant.
Homogeneity Assumption: Now we assume that the underlying rank-2 Backlund
transformation is homogeneous.
By this assumption, dA must be zero. The previous two congruences imply
P3 “ 0, Q1 “ 0; P4 “ ´L, Q2 “ L; R22 “ ´R
44, R3
3 “ L´R44;
T 22 “ C2 ` 2T 3
3 ` T44 , T 2
2 “ ´D4 ` 2T 33 ` T
44 ;
T 134 “ 0, T 2
34 “ 0, T 312 “ 0, T 3
23 “ L, T 334 “ ´L, T 4
12 “ 0.
Furthermore, we compute
dpdω1q ” T 1
2ω3^ ω4
^ γ ` T 10ω
3^ pω4
` γq ^ ω0 mod ω0, ω1, ω2,
dpdω2q ” T 2
0ω3^ pω4
` γq ^ ω0`
1
Ap2AT 3
3 ` AT44 `D4qω
3^ ω4
^ γ mod ω0, ω1, ω2,
dpdω3q ” T 3
4ω1^ ω2
^ γ ` T 30ω
1^ pω2
` γq ^ ω0 mod ω0, ω3, ω4,
dpdω4q ” T 4
0ω1^ pω2
` γq ^ ω0`
1
ApAD4 ´ AT
44 ´ C2qω
1^ γ ^ ω2 mod ω0, ω3, ω4.
This implies that
T 10 “ T 2
0 “ T 30 “ T 4
0 “ T 12 “ T 3
4 “ 0,
T 44 “ ´
1
Ap2AT 3
3 `D4q, T 44 “
1
ApAD4 ´ C2q.
Now, all torsion coefficients are expressed in terms of the constant A and the functions
L, T 33 , C1, C2, T 3
3 , D3, D4, T 12 , T 3
4 , R12, R3
4, R44.
Lemma 5.10. Assume homogeneity. On G1, wherever pD3, D4q ‰ 0 (resp., pC1, C2q ‰
0), one must have R12 “ T 1
2 “ 0 (resp., T 34 “ R3
4 “ 0).
90
Proof. By the structure equations,
Apdpdω1qq ” R1
2γ ^ ω0^ pD3ω
3`D4ω
4q
` T 12 ω
0^ ω0
^ pD3ω3`D4pω
4` γqq mod ω1, ω2,
Apdpdω3qq ” R3
4γ ^ ω0^ pC1ω
1` C2ω
2q
` T 34ω
0^ ω0
^ pC1ω1` C2pω
2` γqq mod ω3, ω4.
The conclusion is immediate.
Now we proceed to find how (infinitesimally) G1 acts on the torsion functions.
First, expanding dpdγq “ 0, then reducing modulo tω0, γ, ω1, ω2, ω3u and tω0, γ, ω1,
ω2, ω4u, respectively, we obtain
dD4 ” pα ` φqD4 mod ω0, ω0, γ, ω1, ..., ω4, (5.30)
dD3 ” 2αD3 ` β2D4 mod ω0, ω0, γ, ω1, ..., ω4. (5.31)
Expanding dpdγq “ 0, then reducing modulo tω0, γ, ω1, ω3, ω4u and tω0, γ, ω2, ω3, ω4u,
respectively, we obtain
dC2 ” pα ` φqC2 mod ω0, ω0, γ, ω1, ..., ω4, (5.32)
dC1 ” 2αC1 ` β1C2 mod ω0, ω0, γ, ω1, ..., ω4. (5.33)
In particular, (5.30)-(5.33) imply that, if pD3, D4q (resp., pC1, C2q) is nonzero, then
it is nonzero along the entire group fibers.
Expanding dpdω1q “ 0, then reducing modulo tω0, ω0, ω1, ω3, ω4u, we obtain
dR12 ” p2φ´ αqR
12 mod ω0, ω0, γ, ω1, ..., ω4; (5.34)
expanding dpdω1q “ 0, then reducing modulo tω0, ω1, ω3, ω4, γu, we obtain
dT 12 ” 2φT 1
2 mod ω0, ω0, γ, ω1, ..., ω4. (5.35)
Expanding dpdω3q “ 0, then reducing modulo tω0, ω0, ω1, ω2, ω3u, we obtain
dR34 ” p2φ´ αqR
34 mod ω0, ω0, γ, ω1, ..., ω4; (5.36)
91
expanding dpdω3q “ 0, then reducing modulo tω0, ω1, ω2, ω3, γu, we obtain
dT 34 ” 2φT 3
4 mod ω0, ω0, γ, ω1, ..., ω4. (5.37)
We also have the following four congruences:
dpdω0q ” ´dpα ` φq ^ ω0
` Lφ^ γ ^ ω0´ dL^ γ ^ ω0 mod ω0, ω1, ..., ω4,
dpdω0q ” ´dpα ` φq ^ ω0
´ Lφ^ γ ^ ω0` dL^ γ ^ ω0 mod ω0, ω1, ..., ω4,
dpdω1q ” ´dL^ γ ^ ω1
` dR44 ^ γ ^ ω
1´ dα ^ ω1
` pL´R44qφ^ γ ^ ω
1`R1
2γ ^ β1 ^ ω1 mod ω0, ω0, ω2, ω3, ω4,
dpdω3q ” dL^ γ ^ ω3
´ dR44 ^ γ ^ ω
3´ dα ^ ω3
` pR44 ´ Lqφ^ γ ^ ω
3`R3
4γ ^ β2 ^ ω3 mod ω0, ω0, ω1, ω2, ω4.
As a consequence,
dL ” φL mod ω0, ω0, γ, ω1, ..., ω4, (5.38)
and
dR44 ” φR4
4 `1
2pR1
2β1 ´R34β2q mod ω0, ω0, γ, ω1, ..., ω4. (5.39)
Finally, by computing
dpdω1q ” ´dα ^ ω1
` pT 33 pα ` φq ´ dT 3
3 q ^ ω0^ ω1 mod ω0, γ, ω2, ω3, ω4,
dpdω3q ” ´dα ^ ω3
` p´T 33 pα ` φq ` dT 3
3 ´ T34 β2q ^ ω
0^ ω3 mod ω0, γ, ω1, ω2, ω4,
dpdω1q ” ´dα ^ ω1
` pT 33 pφ` αq ´ dT 3
3 ´ T12 β1q ^ ω
0^ ω1 mod ω0, γ, ω2, ω3, ω4,
dpdω3q ” ´dα ^ ω3
` p´T 33 pα ` φq ` dT 3
3 q ^ ω0^ ω3 mod ω0, γ, ω1, ω2, ω4,
we obtain
dT 33 ” pα ` φqT
33 `
1
2β2T
34 mod ω0, ω0, γ, ω1, ..., ω4, (5.40)
and
dT 33 ” pα ` φqT
33 ´
1
2β1T
12 mod ω0, ω0, γ, ω1, ..., ω4. (5.41)
92
Now we proceed to cases.
Case of either D4 ‰ 0 or C2 ‰ 0.
Lemma 5.11. Assuming homogeneity, the case when D4 ‰ 0 is empty.
Proof. If D4 is nonzero, then, by Lemma 5.10, R12 and T 1
2 are zero. By (5.30) and
(5.31), one can always reduce to the subbundle G2 Ă G1 defined by
D4 “ 1, D3 “ 0.
By (5.40), (5.41) and (5.32), the functions T 33 , T
33 and C2 are constants along the
fibers of G2 Ñ N ; hence, they are constants, by the homogeneity assumption.
On G2, the 1-forms φ` α and β2 are semi-basic; thus, we can write
α “ ´φ`H0ω0` H0ω
0`Hγγ `
4ÿ
i“1
Hiωi, β2 “ N0ω
0` N0ω
0`Nγγ `
4ÿ
i“1
Niωi,
where the new coefficient functions are defined on G2. By applying d2 “ 0 to the
structure equations, we find
dH0 ” 0,
dH0 ” 0,
dHγ ” Hγφ,
dH1 ” ´H1φ` pH2 ´ Lqβ1,
dH2 ” H2φ,
dH3 ” ´H3φ,
dH4 ” H4φ;
dN0 ” ´2N0φ,
dN0 ” ´2N0φ,
dNγ ” ´Nγφ,
dN1 ” ´3N1φ`N2β1,
dN2 ” ´N2φ,
dN4 ” ´N4φ,
where all congruences are modulo the semi-basic 1-forms. (Note that N3 never ap-
pears in the structure equations.) By homogeneity, H0, H0 are constants.
Now, Hγ, H2, H3, H4, N0, N0, Nγ, N2, N4, L, C1, T34 , R
34, R
44 are relative invariants4
of G2 with nonzero weights in φ only. If they all vanish, then it can be verified that
4 Let G Ă GLpn,Rq be a Lie subgroup, and let G be a G-structure over M . A G-equivariantfunction f : G Ñ R, where G acts on R linearly, is called a relative invariant of G.
93
the structure equations are incompatible with the identity d2 “ 0. On the other
hand, instead of asking which of these relative invariants are nonzero and going into
various cases, we can always choose a nonzero function U on G2, expressed in terms
of these relative invariants, satisfying
dU ” Uφ mod ω0, ω0, γ, ω1, ..., ω4.
Then there exist constants hγ, h2, h3, h4, n0, n0, nγ, n2, n4 such that
Hγ “ hγU,
H2 “ h2U,
H3 “ h3U´1,
H4 “ h4U,
N0 “ n0U´2,
N0 “ n0U´2,
Nγ “ nγU´1,
N2 “ n2U´1,
N4 “ n4U´1,
and constants `, c1, t34, r
34, r
44 such that
L “ `U, C1 “ c1U´2, T 3
4 “ t34U2, R3
4 “ r34U
3, R44 “ r4
4U.
Since U is expressed in terms of the relative invariants and has weight 1 in φ, we can
perform a structure reduction that leads to a subbundle G3 defined by U “ 1.
On G3, φ is semi-basic, namely,
φ “ Z0ω0` Z0ω
0` Zγγ `
4ÿ
i“1
Ziωi,
for Z0, Z0, ..., Z4 defined on G3. We find that
dZ0 ” 0, dZ0 ” 0, dZγ ” 0,
dZ1 ” Z2β1, dZ2 ” 0, dZ3 ” 0, dZ4 ” 0,
all congruences being modulo the semi-basic 1-forms. By homogeneity, Z0, Z0, Zγ,
Z2, Z3, Z4 are constants. Taking this into account, it can be verified that the structure
equations are incompatible with d2 “ 0.
94
Consequently, D4 must be zero on G1. Moreover, note that the equations (5.25)-
(5.27) allows us to switch pω0, ω1, ω2q with pω0, ω3, ω4q; applying this, pC1, C2, C3, C4q
exchanges with pD3, D4, D1, D2q; and pB1, B2q exchanges with pB3, B4q. Since we are
in a case when pB1, B2q “ pB3, B4q, we can conclude from Lemma 5.11 that
Lemma 5.12. Assuming homogeneity, C2 must be zero on G1.
Therefore, the functions D3 and C1 are relative invariants of G1, both having
weight 2 in α.
Case of D3 ‰ 0.
Without loss of generality, we can assume that D3 ‰ 0 on G1. By Lemma 5.10,
R12, T
12 must be zero.
Depending on the sign of D3, we can reduce to the subbundle G2 defined by either
D3 “ 1 or D3 “ ´1. On G2, there exist functions H0, H0, Hγ, H1, ..., H4 such that
α “ H0ω0` H0ω
0`Hγγ `
4ÿ
i“1
Hiωi, (5.42)
Moreover, the torsion functions on G1 restrict to G2 to satisfy
dL ” φL, dT 33 ” T 3
3 φ, dT 33 ” T 3
3 φ`1
2T 3
4 β2,
dC1 ” 0, C2 “ 0, D3 “ ˘1, D4 “ 0,
T 12 “ 0, dT 3
4 ” 2T 34 φ, R1
2 “ 0, dR34 ” 2R3
4φ, dR44 ” R4
4φ´1
2R3
4β2,
(5.43)
where all congruences are modulo the semi-basic 1-forms. By homogeneity, C1 must
be a constant; L, T 33 , T
34 , R
34 are now relative invariants. We have two cases:
I. L, T 33 , T
34 , R
34 are all zero;
II. not all of L, T 33 , T
34 , R
34 are zero.
95
Now consider the case when G2 is defined by D3 “ 1. A “`” sign will be used to
indicate that we are in this case.
`I. If L, T 33 , T
34 , R
34 are identically zero on G2, then, by (5.43), T 3
3 , R44 are relative
invariants. It is easy to verify that
dH0 ” H0φ,
dH0 ” H0φ,
dHγ ” Hγφ,
dH1 ” H2β1,
dH2 ” H2φ,
dH3 ” H4β2,
dH4 ” H4φ,
(5.44)
modulo the semi-basic 1-forms. We can always choose U to be a function defined on
G2, satisfying
dU ” Uφ, mod ω0, ω0, γ, ω1, ..., ω4,
in the following manner: noting that T 33 , R
44, H0, Hγ, H0, H2, H4 are relative invariants
with nonzero weights in φ only, if they are all zero, we simply choose U “ 0; otherwise,
we choose U be an appropriate combination of these relative invariants satisfying the
equation above and the condition U ‰ 0. There are thus two subcases to consider:
`I1. T 33 , R
44, H0, Hγ, H0, H2, H4 are all zero;
`I2. not all of T 33 , R
44, H0, Hγ, H0, H2, H4 are zero.
`I1. In this case, by (5.44) and the homogeneity assumption, H1 and H3 are con-
stants on G2. We compute
dpdγq “ ´dφ^ γ ` p1´ C1qω1^ ω3
^ γ
´2
ApAC1H3 `H1qω
3^ ω1
^ ω0`
2
ApAH1 ` C1H3qω
3^ ω1
^ ω0.
Since A ą 1, we must have
H1 “ C1H3 “ 0; (5.45)
in particular, either C1 or H3 is zero.
96
Lemma 5.13. C1 is nonzero.
Proof. If C1 “ 0 on G2, then
dpdω0q “ ´dφ^ ω0
` ω3^ ω1
^ ω0, (5.46)
dpdω0q “ ´dφ^ ω0
` ω3^ ω1
^ ω0, (5.47)
dpdγq “ ´dφ^ γ ` ω1^ ω3
^ γ. (5.48)
The equation (5.46) and (5.47) imply that, on G2, there exists a function K such
that
dφ “ ω3^ ω1
`Kω0^ ω0,
which, however, is incompatible with (5.48). The conclusion follows.
By Lemma 5.13 and (5.45), we must have H3 “ 0 on G2. Applying d2 “ 0 to the
structure equations yields
dpdω0q “ ´dφ^ ω0
` p1´ C1qω3^ ω1
^ ω0, (5.49)
dpdω0q “ ´dφ^ ω0
` p1´ C1qω3^ ω1
^ ω0, (5.50)
dpdγq “ ´dφ^ γ ` p1´ C1qω1^ ω3
^ γ. (5.51)
Among these three equations, (5.49) and (5.50) imply that there exists a function K
on G2 satisfying
dφ “ p1´ C1qω3^ ω1
`Kω0^ ω0;
in order for (5.51) to hold, we must have
C1 “ 1, K “ 0.
As a result, φ is (determined and) integrable. Restricting to a leaf of xφy, we have
φ “ 0. Such a leaf then can be regarded as a bundle (over an open subset U Ă N)
97
on which the following structure equations are satisfied (A ą 1 being a constant)
dω0 “ Aω1 ^ ω2 ` ω3 ^ pω4 ` γq,
dω0 “ ω1 ^ pω2 ` γq ` Aω3 ^ ω4,
dγ “ ω1 ^
ˆ
ω0 ´1
Aω0
˙
` ω3 ^
ˆ
ω0 ´1
Aω0
˙
,
dω1 “ 0,
dω2 “ ´β1 ^ ω1 ´
1
Aω0 ^ ω3,
dω3 “ 0,
dω4 “ ´β2 ^ ω3 ´
1
Aω0 ^ ω1.
(5.52)
Remark 12. In terms of the method of equivalence, one can check that the structure
equation (5.52) has constant torsion and involutive tableau, the Cartan characters of
the tableau being p2, 0, ..., 0q. By a theorem of Cartan, any two Backlund transfor-
mations in this case are equivalent; the symmetry of such a Backlund transformation
depends on 2 functions of 1 variable.
We have thus proven:
Proposition 5.3. Up to equivalence, there is a unique local model for a rank-2
homogeneous Backlund transformation in Case p`I1q.
Remark 13. We will see later that the Backlund transformation corresponding to
(5.52) is an auto-Backlund transformation of the linear equation zxy “ z.
`I2. In this case, by the construction of U , there exist constants t33, r44, h0, h0, hγ, h2, h4,
such that
T 33 “ t33U, R4
4 “ r44U,
H0 “ h0U, H0 “ h0U, Hγ “ hγU, H2 “ h2U, H4 “ h4U.
Moreover, it is easy to see that one can reduce to the subbundle G3 defined by U “ 1.
98
On G3, there exist functions Z0, Z0, Zγ, Z1, ..., Z4 such that
φ “ Z0ω0` Z0ω
0` Zγγ `
4ÿ
i“1
Ziωi,
By applying d2 “ 0 to the structure equations, one can verify that
dZ0 ” 0,
dZ0 ” 0,
dZγ ” 0,
dZ1 ” Z2β1,
dZ2 ” 0,
dZ3 ” Z4β2,
dZ4 ” 0,
(5.53)
modulo semi-basic 1-forms. Therefore, by homogeneity, Z0, Z0, Zγ, Z2, Z4 are con-
stants. Using the structure equations, we find
dpdγq ” ´2h2ω2^ ω3
^ ω0´ 2h4ω
4^ ω3
^ ω0 mod ω0, γ, ω1.
Evidently, h2 and h4 must be zero. By (5.44) and the homogeneity assumption, H1
and H3 are constants.
By expanding
dpdω0q, dpdω0
q, dpdγq, dpdω2q, dpdω4
q all mod ω1, ω3,
dpdω1q mod ω3, dpdω3
q mod ω1
we find that either Z2, Z4 are both zero, or h0, h0, hγ, r44, t
33 are all zero.
However, noting that h2, h4 are already zero, we cannot have h0, h0, hγ, r44, t
33 to
be all zero in the current case (by the assumption for Case (`I2)).
Now assume that Z2, Z4 are both zero. In this case, Z1, Z3 become invari-
ants, by (5.53), and are thus constants by the homogeneity assumption. Expanding
dpdω0q, dpdω0q, dpdγq, dpdω1q, dpdω3q and
dpdω2q mod ω1, dpdω4
q mod ω3
99
we find
h0 “ h0 “ hγ “ Z0 “ Z0 “ Zγ “ H1 “ H3 “ r44 “ t33 “ 0, C1 “ 1.
Together with the condition h2 “ h4 “ 0, this brings us back to Case (`I1).
`II. In this case, one can always choose a function U , defined on G2 and expressed
in terms of L, T 33 , T
34 and R3
4, that satisfies
dU ” Uφ mod ω0, ω0, γ, ω1, ..., ω4.
Following from this and (5.43), there exist constants `, t33, t34, r
34, not all zero, such
that
L “ `U, T 33 “ t33U, T 3
4 “ t34U2, R3
4 “ r34U
2.
One can reduce to the subbundle G3 defined by U “ 1. On G3, there exist functions
Z0, Z0, Zγ, Z1, ..., Z4 such that
φ “ Z0ω0` Z0ω
0` Zγγ `
4ÿ
i“1
Ziωi.
How the torsion functions vary on the fibers of G3 Ñ N can be found, as usual, by
applying d2 “ 0 to the structure equations. Moreover, by taking into account the
homogeneity assumption, we find
(i) H0, H2, H4, Z0, Z0, Z2, Z4 are constants;
(ii) there exist constants h0, zγ, hγ such that
H0 “ h0 ´ T33 , Hγ “ hγ `R
44, Zγ “ zγ ´R
44; (5.54)
(iii) H1, H3, Z1, Z3 satisfy, modulo the semi-basic 1-forms,
dH1 ” pH2 ´ `qβ1,
dH3 ” pH4 ` `qβ2,
dZ1 ” β1Z2,
dZ3 ” β2Z4.(5.55)
100
Calculation yields
dpdγq ” t34ω0^ ω4
^ ω0 mod ω1, ω2, ω3, γ,
dpdγq ”r3
4
Aω0^ γ ^ ω4 mod ω1, ω2, ω3, ω0.
Hence, t34 and r34 must be zero. By (5.43) and homogeneity, it follows that T 3
3 and
R44 are constants on G3. Using these and expanding the following expressions
dpdω0q, dpdω0
q, dpdγq, dpdω2q, dpdω4
q all mod ω1, ω3,
dpdγq mod γ, dpdω2q mod ω3, dpdω3
q mod ω1,
and
dpdω2q mod ω1, ω2, dpdω2
q mod ω3, ω4,
we find (with MapleTM):
Z2 “ Z4 “ H1 “ 0.
This, with (5.55), implies that Z1, Z3 are constants and that H2 “ `. Taking these
into account, one can show, by applying the identity d2 “ 0 to the structure equa-
tions, that ` and t33 must be zero. This violates the assumption of the current case.
This completes analysis for the case of D3 ą 0 on G1.
The case when D3 ă 0 is similar to the case when D3 ą 0. The only nonempty
case is (´I1), which leads to C1 “ ´1 and the structure equations
dω0 “ ´φ^ ω0 ` Aω1 ^ ω2 ` ω3 ^ pω4 ` γq,
dω0 “ ´φ^ ω0 ` ω1 ^ pω2 ` γq ` Aω3 ^ ω4,
dγ “ ´φ^ γ ´ ω1 ^
ˆ
ω0 ´1
Aω0
˙
´ ω3 ^
ˆ
ω0 ´1
Aω0
˙
,
dω1 “ 0,
dω2 “ ´β1 ^ ω1 ´ φ^ ω2 `
1
Aω0 ^ ω3,
dω3 “ 0,
dω4 “ ´β2 ^ ω3 ´ φ^ ω4 `
1
Aω0 ^ ω1,
(5.56)
101
where φ is integrable.
Remark 14. One can show that the structure equations (5.56) corresponds to an
auto-Backlund transformation of the partial differential equation zxy “ z.
5.2.2 Integration of the structure equations
Equation (5.52). Replace β1 by ω4 and β2 by ω2 in (5.52). It is easy to check
that d2 “ 0 automatically holds. We may thus regard the result as the structure
equations on some open U Ă N .
Since ω1, ω3 are closed, there locally exist functions x, y such that
ω1“ dx, ω3
“ dy.
Since dω0 and dω0 are both closed and both congruent to zero modulo dx, dy, there
exist functions z1, p1, q1, z2, p2, q2 such that
ω0“ dz1 ´ p1dx´ q1dy,
ω0“ dz2 ´ p2dx´ q2dy.
Then the equation of dω0 implies
`
Aω2´ dp1
˘
^ dx``
ω4` γ ´ dq1
˘
^ dy “ 0. (5.57)
Similarly, the equation of dω0 implies
`
ω2` γ ´ dp2
˘
^ dx``
Aω4´ dq2
˘
^ dy “ 0. (5.58)
Following from (5.57) and (5.58), there exist functions s1, s2, s3, t1, t2, t3 satisfying
$
&
%
ω2 “1
Apdp1 ` s1dx` s2dyq,
ω4 ` γ “ dq1 ` s2dx` s3dy;
(5.59)
and$
&
%
ω2 ` γ “ dp2 ` t1dx` t2dy,
ω4 “1
Apdq2 ` t2dx` t3dyq.
(5.60)
102
Writing γ as pω2`γq´ω2 and pω4`γq´ω4, then using (5.59) and (5.60), we obtainˆ
dp2 ´1
Adp1
˙
`
´
t1 ´s1
A
¯
dx`´
t2 ´s2
A
¯
dy “ (5.61)
ˆ
dq1 ´dq2
A
˙
`
ˆ
s2 ´t2A
˙
dx`
ˆ
s3 ´t3A
˙
dy.
Furthermore, the equation of dω2 leads to
pds1 ` dq2 ` t3dyq ^ dx` pds2 ` dz1 ´ p1dxq ^ dy “ 0; (5.62)
the equation of dω4 leads to
pdt2 ` dz2 ´ q2dyq ^ dx` pdt3 ` dp1 ` s1dxq ^ dy “ 0. (5.63)
Applying d2 “ 0 to (5.62) and (5.63) implies
dpp1 ` t3q ^ dx^ dy “ dpq2 ` s1q ^ dx^ dy “ 0.
As as result, there exist functions fpx, yq, gpx, yq defined on U such that
#
p1 ` t3 “ fpx, yq,
q2 ` s1 “ gpx, yq.
Now (5.62) can be written as
ˆ
dps2 ` z1q ´
ˆ
Bg
By` f
˙
dx
˙
^ dy “ 0,
which implies that, by fixing an x0,
s2 “ ´z1 `
ż x
x0
ˆ
Bg
Bypσ, yq ` fpσ, yq
˙
dσ ` hpyq,
for a function hpyq. With a similar argument, one can obtain, by fixing a y0,
t2 “ ´z2 `
ż y
y0
ˆ
Bf
Bxpx, τq ` gpx, τq
˙
dτ ` hpxq,
103
for a function hpxq. Since we have
ω0“ dz1 ´ p1dx´ q1dy,
ω1^ ω2
“ dx^1
Apdp1 ` s2dyq,
ω3^ pω4
` γq “ dy ^ pdq1 ` s2dxq,
it is clear that x, y, p1, q1, z1 are local coordinates on M and that the PDE (up to
contact equivalence) corresponding to pM, Iq must be of the form
B2u
BxBy“ u´
ż x
x0
ˆ
Bg
Bypσ, yq ` fpσ, yq
˙
dσ ` hpyq,
for some functions fpx, yq, gpx, yq and hpyq. Similarly, the PDE (up to contact equiv-
alence) corresponding to pM, Iq must be of the form
B2v
BxBy“ v ´
ż y
y0
ˆ
Bf
Bxpx, τq ` gpx, τq
˙
dτ ` hpxq,
for the fpx, yq, gpx, yq as above, and some hpyq.
We claim that one can make an appropriate choice of coordinates such that
f, g, h, h are all zero. As a result, each of pM, Iq and pM, Iq corresponds (up to
equivalence) to the following linear PDE:
zxy “ z.
To see why such a choice is possible, consider the following change of variables
z1 ÞÑ z1 ` αpx, yq, z2 ÞÑ z2 ` βpx, yq, (5.64)
where αpx, yq, βpx, yq are analytic functions to be determined. This implies the
corresponding change of the variables (in order to preserve ω0, ω0,...,ω4)
p1 ÞÑ p1 ` αx,q1 ÞÑ q1 ` αy,s1 ÞÑ s1 ´ αxx,s2 ÞÑ s2 ´ αxy,s3 ÞÑ s3 ´ αyy,
p2 ÞÑ p2 ` βx,q2 ÞÑ q2 ` βy,t1 ÞÑ t1 ´ βxx,t2 ÞÑ t2 ´ βxy,t3 ÞÑ t3 ´ βyy.
104
In particular,
p1 ` t3 ÞÑ pαx ´ βyyq ` p1 ` t3, q2 ` s1 ÞÑ pβy ´ αxxq ` q2 ` s1.
Thus, by solving for α, β in the following system
"
βyy ´ αx “ fpx, yq,αxx ´ βy “ gpx, yq,
(5.65)
one can find a set of coordinates under which f, g become zero. The system (5.65)
can be turned into the form
"
βyy ´ αx “ 0,αxx ´ βy “ gpx, yq,
(5.66)
by making the change of variable α ÞÑ α´şx
x0fpτ, yqdτ . The system (5.66) is easily
solved once we have solved the equation
βxyy ´ βy “ gpx, yq,
which is essentially
σxy ´ σ “ gpx, yq, (5.67)
after the substitution σ “ βy. By Cartan-Kahler theory, a regular solution of (5.67)
depends on 2 functions of 1 variable. This justifies the normalization: f “ g “ 0.
In addition, under the transformation (5.64), we have
s2 ` z1 ÞÑ s1 ` z1 ` pα ´ αxyq, t2 ` z2 ÞÑ t2 ` z2 ` pβ ´ βxyq.
It is clear that, since f, g are now zero, in order for h, h to become zero after (5.64)
without affecting f and g, we need α, β to satisfy the PDE system
$
’
’
&
’
’
%
βyy ´ αx “ 0,αxx ´ βy “ 0,α ´ αxy “ hpyq,
β ´ βxy “ hpxq.
105
An obvious solution is α “ hpyq and β “ hpxq. This justifies normalizing f, g, h, h
to be zero.
Now, since t3 “ ´p1, s1 “ ´q2, s2 “ ´z1, t2 “ ´z2, the left-hand-side of (5.61)
becomes
γ “ dp2 ´1
Adp1 `
ˆ
t1 `1
Aq2
˙
dx´
ˆ
z2 ´1
Az1
˙
dy. (5.68)
On the other hand, the equation of dγ implies
dγ “´q2
A´ q1 ` p2 ´
p1
A
¯
dx^ dy
` dx^
ˆ
dz1 ´1
Adz2
˙
` dy ^
ˆ
dz2 ´1
Adz1
˙
. (5.69)
Since dγ is closed, it is evident (noting that A is a constant) that locally there exists
a function kpx, yq such that
q2
A´ q1 ` p2 ´
p1
A“ kpx, yq, (5.70)
which is precisely the extra equation (the other two being x1 “ x2 “ x, y1 “ y2 “ y)
needed to define a 7-submanifold N ĂM ˆ M .
Using (5.69), (5.70) and the exterior derivative of (5.68), we obtain
d
ˆ
t1 `1
Aq2 ` z1 ´
1
Az2
˙
^ dx` kpx, yqdy ^ dx “ 0.
Following from this,
t1 “ ´1
Aq2 ´ z1 `
1
Az2 ´Kpx, yq,
where Kpx, yq is a function satisfying ByKpx, yq “ kpx, yq. The function kpx, yq is
not arbitrary. In fact, expanding (5.61), using the expressions for t1, s1, t2, s2 and t3,
and using (5.70), one can obtain
dpkpx, yqq ´Kpx, yqdx`
ˆ
´z2 `1
Az1 ´ s3 ´
1
Ap1
˙
dy “ 0.
106
Consequently, kx “ K holds; in particular, we have kxy “ k and
s3 “ kypx, yq ` z2 ´1
Az1 `
1
Ap1.
Again, we ask whether the function kpx, yq can be normalized by a change of
local coordinates. It amounts to solving the following linear system of PDEs
$
’
’
’
’
’
’
&
’
’
’
’
’
’
%
βyy ´ αx “ 0,
αxx ´ βy “ 0,
α ´ αxy “ 0,
β ´ βxy “ 0,
A´1pβy ´ Aαy ` Aβx ´ αxq “ kpx, yq.
(5.71)
It is clear that kxy “ k is an integrability condition for (5.71).
Using the Frobenius Theorem, one can show that solutions of (5.71) exist and
depend on 5 constants. Consequently, a Backlund transformation corresponding to
the structure equations (5.52) is contact equivalent to one that relates solutions of
the equation zxy “ z. Such a Backlund transformation is locally determined by the
equation
q2
A´ q1 ` p2 ´
p1
A“ 0, pconst. A ą 1q; (5.72)
and the 1-form
γ “ dp2 ´1
Adp1 `
ˆ
´z1 `1
Az2
˙
dx´
ˆ
z2 ´1
Az1
˙
dy. (5.73)
Equation (5.56). The integration of (5.56) is similar to that of (5.52). Replace β1
by ω4 and β2 by ´ω2. Applying d2 “ 0 to (5.56) yields only identities. We can show
that there exist functions x, y; z1, p1, q1; z2, p2, q2; s1, s2, s3; t1, t2, t3 on U satisfying
(5.59), (5.60) and (5.61). After a contact transformation of the coordinates, without
changing x, y, we obtain
t3 “ p1, s1 “ ´q2, s2 “ z1, t2 “ z2, t1 “ ´q2
A` z1 ´
1
Az2
107
and the equation
q2
A´ q1 ` p2 ´
p1
A“ 0. (5.74)
Now γ can be written as
γ “ dp2 ´1
Adp1 `
ˆ
z1 ´1
Az2
˙
dx`
ˆ
z2 ´1
Az1
˙
dy. (5.75)
The corresponding Backlund transformation, up to contact equivalence, relates so-
lutions of the equation zxy “ ´z. We can perform an additional contact transforma-
tion, sending x to ´x, so that p1 is replaced by ´p1, p2 by ´p2, and so on. Under
the new coordinates, (5.74) becomes
q2
A´ q1 ´ p2 `
p1
A“ 0, (5.76)
and γ becomes
γ “ ´dp2 `1
Adp1 ´
ˆ
z1 ´1
Az2
˙
dx`
ˆ
z2 ´1
Az1
˙
dy. (5.77)
It is easy to see that the corresponding Backlund transformation is an auto-Backlund
transformation of the equation zxy “ z (up to equivalence).
5.3 Assuming Genericity Condition 1
If a rank-2 Backlund transformation satisfies only the first genericity condition, it
then admits a coframing pω0, ω0, γ, ω1, ..., ω4q defined on an open subset U Ă N and
satisfying the following congruences
dω0” ω1
^ ω2` ω3
^ ω4 mod ω0, (5.78)
dω0” ω1
^ ω2` ω3
^ ω4 mod ω0, ω0, γ, (5.79)
dγ ” A1ω1^ ω2
` A2ω3^ ω4 mod ω0, ω0, γ. (5.80)
108
We can further refine such a coframing by performing the following steps successively:
1) add a suitable multiple of ω0 to γ to arrange A2 “ ´A1;
2) add appropriate multiples of ω0 to ω1, .., ω4 such that the congruence (5.79)
still holds when modulo only ω0 and γ;
3) add appropriate multiples of γ to ω3, ω4 such that the following congruences
hold (note that dω0 cannot have a ω0 ^ γ term since ω0 ^ pdω0q3 “ 0.):
dω0” ω1
^ ω2` ω3
^ ω4` pB3ω
3`B4ω
4q ^ γ mod ω0, (5.81)
dω0” ω1
^ ω2` ω3
^ ω4` pB1ω
1`B2ω
2q ^ γ mod ω0, (5.82)
where B1, ..., B4 are functions defined on U .
Whether the functions B1, ..., B4 are all zero is an intrinsic property. To be spe-
cific, it is easy to see that Bi “ 0 pi “ 1, ..., 4q precisely when the rank-2 distributions
pCpxω0yqqK and pCpxω0yqqK are everywhere non-transversal. In this case, an analo-
gous result as Proposition 5.2 holds.
From now on, we will assume that Bi pi “ 1, ..., 4q are not all zero.
For (5.80), locally there are two possibilities:
i. A1 vanishes on an open subset U Ă N ;
ii. A1 is nonvanishing on an open subset U Ă N ,
These two cases are intrinsically distinguished by whether the first derived system
of B1 Ă T ˚U has rank 1 or 2.
From now on, we will assume Case ii. In this case, one can scale γ to arrange
A1 “ 1, so that the following congruence holds:
dγ ” ω1^ ω2
´ ω3^ ω4 mod ω0, ω0, γ. (5.83)
Definition 5.4. Let pN,B; π, πq be a rank-2 Backlund transformation satisfying only
the first genericity condition. Moreover, suppose that the first derived system of B1
109
has rank 1. A coframing pω0, ω0, γ, ω1, ..., ω4q defined on an open subset U Ă N
satisfying
rrω0ss “ rrπ˚θss, rrω0
ss “ rrπ˚θss, rrω0, ω0, γss “ B1
and the equations (5.81), (5.82) and (5.83) will be called 0-adapted.
In terms of a 0-adapted coframing pω0, ω0, γ, ω1, ..., ω4q, locally the systems
Ξ10 “ rrω0, ω0, γ, ω1, ω2
ss and Ξ01 “ rrω0, ω0, γ, ω3, ω4
ss
are well-defined up to ordering. From now on, we only consider those local 0-adapted
coframings that respect a fixed order of these two systems. Next, we show that such
coframings are the local sections of a G-structure with G Ă GLp7,Rq being the
subgroup consisting of elements of the form
g “
¨
˝
rI3 0 00 a 00 0 b
˛
‚, r P R; a, b P GLp2,Rq; detpaq “ detpbq “ r ‰ 0, (5.84)
where I3 is the 3ˆ 3 identity matrix.
Lemma 5.14. Let pN,Bq be a rank-2 Backlund transformation satisfying only the
first genericity condition. Let G Ă GLp7,Rq be as above. Any 0-adapted coframing
defined on an open subset U Ă N and respecting a fixed order of the systems Ξ10,Ξ01
is a section U Ñ G, where G is a G-structure on N .
Proof. It is easy to see that, if ω is a 0-adapted coframing in the sense above,
then ω ¨ g is also a 0-adapted coframing for any function g : U Ñ G.
Now we prove the converse. Suppose that ω is a 0-adapted coframing defined on
U . We determine the most general form of a function g : U Ñ GLp7,Rq so that ω ¨ g
remains 0-adapted.
First note that ω0 and ω0 are determined up to scaling. Moreover, since a 0-
adapted coframing respects a fixed order of the systems Ξ10 and Ξ01, ω1 ^ ω2 and
110
ω3 ^ ω4 (both modulo ω0, ω0 and γ) are determined up to scaling. As a result, a
transformation g P GLp7,Rq that makes ω ¨ g a 0-adapted coframing must take the
form
g “
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
r 0 0 0 0 0 00 r 0 0 0 0 0c1 c2 r 0 0 0 0f1 f5 f9 a11 a12 0 0f2 f6 f10 a21 a22 0 0f3 f7 f11 0 0 b11 b12
f4 f8 f12 0 0 b21 b22
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
,r “ detpaijq
“ detpbijq ‰ 0.
Any such g can be written as the product of of two matrices of the same form,
one, say g1, with all the fi being zero, the other, say g2, with r “ 1 and paiq, pbiq
both being the 2ˆ 2 identity matrix.
It is clear that ω ¨ g1 is 0-adapted.
In order for ω ¨ g2 to be 0-adapted, by (5.81) and (5.82), f9, f10, f11, f12 must be
zero; by (5.83), c1 and c2 must be the negative of each other, say,
c1 “ ´c2 “ c.
Now (5.81) implies that
pB3ω3`B4ω
4q ^ γ “ ω0
^ pf5ω2´ f6ω
1` f7ω
4´ f8ω
3q
` pB3ω3` B4ω
4q ^ pγ ´ cω0
q ` pB3f7 ` B4f8qω0^ γ.
for some functions B3, B4. Reducing modulo γ, it is clear that f5, f6, f7, f8 are zero.
Immediately, either c “ 0 or B3 “ B4 “ B3 “ B4 “ 0. Similarly, (5.82) implies that
f1, f2, f3, f4 are all zero. In addition, either c “ 0 or B1 “ B2 “ 0.
To conclude, we have either Bi “ 0 pi “ 1, ..., 4q or c “ 0. By assumption, Bi
pi “ 1, ..., 4q are not all zero. It follows that c “ 0 and g is of the form (5.3).
Furthermore, one can show that, on G,
ˆ
B1
B2
˙
pu ¨ gq “ aTˆ
B1
B2
˙
puq,
ˆ
B3
B4
˙
pu ¨ gq “ bTˆ
B3
B4
˙
puq.
111
Using this transformation, one can reduce to one of the following two cases5:
Case 1: B1 “ B3 “ 1, B2 “ B4 “ 0;
Case 2: B2, B3, B4 “ 0, B1 “ 1.
We now focus on Case 1; the analysis for Case 2 is work in progress.
5.3.1 Case: pB1, B2q “ pB3, B4q “ p1, 0q
Let G and G be as above. It is easy to see that the subbundle G1 Ă G defined by
B1 “ B3 “ 1, B2 “ B4 “ 0 is a G1-structure on N , where G1 Ă G is 3-dimensional
and consists of those elements of G that satisfy the extra conditions:
a22 “ b22 “ r, a12 “ b12 “ 0, a11 “ b11 “ 1.
The structure equations on G1 can be written as
d
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
ω0
ω0
γω1
ω2
ω3
ω4
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
“ ´
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
φ 0 0 0 0 0 00 φ 0 0 0 0 00 0 φ 0 0 0 00 0 0 0 0 0 00 0 0 α φ 0 00 0 0 0 0 0 00 0 0 0 0 β φ
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
^
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
ω0
ω0
γω1
ω2
ω3
ω4
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
`
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
Ω0
Ω0
ΓΩ1
Ω2
Ω3
Ω4
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
,
where, after adding a linear combination of the semi-basic 1-forms to φ, one can
arrange
Ω0“ ω1
^ ω2` ω3
^ ω4` ω3
^ γ ` pP0ω0`Kγ ` Piω
iq ^ ω0,
Ω0“ ω1
^ ω2` ω3
^ ω4` ω1
^ γ ` pQ0ω0` Lγ `Qiω
iq ^ ω0,
Γ “ ω1^ ω2
´ ω3^ ω4
` Ciωi^ ω0
`Diωi^ ω0.
5 One can verify that the classical Backlund transformation (1.5) relating solutions of the hyper-bolic Tzitzeica equation (1.4) belongs Case 1. By computing the corresponding structure invariants,one can show that it is non-homogeneous.
112
Lemmas 5.2 and 5.3 apply to the current case without change. Following from
this, Ω1, ...,Ω4 have expressions (5.17)-(5.20) where T kij “ ´Tkji for all k, i, j “ 1, ..., 4,
and T 123, T
124, T
223, T
224, T
314, T
324, T
414, T
424 are zero.
Furthermore, one can add a linear combination of the semi-basic 1-forms to α to
arrange
T 21 “ T 2
1 “ R21 “ T 2
12 “ T 213 “ T 2
14 “ 0.
By adjusting β, we can arrange
T 43 “ T 4
3 “ R43 “ T 4
13 “ T 423 “ T 4
34 “ 0
Finally, by adding a suitable multiple of γ to φ, we can arrange
K “ ´L.
The torsion cannot be absorbed further.
Applying d2 “ 0 to the structure equations, we obtain the congruences
dpdω0q ” p´P3 ´ 1´ T 4
12 ´ T113qω
3^ ω1
^ ω2` p´P4 ´ T
114 ` T
312qω
4^ ω1
^ ω2
` p´P4 ´ L´ T334 ´R
33 ´R
44qω
4^ ω3
^ γ ` p´P1 ` T313 ´ T
234qω
1^ ω3
^ ω4
` p´P2 ` T323 ` T
134qω
2^ ω3
^ ω4` pL` T 3
12 `R11 `R
22qγ ^ ω
1^ ω2
` p´P0 ` T11 ` T
22 qω
0^ ω1
^ ω2` p´P0 ´D4 ` T
33 ` T
44 qω
0^ ω3
^ pω4` γq
` p´P1 ` T313qω
1^ ω3
^ γ ` p´P2 ` T323qω
2^ ω3
^ γ mod ω0,
and
dpdω0q ” p´Q1 ` 1` T 3
13 ´ T234qω
1^ ω3
^ ω4` p´Q4 ´ T
114 ` T
312qω
4^ ω1
^ ω2
` p´Q2 ` L´ T112 ´R
11 ´R
22qω
2^ ω1
^ γ ` p´Q3 ´ T412 ´ T
113qω
3^ ω1
^ ω2
` p´Q2 ` T323 ` T
134qω
2^ ω3
^ ω4` p´L` T 1
34 `R33 `R
44qγ ^ ω
3^ ω4
` p´Q0 ` T33 ` T
44 qω
0^ ω3
^ ω4` p´Q0 ` T
11 ` T
22 ´ C2qω
0^ ω1
^ pω2` γq
` p´Q3 ´ T113qω
3^ ω1
^ γ ` p´Q4 ´ T114qω
4^ ω1
^ γ mod ω0.
113
This implies the following relations
T 112 “ ´Q2 ` 2L, T 1
13 “ ´P3 ´ 1, T 114 “ ´P4, T 1
34 “ 0; T 234 “ 0,
T 334 “ ´P4 ´ 2L, T 3
13 “ Q1 ´ 1, T 323 “ Q2, T 3
12 “ 0, T 412 “ 0,
T 22 “ ´T
11 `Q0 ` C2, T 4
4 “ Q0 ´ T33 , T 2
2 “ P0 ´ T11 , T 4
4 “ ´T33 ` P0 `D4,
Q4 “ P4, P1 “ Q1 ´ 1, P2 “ Q2, Q3 “ P3 ` 1,
R22 “ ´L´R
11, R4
4 “ L´R33.
Taking this into account, we compute
dpdω1q ” T 1
0 ω0^ ω3
^ pω4` γq ` T 1
2 γ ^ ω3^ ω4 mod ω0, ω1, ω2,
dpdω2q ” T 2
0 ω0^ ω3
^ pω4` γq ` pC4 ´ T
11 `Q0qγ ^ ω
3^ ω4 mod ω0, ω1, ω2,
dpdω3q ” T 3
0ω0^ ω1
^ pω2` γq ` T 3
4 γ ^ ω1^ ω2 mod ω0, ω3, ω4,
dpdω4q ” T 4
0ω0^ ω1
^ pω2` γq ` p´D2 ` P0 ´ T
33 qγ ^ ω
1^ ω2 mod ω0, ω3, ω4
and
dpdγq ” p´D2 ´ C2 ´Q2qω2^ ω3
^ ω4` p´C3 ´D3 ` P3 ` 1qω3
^ ω1^ ω2
` p´D4 ´ C4 ` P4qω4^ ω1
^ ω2` p´D1 ´ C1 ´Q1 ` 1qω1
^ ω3^ ω4
mod ω0, ω0, γ.
As a result,
T 10 “ T 2
0 “ T 12 “ T 3
0 “ T 40 “ T 3
4 “ 0,
T 11 “ Q0 ´ C4, T 3
3 “ P0 ´D2,
Q2 “ ´D2 ´ C2, P4 “ D4 ` C4, Q1 “ 1´ C1 ´D1, P3 “ D3 ` C3 ´ 1.
Now the torsion coefficients are expressed in terms of the 19 functions : Ci pi “
1, ..., 4q, Di pi “ 1, ..., 4q, T 11 , T
12 , R1
1, R12, T 3
3 , T34 , R3
3, R34, P0, Q0 and L. Applying
d2 “ 0 to the structure equations, we find the G1-action on these torsion functions
to be, infinitesimally,
d
ˆ
C1
C2
˙
”
ˆ
0 α0 φ
˙ˆ
C1
C2
˙
, d
ˆ
C3
C4
˙
”
ˆ
0 β0 φ
˙ˆ
C3
C4
˙
,
114
d
ˆ
D1
D2
˙
”
ˆ
0 α0 φ
˙ˆ
D1
D2
˙
, d
ˆ
D3
D4
˙
”
ˆ
0 β0 φ
˙ˆ
D3
D4
˙
,
d
ˆ
T 11
T 12
˙
”
ˆ
φ α0 2φ
˙ˆ
T 11
T 12
˙
, d
ˆ
T 33
T 34
˙
”
ˆ
φ β0 2φ
˙ˆ
T 33
T 34
˙
,
d
ˆ
R11
R12
˙
”
ˆ
φ α0 2φ
˙ˆ
R11
R12
˙
, d
ˆ
R33
R34
˙
”
ˆ
φ β0 2φ
˙ˆ
R33
R34
˙
,
and
dL ” Lφ, dP0 ” P0φ, dQ0 ” Q0φ,
where all congruences are modulo the semi-basic 1-forms ω0, ω0, γ, ω1, ..., ω4.
Clearly, each element in R :“ tC2, C4, D2, D4, T12 , T
34 , R
12, R
34, L, P0, Q0u is a rela-
tive invariant with nonzero weights only in φ. There are two cases:
I. All functions in R are all zero;
II. Not all functions in R are zero.
Homogeneity Assumption: Now we assume that the underlying rank-2 Backlund
transformation is homogeneous.
I. In this case, by the homogeneity assumption, C1, C3, D1, D3 must be constants;
then T 11 , R
11, T
33 , R
33 become relative invariants, with nonzero weights only in φ.
We apply d2 “ 0 to the equations of dω0 and dω0, obtaining
dpdω0q “ ´dφ^ ω0
´R11pC1 `D1qγ ^ ω
1^ ω0
` ppC3 `D3 ´ 1qR33 ´ T
33 qγ ^ ω
3^ ω0
` p2C1 `D1 ` C3qω0^ ω1
^ ω3´ pC1 `D1qT
11ω
0^ ω1
^ ω1,
dpdω0q “ ´dφ^ ω0
`R33pD3 ` C3qγ ^ ω
3^ ω0
´ ppD1 ` C1 ´ 1qR11 ` T
11 qγ ^ ω
1^ ω0
` p2D3 ` C3 `D1qω3^ ω1
^ ω0` pD3 ` C3qT
33ω
0^ ω3
^ ω0.
From this we deduce
T 11 “ R1
1, T 33 “ ´R
33, C1 ` C3 “ ´pD1 `D3q.
115
By writing D3 as ´pC1 ` C3 `D1q, we have
dφ “ ´Wω0^ ω0
´ pC1 `D1qγ ^ pR11ω
1`R3
3ω3q (5.85)
´ pC1 `D1qpR11ω
0^ ω1
´R33ω
0^ ω3
q ` p2C1 `D1 ` C3qω1^ ω3,
for a function W .
Taking this into account, we compute
dpdγq “ Wω0^ ω0
^ γ ` C1R11γ ^ ω
1^ pω0
´ ω0q ` C1R
11ω
0^ ω0
^ ω1
` pC1 ` C3 `D1qR33γ ^ ω
3^ pω0
´ ω0q ´ pC1 ` C3 `D1qR
33ω
0^ ω0
^ ω3
` p2pC1q2´ 2C1C3 ` 2C1D1 ´ 2C3D1 ` C1 ` C3qω
1^ ω3
^ ω0 (5.86)
` p2pC1q2` 2C1C3 ` 6C1D1 ` 2C3D1 ` 4pD1q
2´ C1 ´ C3qω
1^ ω3
^ ω0
´ 2p2C1 `D1 ` C3qω1^ ω3
^ γ.
Therefore, W “ 0, and, by the vanishing of the coefficients of ω1^ω3^γ, ω1^ω3^ω0
and ω1 ^ ω3 ^ ω0, we must have
´C1 “ D1 “ C3.
As a result, φ is closed, by (5.85).
To continue, we need to ask whether the functions R11 and R3
3 are both zero. This
leads to two subcases:
I1. If R11, R
33 are zero, then, after replacing the constant C3 by λ and restricting to
a leaf of φ, the structure equations become
dω0“ ω1
^ ω2` ω3
^ pω4` γ ´ ω0
q,
dω0“ ω1
^ pω2` γ ` ω0
q ` ω3^ ω4,
dγ “ ω1^ ω2
´ ω3^ ω4
´ λpω1´ ω3
q ^ pω0´ ω0
q,
dω1“ 0, (5.87)
dω2“ ´α ^ ω1
` λω0^ ω3,
dω3“ 0,
dω4“ ´β ^ ω3
` λω0^ ω1.
116
Applying d2 “ 0 to dω0, dω0, dγ, dω1 and dω3 leads to identities. Applying d2 “ 0
to dω2 and dω4 yields
pdα ´ λω2^ ω3
q ^ ω1“ 0,
pdβ ´ λω4^ ω1
q ^ ω3“ 0.
I2. Suppose that R11, R
33 are not both zero. If R1
1 is nonzero, then let U be R11,
otherwise, let U be R33. Such a function U , being nonzero, satisfies
dU ” Uφ, mod ω0, ω0, γ, ω1, ..., ω4.
By homogeneity, there exist constants r11, r
33 (one of which equals to 1), such that
R11 “ r1
1U, R33 “ r3
3U.
Let G2 Ă G1 be defined by U “ 1. On G2, there exist functions H0, H0, Hγ, H1, ..., H4
such that
φ “ H0ω0` H0ω
0`Hγγ `
4ÿ
i“1
Hiωi.
Applying d2 “ 0 to the structure equations, it is easy to find
dH0 ” 0,
dH0 ” 0,
dHγ ” 0,
dH1 ” αH2,
dH2 ” 0,
dH3 ” βH4,
dH4 ” 0,
where all congruences are modulo the semi-basic 1-forms. As a result of this and the
homogeneity assumption, H0, H0, Hγ, H2, H4 are constants. Moreover, we have
dpdω1q ” ´r1
1pH2ω2`H3ω
3`H4ω
4q ^ γ ^ ω1 mod ω0, ω0,
dpdω3q ” ´r3
3pH1ω1`H2ω
2`H4ω
4q ^ γ ^ ω3 mod ω0, ω0.
Since we have assumed that r11 and r3
3 are not both zero, it follows that
H2 “ H4 “ 0.
117
As a result, H1 and H3 must be constants. Using this and expanding dpdω0q, dpdω0q,
dpdγq, dpdω1q, dpdω3q, and
dpdω2q mod ω1, dpdω4
q mod ω3,
we find that H0, H0, Hγ, H1 and H3 must all be zero.
Taking these into account, replacing r11 by λ and r3
3 by µ; the structure equations
now take the form
dω0“ ω1
^ ω2` ω3
^ pω4` γ ´ ω0
q,
dω0“ ω1
^ pω2` γ ` ω0
q ` ω3^ ω4,
dγ “ ω1^ ω2
´ ω3^ ω4,
dω1“ λpγ ` ω0
q ^ ω1, (5.88)
dω2“ ´α ^ ω1
´ λpγ ` ω0q ^ ω2,
dω3“ µpγ ´ ω0
q ^ ω3,
dω4“ ´β ^ ω3
´ µpγ ´ ω0q ^ ω4,
where either λ or µ equals to 1. Applying d2 “ 0 to the equations of dω0, dω0, dγ, dω1
and dω3 yields identities; applying it to the equations of dω2 and dω4 yields
pdα ` λpγ ` ω0q ^ p2α ` ω2
qq ^ ω1“ 0,
pdβ ` µpγ ´ ω0q ^ p2β ´ ω4
qq ^ ω3“ 0.
II. In this case, we can find a nonzero function U , expressed in terms of the functions
in R, satisfying
dU ” Uφ, mod ω0, ω0, γ, ω1, ..., ω4.
By the homogeneity assumption, there exist constants c2, c4, d2, d4, `, p0, q0, t12, r12, t
34, r
34
such that
L “ `U,
P0 “ p0U,
Q0 “ q0U,
C2 “ c2U,
C4 “ c4U,
D2 “ d2U,
D4 “ d4U,
T 12 “ t12U
2,
R12 “ r1
2U2,
T 34 “ t34U
2,
R34 “ r3
4U2.
118
Since U is a relative invariant with weight 1 in φ, we can perform a structure reduction
to G2 Ă G1 defined by
U “ 1.
On G2, there exist functions H0, H0, Hγ, H1, ..., H4 such that
φ “ H0ω0` H0ω
0`Hγγ `
4ÿ
i“1
Hiωi.
By applying d2 “ 0 to the structure equations, we find
dH0 ” 0,
dH0 ” 0,
dHγ ” 0,
dH1 ” αH2,
dH2 ” 0,
dH3 ” βH4,
dH4 ” 0,
where all congruences are modulo the semi-basic 1-forms. By the homogeneity as-
sumption, H0, H0, Hγ, H2, H4 are constants.
Now, depending on whether the structure can be further reduced, we need to
consider three subcases:
II1. c2, d2, t12, r
12, H2 are all zero;
II2. c4, d4, t34, r
34, H4 are all zero;
II3. Not all of c2, d2, t12, r
12, H2, and not all of c4, d4, t
34, r
34, H4 are zero.
II1. In this case, by homogeneity, C1, D1, T11 , R
11, H1 must be constants. Taking this
into account and using the structure equations, we expand dpdω0q, dpdω0q, dpdγq,
dpdω1q, dpdω3q and reduce modulo ω3; expand dpdω2q, modulo ω1, ω3; and expand
dpdω4q, modulo ω3, ω4. Using MapleTM, we verify that these calculations imply
` “ p0 “ q0 “ c4 “ d4 “ 0, r34 “ ´t
34.
By the assumption of the current case, the only possibility is when t34 is nonzero. As
a result, one can reduce to the subbundle G3 defined by
T 33 “ 0.
119
On G3, the functions C3, D3, R33, H3 reduce to constants. Moreover, there exist func-
tions N0, N0, Nγ, N1, ..., N4 on G3 such that
β “ N0ω0` N0ω
0`Nγγ `
4ÿ
i“1
Niωi.
By differentiation and using the homogeneity assumption, we find that among these
new functions, N0, N0, Nγ, N2, N4 are constants, and that
dN1 ” αN2, mod ω0, ω0, γ, ω1, ..., ω4.
Moreover, we have
dpdω4q ” ´N2t
34 ω
0^ ω2
^ ω4 mod ω0, γ, ω1, ω3.
Since t34 is nonzero (by assumption), N2 must be zero, and N1 is therefore a constant.
No further structure reduction is possible.
Now, expanding using MapleTM dpdω0q, dpdω0q, dpdγq, dpdω1q, dpdω3q and
dpdω2q mod ω1, dpdω4
q mod ω1.
we find that either
tHγ, `, C1, C3, D1, D3, H0, H0, H1, H3,H4, N0, N1, N2, R33, c4, d4, p0, q0u “ t0u,
N0 “ ´Nγ, N4 “ 1, R11 “ T 1
1 , r34 “ ´t
34;
or
t`, C1, C3, D1, D3, H0,H1, H3, N0, N1, N2, R11, R
33, T
11 , c4, d4, p0, q0u “ t0u,
Hγ “ ´H0 “1
2H4, Nγ “ ´N0 “
1
2, N4 “ 1, r3
4 “ ´t34.
120
By a change of notation, the former case provides the structure equations
dω0“ ω1
^ ω2` ω3
^ pω4` γ ´ ω0
q,
dω0“ ω1
^ pω2` γ ` ω0
q ` ω3^ ω4,
dγ “ ω1^ ω2
´ ω3^ ω4,
dω1“ λpω0
` γq ^ ω1, (5.89)
dω2“ ´α ^ ω1
´ λpω0` γq ^ ω2,
dω3“ µpω0
´ γq ^ ω4,
dω4“ τpω0
´ γq ^ ω3` ω3
^ ω4,
where λ, µ, τ are constants with µ being nonzero. These structure equations verify
the identity d2 “ 0 except that
dpdω2q “ ´pdα ` λpω0
` γq ^ p2α ` ω2qq ^ ω1.
The latter case provides the structure equations
dω0“ ´λpγ ` 2ω4
q ^ ω0` ω1
^ ω2` ω3
^ pω4` γ ´ ω0
q,
dω0“ ´λpγ ´ ω0
` 2ω4q ^ ω0
` ω1^ pω2
` γ ` ω0q ` ω3
^ ω4,
dγ “ λpω0´ 2ω4
q ^ γ ` ω1^ ω2
´ ω3^ ω4,
dω1“ 0, (5.90)
dω2“ ´α ^ ω1
´ λpγ ´ ω0` 2ω4
q ^ ω2,
dω3“ ´µpγ ´ ω0
q ^ ω4,
dω4“ ´
1
2pγ ´ ω0
` 2ω4q ^ ω3
´ λpγ ´ ω0q ^ ω4,
where λ, µ are constants with µ being nonzero. These structure equations verify the
identity d2 “ 0 except that
dpdω2q “ p´dα ` λα ^ pγ ´ ω0
` 2ω4qq ^ ω1.
121
II2. The analysis of this case is similar to that of Case (II1), so we omit the details.
In fact, the only two possible solutions in this case can be turned into (5.89) and
(5.90), respectively, by switching pω0, ω1, ω2q and pω0, ω3, ω4q then changing the signs
of ω3 and ω4. Therefore, we will not regard these as new solutions.
II3. In this case, not all of c2, d2, t12, r
12, H2 are zero, and not all of c4, d4, t
34, r
34, H4
are zero. Thus, it is always possible to reduce to an e-structure. Now let
α “M0ω0` M0ω
0`Mγγ `
4ÿ
i“1
Miωi,
β “ N0ω0` N0ω
0`Nγγ `
4ÿ
i“1
Niωi,
where all the new coefficients are constants, by the homogeneity assumption. Ap-
plying d2 “ 0 to the structure equations and computing with MapleTM, we find that
the structure equations must take the form
dω0“ ω1
^ ω2` ω3
^ pω4` γ ´ ω0
q,
dω0“ ω1
^ pω2` γ ` ω0
q ` ω3^ ω4,
dγ “ ω1^ ω2
´ ω3^ ω4,
dω1“ pω0
` γq ^ pV ω1` λω2
q, (5.91)
dω2“ ´pω0
` γq ^ pMω1` V ω2
q ´ ω1^ ω2,
dω3“ pω0
´ γq ^ pWω3` µω4
q,
dω4“ pω0
´ γq ^ pNω3´Wω4
q ` ω3^ ω4,
where M,N, V,W, λ, µ are constants with λ, µ being nonzero. Applying d2 “ 0 to
(5.91) yields identities.
122
5.3.2 Integration of the Structure Equations
Equations (5.87). Suppose that there exists a coframing defined on a small enough
domain U Ă N satisfying the structure equations (5.87). Since ω1 and ω3 are closed,
there exist functions x, y on U such that
ω1“ dx, ω3
“ dy.
The equations of dω0 and dω0 then imply the existence of new functions z, p, q, w, s, t
that satisfy
ω0“ dz ´ pdx´ qdy, ω0
“ dw ´ sdx´ tdy.
By the equation of dω0, we have
´dp^ dx´ dq ^ dy “ dω0“ ´ω2
^ dx´ pω4` γ ´ dz ` pdx` qdyq ^ dy,
which implies that
ω2” dp` hdy mod dx, (5.92)
ω4` γ ” dz ` dq ` p´p` hqdx mod dy, (5.93)
for a new function h. Similarly, by the equation of dω0, we obtain
´ds^ dx´ dt^ dy “ dω0“ ´pω2
` γ ` dw ´ sdx´ tdyq ^ dx´ ω4^ dy,
which implies that
ω2` γ ” ´dw ` ds` pt` gqdy mod dx,
ω4” dt` gdx mod dy,
for a new function g.
Now, using the expressions of ω2, ω4 and their exterior derivatives, we have
dh^ dy ” dω2” λdz ^ dy mod dx,
dg ^ dx ” dω4” λdw ^ dx mod dy.
123
As a result, there exist functions Hpx, yq and Gpx, yq such that
h “ λz `Hpx, yq, g “ λw `Gpx, yq. (5.94)
We claim that one can always arrange H “ G “ 0 by a change of coordinates
that preserves the expressions of ω0, ω0. In fact, consider the change of variable
z ÞÑ z ` ζpx, yq.
By the expression of ω0, we have, correspondingly,
p ÞÑ p` ζx, q ÞÑ q ` ζy;
and, by the expression of ω2,
h ÞÑ h´ ζxy;
finally, by the expression of h,
H ÞÑ H ´ λζ ´ ζxy.
Locally, given an analytic function Hpx, yq, it is always possible to solve the equation
H “ ζxy ` λζ. This shows that we can always put H to be equal to zero by an
appropriate choice of the coordinates. By a similar argument, it is easy to see that
a transformation of the form
w ÞÑ w ` ξpx, yq
leads to
G ÞÑ G´ λξ ´ ξxy.
Hence, we can put G “ 0 by an appropriate choice of the functions w, s and t.
With the simplified expressions of h and g, we have
dγ “ dx^ pdp´ λdz ` λdwq ´ dy ^ pdt´ λdz ` λdwq
` λpz ` w ` q ´ t` p´ sqdx^ dy.
124
Moreover, comparing the expressions of ω4, ω4 ` γ and ω2, ω2 ` γ yields:
γ “ dz ` dq ´ dt´ pp` λw ´ λzqdx`Mdy (5.95)
“ ´dw ` ds´ dp` pt` λw ´ λzqdy `Ndx,
for two new functions M and N . By (5.95), we have
dpz ` w ` q ´ s` p´ tq ´ pp` λw ´ λzqdx´ pt` λw ´ λzqdy “ Ndx´Mdy.
This shows that z ` w ` q ´ t ` p ´ s is a function of x, y only, say, ηpx, yq. The
functions M and N are then:
M “ ´ηy ` t` λw ´ λz, N “ ηx ´ p´ λw ` λz.
Now, we can compute the exterior derivative of γ “ dz ` dq ´ pp ` λw ´ λzqdx `
p´ηy ` t ` λw ´ λzqdy. This is compatible with the equation for dγ if and only if
ηpx, yq satisfies
ηxy ` λη “ 0.
If we make the following change of variables
z ÞÑ z ` ζpx, yq, w ÞÑ w ` ξpx, yq
such that the functions H and G both remain zero, i.e., such that ζ, ξ satisfy the
equations ζxy ` λζ “ 0 and ξxy ` λξ “ 0, then η would change by
η ÞÑ η ` pζ ` ζx ` ζyq ` pξ ´ ξx ´ ξyq.
Now let η “ η ` pξ ´ ξx ´ ξyq. It suffices to ask whether the following system of
partial differential equations has solutions, given that η satisfies ηxy ` λη “ 0:
#
ζxy ` λζ “ 0,
ζ ` ζx ` ζy “ ´η.(5.96)
125
One can apply the Frobenius Theorem to (5.96), and conclude that its solutions
depend on 2 constants. Consequently, one can put η “ 0 by an appropriate choice
of coordinates.
We have shown that the Backlund transformation corresponding to the structure
equations (5.87) is an auto-Backlund transformation relating solutions of the equa-
tion zxy ` λz “ 0 (see (5.92) and (5.94)). It is not hard to see that such a Backlund
transformation locally correspond to the system (up to contact equivalence):
#
z ` zx ` zy ` w ´ wx ´ wy “ 0,
zxy ` λz “ wxy ` λw,
where the second equation is deduced from the first and the vanishing of γ.
Equations (5.88), (5.89), (5.90) and (5.91). All these structure equations except
(5.90) satisfy the property
dpω1^ ω2
q “ dpω3^ ω4
q “ 0.
However, for (5.90), note that ω0´γ´2ω4 is closed. Thus, there exists a function
h such that ω0 ´ γ ´ 2ω4 “ dh. After the change of variables
ηγ “ e´λhγ, η0“e´λhω0, η0
“ e´λhω0,
η1“ ω1, η3
“ ω3, η2“ e´λhω2, η4
“ e´λhω4.
the equations (5.90) become
dη0“ η1
^ η2´ η3
^ pη4` e´λhdhq,
dη0“ η1
^ pη2` ηγq ` η
1^ η0
` η3^ η4,
dηγ “ η1^ η2
´ η3^ η4, (5.97)
dη1“ 0, dη3
“ µeλhdh^ η4,
dη2“ ´α ^ η1, dη4
“1
2e´λhdh^ η3.
Clearly, in (5.97), η1 ^ η2 and η3 ^ η4 are closed.
126
Proposition 5.4. The structure equations (5.88), (5.89), (5.97) and (5.91) all rep-
resent rank-2 auto-Backlund transformations of the wave equation zxy “ 0.
Proof. We only provide details for (5.88). The argument works identically for the
other structure equations.
In (5.88), the pull-back of I corresponds to the system xω0,Φ,Ψy where Φ “
ω1^ω2, Ψ “ ω3^pω4`γ´ω0q are decomposable 2-forms. These forms satisfy dΦ “ 0
and dω0 “ Φ`Ψ. It follows that dΨ “ 0 as well. Hence, there locally exist functions
x, y, p, q such that Φ “ dx^ dp, Ψ “ dy^ dq. It follows that dpω0` pdx` qdyq “ 0.
Thus, locally, there exists a function z such that ω0 “ dz ´ pdx ´ qdy. Clearly, I
corresponds to the wave equation zxy “ 0. The argument for I is similar.
5.4 Summary
We summarize our main results in this chapter in the following
Theorem 5.15. Let pN,Bq be a homogeneous rank-2 Backlund transformation re-
lating two hyperbolic Monge-Ampere systems.
(i) It is impossible for pN,Bq to satisfy all three genericity conditions with pB1, B2q
and pB3, B4q being both nonzero pTheorem 5.7 q.
(ii) Suppose that pN,Bq satisfies only the first two genericity conditions with
ε “ 1 and with the relative invariants pB1, B2q and pB3, B4q being both nonzero. If
pN,Bq does not arise as a 1-parameter family of rank-1 Backlund transformations
in the sense of Lemma 5.9, then pN,Bq is an auto-Backlund transformation of the
linear equation zxy “ z psee (5.72), (5.73), (5.76) and (5.77)q.
(iii) If pN,Bq satisfies only the first genericity condition, with the first derived
system of B1 having rank 2 and pB1, B2q, pB3, B4q being both nonzero, then it is an
auto-Backlund transformation of a linear equation of the form zxy “ λz psee Section
5.3.2q.
127
Of course, Theorem 5.15 does not exhaust all cases of homogeneous rank-2
Backlund transformations relating two hyperbolic Monge-Ampere systems. In par-
ticular, we still need to investigate the following cases:
1. when all three genericity conditions are satisfied, and exactly one of
the relative invariants pB1, B2q and pB3, B4q is zero;
2. when only the first two genericity conditions are satisfied, ε “ ´1 and
both pB1, B2q and pB3, B4q are zero;
3. when only the first two genericity conditions are satisfied, and exactly
one of the relative invariants pB1, B2q and pB3, B4q is zero;
4. when only the first genericity condition is satisfied, with the first
derived system of B1 having rank 1, and exactly one of pB1, B2q, pB3, B4q
is zero;
5. when only the first genericity condition is satisfied, with the first
derived system of B1 having rank 2.
However, based on our classification so far, one may conjecture that those homo-
geneous Backlund transformations (relating two hyperbolic Monge-Ampere systems)
that are ‘genuinely’ rank-2 are quite few.
128
6
Conclusion
The purpose of this short chapter is to briefly summarize the main results in this
thesis and to give several directions for future research.
1. In this thesis, we have proved an upper bound for the generality of generic rank-1
Backlund transformations relating two hyperbolic Monge-Ampere systems (Theorem
3.3). What is a corresponding result in the non-generic case? What is the generality
of Backlund transformations of higher ranks? Given a hyperbolic Monge-Ampere
system, is it related to a system of the same type by a Backlund transformation,
regardless of the rank?
2. We have proved several results that tell us which hyperbolic Monge-Ampere
systems may be related by a rank-1 Backlund transformation of a particular type
(Theorem 4.5 and Proposition 4.12). Is there an efficient way to tell whether a
given hyperbolic Monge-Ampere system is not related to any other hyperbolic Monge-
Ampere system by a rank-1 Backlund transformation?
3. Our classification of homogeneous rank-2 Backund transformations relating two
hyperbolic Monge-Ampere systems so far is summarized in Theorem 5.15. Is there
any non-homogeneous rank-2 Backlund transformation arising in a similar way as
129
the classical one relating solutions of the hyperbolic Tzitzeica equation?
4. In this thesis, our study of Backlund transformations is local. What can be said
about the global structure of Backlund transformations?
5. How to tell whether a Backlund transformation relating two hyperbolic Monge-
Ampere systems can arise as the composition of two Backlund transformations of
lower ranks?
6. Apply the methods used in this thesis to study Backlund transformations relating
equations of broader classes (elliptic, parabolic, etc.).
130
Appendix A
Calculations for Theorem 3.3
This Appendix supplements the proof of Theorem 3.3 by providing more calculation
details. Most calculations below are computed with MapleTM.
First consider the case when, on U , ε1 “ ε2 “ 1. Since P24, P33, P66, P75 never
appear in the equation (3.7), we can set them all to zero. Since P14 and P23 only
appear in the term pP14 ´ P23qω3 ^ ω4, we can set P14 “ 0. For similar reasons, we
can set P13, P55, P56 “ 0. For convenience, we rename A1 as P81 and A4 as P84. Now
there are 42 functions Pij remaining, and they are determined.
For each Pij, there exist functions Pijk defined on U satisfying
dpPijq “ Pijkωk.
We call these Pijk the derivatives of Pij.
Now, applying d2 “ 0 to the equation (3.7), we obtain 106 polynomial equations
expressed in terms of all 42 Pij and 186 of all 252 Pijk. These equations imply:
P01 “ P41 ´ P51, P02 “ P42 ´ P52, P03 “ ´P43 ´ P81 ` P53, P04 “ ´P44 ` P54,
P05 “ ´P45 ´ P84 ` P15, P11 “ ´P51, P12 “ ´P52, P21 “ P84, P22 “ ´1,
P35 “ 0, P36 “ ´1, P61 “ 1, P62 “ ´P81, P73 “ 0, P74 “ ´1.
131
With these relations, all coefficients in (3.7) can be expressed in terms of 27
Pij. Repeating the steps above by defining the derivatives Pijk (now 162 in all) and
applying d2 “ 0 to (3.7), we obtain a system of 91 polynomial equations, which imply
P31 “ ´P32P84 ´ P15 ´ P34 ´ 2P43 ´ 2P45 ` P53 ` P76 ´ P81 ´ P84,
P72 “ ´P71P81 ´ P15 ` P34 ` 2P43 ` 2P45 ´ 3P53 ´ P76 ` P81 ` P84.
Using these relations and repeating the steps above, we obtain
P06 “ P16 ´ P46.
Now all coefficients in (3.7) are expressed in terms of 24 Pij.
Now, corresponding to the remaining 24 Pij are 144 derivatives Pijk. Applying
d2 “ 0 to (3.7) yields a system of 88 equations, expressed in terms of the 24 Pij and
122 of the 144 derivatives Pijk. This system can be solved for Pijk; in the solution,
all Pijk are expressed explicitly in terms of 24 Pij and 64 Pijk that are ‘free’.
Let a “ paαq pα “ 1, ..., 24q stand for the 24 remaining Pij; let b “ pbρq pρ “
1, ..., 64q stand for the 64 ‘free’ Pijk. We already have
dωi “ ´1
2Cijkpaqω
j^ ωk, (A.1)
daα “ Fαi pa, bqω
i, (A.2)
for some real analytic functions Fαi and Ci
jk satisfying Cijk ` C
ikj “ 0.
Now compute the exterior derivatives
dpFαi pa, bqω
iq, α “ 1, ..., 24,
and take into account (A.1) and (A.2). From this we obtain 2-forms Ωα that are
linear combinations of dbρ^ωi and ωi^ωj. Let Ωα denote the part of Ωα consisting
of linear combinations of dbρ^ωi only. Replacing dbρ in Ωα by Gρiω
i defines a linear
map
φ : HompR6,R64q Ñ Λ2
pR6q˚b R24
132
at each point of U .
Let rΩs denote the equivalence class of pΩαq in the cokernel of φ. One can show
that rΩs must vanish and that its vanishing leads to a system of 35 equations for a
and b. This system can be solved for 12 of the 64 components of b. Apply such a
solution and update aα, bρ and the functions Fαi accordingly.
It is not hard to verify, using MapleTM, that the updated aα pα “ 1, ..., 24q, bρ
pρ “ 1, ..., 52q, Cijk and Fα
i satisfy the conditions pAq-pCq in Step 1.
For Steps 2 and 3, calculation shows that the tableaux of free derivatives has
Cartan characters
ps1, s2, s3, s4, s5, s6q “ p24, 22, 6, 0, 0, 0q
and the dimension of its first prolongation
δ “ 64 ă s1 ` 2s2 ` 3s3 ` 4s4 ` 5s5 ` 6s6 “ 86.
The cases when, on U , ε1 and ε2 take other values follow similar steps. In each
of these cases, the last nonzero Cartan character, computed at a correspond stage,
is s3 “ 6.
133
Appendix B
Invariants of an Euler-Lagrange System
This Appendix supplements the proof of Proposition 3.4.
We start with the G1-structure π : G1 Ñ M of a hyperbolic Monge-Ampere
system pM, Iq (see [BGG03] or Section 4.1). Assume that S2 “ 0 (i.e., the Euler-
Lagrange case).
Recall that the 2ˆ2-matrix S1 : G1 Ñ glp2,Rq is equivariant under the G1-action.
By (4.5) and (4.6), it is easy to see that, when detpS1puqq ą 0 (resp., detpS1puqq ă 0)
at u P G1, the same is true for detpS1pu ¨ gqq for all g P G1, and the matrix S1puq lies
in the same G1-orbit as diagp1, 1q (resp., diagp1,´1q).
Now assume that detpS1q ą 0 holds on π´1U Ă G for some domain U Ă M .
By the discussion above, we can reduce to a subbundle H Ă G1 defined by S1 “
diagp1, 1q.
It is easy to see that H is an H-structure on U where
H “
$
&
%
¨
˝
ε 0 00 A 00 0 εA
˛
‚
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ε “ ˘1, A P GLp2,Rq, detpAq “ ε
,
.
-
Ă G1
is a (disconnected) 3-dimensional Lie subgroup. Let the restriction of π : G1 Ñ M
134
to H be denoted by the same symbol π.
One can verify that, restricted to H, the 1-forms φ7 ´ φ3, φ6 ´ φ2, φ5 ´ φ1 and
φ0 in equation (4.1) become semi-basic relative to π : H Ñ U . Hence, there exist
functions Qij defined on H such that
φ7 “ φ3 `Q7iωi, φ6 “ φ2 `Q6iω
i,
φ5 “ φ1 `Q5iωi, φ0 “ Q0iω
i,(B.1)
where the summations are over i “ 0, 1, ..., 4. There are ambiguities in these Qij
as we can modify them without changing the form of the structure equation (4.1).
Using such ambiguities, we can arrange that
Q71 “ Q73 “ Q62 “ Q64 “ Q51 “ Q52 “ Q53 “ Q54 “ 0; (B.2)
the remaining Qij are then determined.
Applying d2 “ 0 to (4.1) and reducing appropriately, we obtain
d2ω1” pQ63 ´Q04qω
0^ ω3
^ ω4 mod ω1, ω2,
d2ω2” pQ03 ´Q74qω
0^ ω3
^ ω4 mod ω1, ω2,
d2ω3” pQ02 `Q61qω
0^ ω1
^ ω2 mod ω3, ω4,
d2ω4” p´Q01 ´Q72qω
0^ ω1
^ ω2 mod ω3, ω4.
This implies that
Q61 “ ´Q02, Q63 “ Q04, Q72 “ ´Q01, Q74 “ Q03.
Now all coefficients in the structure equation (4.1) are expressed in terms of Q0i
pi “ 0, 1, ..., 4q and Qj0 pj “ 5, 6, 7q. By applying d2 “ 0 to (4.1), it is not hard to
verify that, reduced modulo ω0, ω1, ..., ω4, the following congruences hold:
d
ˆ
Q01 Q03
Q02 Q04
˙
”
ˆ
φ1 φ3
φ2 ´φ1
˙ˆ
Q01 Q03
Q02 Q04
˙
, dpQ00q ” 0,
d
¨
˝
Q50
Q60
Q70
˛
‚”
¨
˝
0 φ3 ´φ2
2φ2 ´2φ1 0´2φ3 0 2φ1
˛
‚
¨
˝
Q50
Q60
Q70
˛
‚. (B.3)
135
The equations (B.3) tell us how the remaining Qij transform under the action by
the identity component of H. Moreover, it is easy to compute directly from (4.1) to
verify that
ˆ
Q01 Q03
Q02 Q04
˙
pu ¨ h0q “
ˆ
´Q01 Q03
Q02 ´Q04
˙
puq, Q00pu ¨ h0q “ ´Q00puq, (B.4)
¨
˝
Q50
Q60
Q70
˛
‚pu ¨ h0q “
¨
˝
´Q50
Q60
Q70
˛
‚puq, h0 “ diagp´1,´1, 1, 1,´1q P H
hold for any u P H.
Note that H is generated by its identity component and h0. Combining (B.3)
and (B.4), it is easy to see that Q01Q04´Q02Q03 and |Q00| are local invariants of the
underlying Euler-Lagrange system.
Moreover, using (B.3) and (B.4), it is easy to see that the H-orbit of
qpuq :“
ˆ
Q01 Q03
Q02 Q04
˙
puq, u P H
consists of all 2-by-2 matrices with the same determinant as qpuq. Now we are ready
to prove the
Lemma B.1. If detpqq ‰ 0 on H, then there is a canonical way to define a coframing
on U .
Proof. If the function L :“ detpqq is nonvanishing on U , one can reduce to the
subbundle H1 of H defined by q “ diagpL, 1q. It is easy to see that each fiber of H1
over U contains a single element.
Remark 15. As a result of Lemma B.1, if detpqq ‰ 0 on U , then the corresponding
hyperbolic Euler-Lagrange system has a symmetry of dimension at most 5. This is
a consequence of applying the Frobenius Theorem.
136
Now we proceed to complete proving Proposition 3.4. Recall that the coframing
pη0, η1, ..., η4q and the φα in (3.19) verify the equation (4.1), S1 “ diagp1, 1q, and
S2 “ 0. Moreover, we have chosen the φα to satisfy (B.2), where Qij are computed
using (B.1). By (3.19), it is immediate that
Q00 “ Q02 “ 0, Q01 “ ´Q04 “1?
2, Q03 “ 1,
Q70 “ 1, Q60 “ ´1, Q50 “?
2.
Clearly, detpqq “ Q01Q04 ´ Q02Q03 “ ´1
2‰ 0. By Lemma B.1 and Remark 15,
the hyperbolic Euler-Lagrange system considered in Proposition 3.4 has a symmetry
of dimension at most 5. Since such an Euler-Lagrange system is homogeneous, it
follows that its symmetry has dimension 5.
137
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139
Biography
Yuhao Hu was born on Jan 8, 1991 in Jingdezhen, China. During 2008-2012, he
was a student at the Shing-Tung Yau class of mathematics at Zhejiang University,
where he obtained a Bachelor of Science degree in June, 2012. After graduation, he
went to Duke University for PhD studies in mathematics. He conducted research in
differential geometry and the geometry of differential equations under the supervision
of Professor Robert Bryant. After graduating from Duke, he will work as a postdoc
with Professor Jeanne Clelland at the University of Colorado at Boulder. As a hobby,
he plays the piano.
140