differential geometry and the …lott/chernlecture1.pdfdifferential geometry and the quaternions...

DIFFERENTIAL GEOMETRY ANDTHE QUATERNIONS

Nigel Hitchin (Oxford)

The Chern Lectures

Berkeley April 9th-18th 2013

3

2

• 26th December 1843

16th October 1843

ON RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS1

SHIING-SHEN CHERN

Introduction. It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretic reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no ana-logue for orthogonal groups in n ( > 4 ) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifolds of four di-mensions, because their tangent spaces possess a geometry of this kind. I t is the purpose of this note to give a study of a compact orient-able Riemannian manifold of four dimensions at each point of which is attached a three-dimensional spherical space. This necessitates a more careful study of spherical geometry than hitherto given in the literature, except, so far as the writer is aware, in a paper by E. Study [2] . 2 0ur main result consists of two formulas, which express two topological invariants of a compact orientable differentiable manifold of four dimensions as integrals over the manifold of differ-ential invariants constructed from a Riemannian metric previously given on the manifold. These two topological invariants have a linear combination which is the Euler-Poincaré characteristic.

1. Three-dimensional spherical geometry. We consider an ori-ented Euclidean space of four dimensions E 4 with the coordinates xo, Xi, x2, X3. In EA let S* be the oriented unit hypersphere defined by the equation

2 2 2 2 (1) Xo + Xl + X2 + #3 = I-Three-dimensional spherical geometry is concerned with properties on 5 s which remain invariant under the rotation group (that is, the proper orthogonal group) of E 4 leaving the origin fixed.

Received by the editors June 22, 1945. 1 The content of this paper was originally intended to be an illustration in the

author's article, Some new viewpoints in differential geometry in the large, which is due to appear in this Bulletin. Later it appeared more advisable to publish these results separately, but a comparison with the above-mentioned article, in particular §7, is recommended.

2 Numbers in brackets refer to the references cited at the end of the paper.

964

RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 965

We call a frame an ordered set of four mutually perpendicular unit vectors eo, Ci, C2, 63. There exists one and only one rotation carrying one frame to another. The coordinates Xof X\, &2, Xz of a point ï & S 3

with respect to the frame eo, ei, C2, e3 are defined by the equation

(2) % = XQCO + Xtfi + X2t2 + Xst3.

Let eo*, ei*, e2*, e3* be a frame related to eo, ei, e2, e3 by means of the rela-tions

3

(3) e«* = ]F) daftp, a = 0, 1, 2, 3,

where (aap) is a proper orthogonal matrix, and let #0*, #1*, #2*, #3* be the coordinates of the same point x with respect to the frame eo*, ei*, e2*, e3*. Then we have

3 (3a) x* = X) a<*p%p, ot = 0, 1, 2, 3.

The properties of spherical geometry are those which, when expressed in terms of coordinates with respect to a frame, remain invariant under change of the frame.

Let # o , #1» ^ 2 , # 3 be the coordinates of a point £ with respect to a

frame Co, ei, e2, e3, as defined by (2). To these coordinates we associate a unit quaternion (4) X = XQ + xii + X2J + xzk, N(X) = 1,

where N(X) denotes the norm of X. Let (4a) X* = x* + x*i+ x*j + x?k. Then the following theorem is well known [l ] :

THEOREM 1. The proper orthogonal group (3a) can be expressed in the quaternion notation in the form

(5) X* = AXB,

where A, B are unit quaternions. It contains the two subgroups

(6a) X* = AX, (6b) X* = XB,

called the subgroups of left and right translations respectively. A left translation is a right translation when and only when it is X* = ± X.

I t is important to give a distinction between the left and right

• locally µ = µ1ω1 + µ2ω2 + µ3ω3

• if µ = 0 distinguished almost complex structure

1

µ(µ1I + µ2J + µ3K)

• PROP: This is integrable.

F.Battaglia, Circle actions and Morse theory on quaternion-

Kahler manifolds, J.Lond.Math.Soc 59 (1999) 345 – 358.

S-S.Chern, On Riemannian manifolds of four dimensions,

Bull. Amer. Math. Soc. 51 (1945) 964–971.

• PROP: M carries a canonical complex quaternionic8

“Quaternions came from Hamilton after his best work had been

done, and though beautifully ingenious, they have been an un-

mixed evil to those who have touched them in any way”

Lord Kelvin 1890

GEOMETRY OVER THE QUATERNIONS

2

• q ∈ H quaternions q = x0 + ix1 + jx2 + kx3

• algebraic variety? f(q1, . . . , qn) = 0

• q2 + 1 = 0: 2-sphere q = ix1 + jx2 + kx3, x21 + x22 + x23 = 1

5

• submanifold M ⊂ Hn

• TxM ⊂ Hn

• TxM quaternionic for all x ∈ M ⇒ M = Hm ⊂ Hn

6

INTRINSIC DIFFERENTIAL GEOMETRY

3

• quaternionic structure on the tangent bundle T

• affine connection ∇XY

• zero torsion ∇XY −∇Y X = [X, Y ]

2

• Hn n-dimensional quaternionic vector space

• left action by GL(n,H)

• commutes with right action of H

• GL(n,H) · H∗

3

• metric ⇔ maximal compact subgroup

• Sp(n) · Sp(1) ⊂ GL(n,H) · H∗

• Levi-Civita connection ∇ : unique torsion-free connectionpreserving metric

• Quaternionic Kahler ⇔ ∇ preserves quaternionic structure

4

• GL(n,H) preserves action of H on tangent bundle T

• I, J,K ∈ End(T ) such that I2 = J2 = K2 = IJK = −1

• metric Sp(n) ⊂ GL(n,H)


• Hyperkahler ⇔ ∇ preserves I, J,K

3

• Kahler form ω ∈ Ω1,1

• dω = 0

• locally ω = ddcf = dIdf

• f Kahler potential

• volume form ⇒ U(1) connection on K

3

• SL(n,H) · U(1) preserves action of C on tangent bundle T

• if a torsion-free connection ∇ preserves this structure, it isunique

• complex quaternionic – complex manifold

2

• SL(n,H) · U(1)

• SL(1,H) · U(1) = Sp(1) · U(1) = SU(2) · U(1) = U(2)

• for n = 1 complex quaternionic = Kahler complex surfacewith zero scalar curvature

• n > 1 complex quaternionic is non-metric

7

• Lecture 1 Quaternionic manifolds

• Lecture 2 Hyperkahler moduli spaces

• Lecture 3 Twistors and holomorphic geometry

• Lecture 4 Correspondences and circle actions

6

THE HYPERKAHLER QUOTIENT

2

• hyperkahler manifold M4k

• complex structures I, J, K + metric g

• ⇒ Kahler forms ω1, ω2, ω3

• ωi : T → T ∗, K = ω−11 ω2 etc.

3




• ωi : T → T ∗, K = ω−11 ω2 etc.

3




• ωi : T → T ∗, K = ω−11 ω2 etc.

3

• Lie group G acting on M , fixing ω1, ω2, ω3

• a ∈ g vector field Xa

• d(iXaωi) + iXadωi = LXaωi = 0

• moment map iXaωi = dµai

13





13





13

• µ : M → g∗ ⊗R3

• If G acts properly and freely on µ−1(0) then...

• ... the quotient metric on µ−1(0)/G is hyperkahler...

• ... of dimension dimM − 4dimG

5

EXAMPLE

• M = Hn = Cn + jCn flat hyperkahler manifold

•

ω1 =i

2(dzk ∧ dzk + dwk ∧ dwk)

ω2 + iω3 = dzk ∧ dwk

• G = U(1) action u · (z, w) = (uz, u−1w)

• µ(z, w) = (zkzk − wkwk + c, zkwk) ∈ R×C = R3

3

EXAMPLE


•

ω1 =i





3

EXAMPLE


•

ω1 =i





3

EXAMPLE


•

ω1 =i





3

EXAMPLE


•

ω1 =i





3

EXAMPLE


•

ω1 =i





3

EXAMPLE


•

ω1 =i





3

EXAMPLE


•

ω1 =i





3

EXAMPLE


•

ω1 =i




• µ(z, w) = (zkzk − wkwk, zkwk) + const. ∈ R×C = R3

3

choice

NJH, A. Karlhede, U. Lindstrom & M. Rocek, Hyperkahler met-

rics and supersymmetry, Comm. Math. Phys. 108 (1987),

535–589.

K.Galicki & H.B Lawson Jr. Quaternionic reduction and quater-

nionic orbifolds, Math. Ann. 282 (1988) 121.

• µ(z, w) = (zkzk − wkwk, zkwk) + (1,0) ∈ R×C = R3

• µ−1(0) : z2 − w2 + 1 = 0 and zkwk = 0

• w = 0⇒ projection µ−1(0)→ CPn−1

• µ−1(0)/U(1)

∼= T ∗CPn−1

Calabi metric, Eguchi-Hanson (n=2)

6


• µ−1(0) : z2 − w2 + 1 = 0 and zkwk = 0


• µ−1(0)/U(1)

∼= T ∗CPn−1


6


• µ−1(0) : z2 − w2 + 1 = 0 and zkwk = 0


• µ−1(0)/U(1)

∼= T ∗CPn−1


6


• µ−1(0) : z2 − w2 + 1 = 0 and zkwk = 0


• µ−1(0)/U(1)

∼= T ∗CPn−1


6

HERMITIAN SYMMETRIC SPACES

O. Biquard, P. Gauduchon, Hyperkahler metrics on cotangent

bundles of Hermitian symmetric spaces, in Lecture Notes in Pureand Appl. Math 184, 287–298, Dekker (1996)

• p : T ∗(G/H) → G/H

• ω1 = p∗ω + dd

ch

• h = (f(IR(IX,X))X,X), R curvature tensor, X ∈ T∗

8•

f(u) =1

u

1+ u− 1− log

1 +√1+ u

2

•

f(u) =1

u

1+ u− 1− log

1 +√1+ u

2

EXAMPLE

• M = H + H and G = R

• action t · (q1, q2) = (eitq1, q2 + t)

• µ−1(0) : |z1|2 − |w1|2 = im z2 and z1w1 = w2

• µ−1(0)/R ∼= C2, coordinates (z1, w1)

Taub-NUT metric

4

EXAMPLE

• M = H + H and G = R


• µ−1(0) : |z1|2 − |w1|2 = im z2 and z1w1 = w2


Taub-NUT metric

4

EXAMPLE

• M = H + H and G = R


• µ−1(0) : |z1|2 − |w1|2 = im z2 and z1w1 = w2


Taub-NUT metric

4

EXAMPLE

• M = H + H and G = R


• µ−1(0) : |z1|2 − |w1|2 = im z2 and z1w1 = w2


Taub-NUT metric

4

• V harmonic function on R3

• ∗dV = dα

•

g = V (dx21 + dx22 + dx23) + V −1(dθ + α)2.

• ω1 = V dx2 ∧ dx3 + dx1 ∧ (dθ + α)

3


• ∗dV = dα

•

g = V (dx21 + dx22 + dx23) + V −1(dθ + α)2.

• ω1 = V dx2 ∧ dx3 + dx1 ∧ (dθ + α)

3


• ∗dV = dα

•

g = V (dx21 + dx22 + dx23) + V −1(dθ + α)2.

• ω1 = V dx2 ∧ dx3 + dx1 ∧ (dθ + α)

3


• ∗dV = dα

•

g = V (dx21 + dx22 + dx23) + V −1(dθ + α)2.

• ω1 = V dx2 ∧ dx3 + dx1 ∧ (dθ + α)

3

TAUB-NUT

•

V =1

2r+ c

2

choice

NJH, A. Karlhede, U. Lindstrom & M. Rocek, Hyperkahler met-

rics and supersymmetry, Comm. Math. Phys. 108 (1987),

535–589.

K.Galicki & H.B Lawson Jr. Quaternionic reduction and quater-

nionic orbifolds, Math. Ann. 282 (1988) 121.

QUATERNIONIC KAHLER ANDHYPERKAHLER

6


• Sp(n) · Sp(1) ⊂ GL(n,H) · H∗



11


• Sp(n) · Sp(1) ⊂ GL(n,H) · H∗



11

• principal Sp(1) bundle with connection

16

• T is a module over a bundle of quaternions (e.g. HPn)

• equivalently a rank 3 bundle of 2-forms ω1, ω2, ω3

• ∇ω1 = θ2 ⊗ ω3 − θ3 ⊗ ω2

• curvature K23 = dθ1 − θ2 ∧ θ3 etc.

• in fact K23 = cω1, c constant ∼ scalar curvature

4



• ∇ω1 = θ2 ⊗ ω3 − θ3 ⊗ ω2



4



• ∇ω1 = θ2 ⊗ ω3 − θ3 ⊗ ω2



4

• P = SO(3) frame bundle

• θi well-defined 1-forms on P

• dimP ×R+ = 4n + 4

• define ϕi = d(tθi) (t = R+ coordinate)

• three closed 2-forms ϕ1, ϕ2, ϕ3

5



• dimP ×R+ = 4n + 4



5



• dimP ×R+ = 4n + 4



5

• T (P ×R+) = H ⊕ V

• on H, θi = 0 and dt = 0, ϕi = tcωi

• on V , ϕ1 = dt ∧ θ1 + t2θ2 ∧ θ3 etc.

• algebraic relations for hyperkahler if c > 0

Lorentzian version Sp(1, n) if c < 0

23

EXAMPLE

• M = HPn quaternionic projective space

• P = S4n+3 ⊂ Hn+1

• P ×R+ = Hn+1\0

2

• P ×R+ = Swann bundle or hyperkahler cone

• G preserves quaternionic Kahler structure ⇒ induced actionon P preserves ϕ1, ϕ2, ϕ3

• Quaternionic Kahler quotient⇔ hyperkahler quotient on Swannbundle

7

• P ×R+ = Swann bundle or hyperkahler cone

• G preserves quaternionic Kahler structure ⇒ induced actionon P preserves ϕ1, ϕ2, ϕ3

• Quaternionic Kahler quotient⇔ hyperkahler quotient on Swannbundle

7

• Kahler form ω ∈ Ω1,1

• dω = 0

• locally ω = ddcf = dIdf

• f Kahler potential

• volume form ⇒ U(1) connection on K

• ..... at zero value of the moment map

4

EXAMPLE

• M = Sp(2,1)/Sp(2)× Sp(1) and G = R

• R = SO(1,1) ⊂ Sp(1,1) ⊂ Sp(2,1)

• Quotient = deformation of hyperbolic metric on B4

• self-dual Einstein

8

Math. Ann. 290, 323-340 (1991)

A n m 9 Springer-Verlag 1991

The hypercomplex quotient and the quaternionic quotient Dominic Joyce Merton College, Oxford, OX1 4JD, UK

Received November 30, 1990

1 Introduction

When a symplectic manifold M is acted on by a compact Lie group of isometries F, then a new symplectic manifold of dimension d i m M - 2 d i m F can be defined, called the Marsden-Weinstein reduction of M by F [MW]. Kfihler manifolds are important examples of symplectic manifolds, and in this case the Marsden- Weinstein reduction yields a new K/ihler manifold, which as a complex manifold is the quotient of the set of stable points of M by the complexified action of F. This is called the K/ihler quotient.

Recently, these constructions have been extended to two other classes of manifolds. In the classification of Riemannian manifolds by holonomy [S 2], K/ihler manifolds are manifolds with holonomy U(n), and related to these are hyperk/ihler manifolds with holonomy Sp(n), and quaternionic Kfihler manifolds with holonomy Sp(n)Sp(1). A quotient process for hyperk/ihler manifolds has been described by Hitchin et al. [HKLR] that reduces dimension by 4dimH, and this was generalised by Galicki and Lawson [-GL] to a quotient for quaternionic K/thler manifolds.

Now in parallel with the classification of Riemannian manifolds by holonomy there is a theory [B] that classifies manifolds with torsion-free connections by holonomy. K/ihler, hyperk/ihler, and quaternionic K/ihler manifolds have ana- logues in this theory: the analogue of a Kfihler manifold is a complex manifold [with holonomy GL(n, C)], the analogue ofa hyperkfihler manifold is a hypercomplex manifold [with holonomy GL(n, ~-I)], and the analogue of a quaternionic K/ihler manifold is a quaternionic manifold [with holonomy GL(n, I-I)GL(I, ~)].

The purpose of this paper is to present quotient constructions for hypercomplex and quaternionic manifolds that are analogous to those already known for hyperk/~hler and quaternionic K/ihler manifolds. There is an essential difference between the new constructions and the known ones, which will now be explained.

The Marsden-Weinstein reduction and the other reductions above are two- stage processes. First, a moment map is defined, which is a map from the manifold

Math. Ann. 290, 323-340 (1991)

A n m 9 Springer-Verlag 1991

The hypercomplex quotient and the quaternionic quotient Dominic Joyce Merton College, Oxford, OX1 4JD, UK

Received November 30, 1990

1 Introduction

When a symplectic manifold M is acted on by a compact Lie group of isometries F, then a new symplectic manifold of dimension d i m M - 2 d i m F can be defined, called the Marsden-Weinstein reduction of M by F [MW]. Kfihler manifolds are important examples of symplectic manifolds, and in this case the Marsden- Weinstein reduction yields a new K/ihler manifold, which as a complex manifold is the quotient of the set of stable points of M by the complexified action of F. This is called the K/ihler quotient.

Recently, these constructions have been extended to two other classes of manifolds. In the classification of Riemannian manifolds by holonomy [S 2], K/ihler manifolds are manifolds with holonomy U(n), and related to these are hyperk/ihler manifolds with holonomy Sp(n), and quaternionic Kfihler manifolds with holonomy Sp(n)Sp(1). A quotient process for hyperk/ihler manifolds has been described by Hitchin et al. [HKLR] that reduces dimension by 4dimH, and this was generalised by Galicki and Lawson [-GL] to a quotient for quaternionic K/thler manifolds.

Now in parallel with the classification of Riemannian manifolds by holonomy there is a theory [B] that classifies manifolds with torsion-free connections by holonomy. K/ihler, hyperk/ihler, and quaternionic K/ihler manifolds have ana- logues in this theory: the analogue of a Kfihler manifold is a complex manifold [with holonomy GL(n, C)], the analogue ofa hyperkfihler manifold is a hypercomplex manifold [with holonomy GL(n, ~-I)], and the analogue of a quaternionic K/ihler manifold is a quaternionic manifold [with holonomy GL(n, I-I)GL(I, ~)].

The purpose of this paper is to present quotient constructions for hypercomplex and quaternionic manifolds that are analogous to those already known for hyperk/~hler and quaternionic K/ihler manifolds. There is an essential difference between the new constructions and the known ones, which will now be explained.

The Marsden-Weinstein reduction and the other reductions above are two- stage processes. First, a moment map is defined, which is a map from the manifold

QUATERNIONIC KAHLER ANDCOMPLEX QUATERNIONIC

2

• M quaternionic Kahler

• locally defined 2-forms ω1,ω2,ω3 span a subbundle E ⊂ Λ2T ∗

• invariant closed 4-form Ω = ω21 + ω2

2 + ω23

• stabilizer Sp(n) · Sp(1)

4

• action of G preserving Ω (and therefore the metric)

• iXaΩ = dµa

• 2-form µa

• moment form µ ∈ Λ2T ∗ ⊗ g∗

5

• vector field X ⇒ 1-form X

• (dX)+ = component in E ⊂ Λ2T ∗

• = µ up to a constant multiple

6



1

µ(µ1I + µ2J + µ3K)


F.Battaglia, Circle actions and Morse theory on quaternion-Kahler manifolds, J.Lond.Math.Soc 59 (1999) 345 – 358.

7



1

µ(µ1I + µ2J + µ3K)



7



1

µ(µ1I + µ2J + µ3K)



7



1

µ(µ1I + µ2J + µ3K)



7



1

µ(µ1I + µ2J + µ3K)



7



1

µ(µ1I + µ2J + µ3K)



• PROP: M carries a canonical complex quaternionic

structure.

7



1

µ(µ1I + µ2J + µ3K)



• PROP: M carries a canonical complex quaternionic

structure.

7

THE CONNECTION

• Levi-Civita connection ∇ holonomy Sp(k) · Sp(1)

• torsion-free, holonomy in GL(k,H) ·H∗:

• 1-form α

• ∇ZY = ∇ZY + α(Z)Y + α(Y )Z − α(IY )IZ − α(IZ)IY

−α(JY )JZ − α(JZ)JY − α(KY )KZ − α(KZ)KY

10

• ∇I = 0?


• ∇µ =

i iXωi ⊗ ωi

• choose local gauge µ = µω1

• dµ = iXω1 µθ2 = iXω3 µθ3 = −iXω2

• ∇I = θ2 ⊗K − θ3 ⊗ J

∇I = 0 ⇔ α = Jθ2/2 = Kθ3/2

12

• ∇I = 0?


• ∇µ =

i iXωi ⊗ ωi



• ∇I = θ2 ⊗K − θ3 ⊗ J

∇I = 0 ⇔ α = Jθ2/2 = Kθ3/2

12

• ∇I = 0?


• ∇µ =

i iXωi ⊗ ωi



• ∇I = θ2 ⊗K − θ3 ⊗ J

∇I = 0 ⇔ α = Jθ2/2 = Kθ3/2

12

• ∇I = 0?


• ∇µ =

i iXωi ⊗ ωi



• ∇I = θ2 ⊗K − θ3 ⊗ J

∇I = 0 ⇔ α = Jθ2/2 = Kθ3/2

12

• ∇I = 0?


• ∇µ =

i iXωi ⊗ ωi



• ∇I = θ2 ⊗K − θ3 ⊗ J

∇I = 0 ⇔ α = Jθ2/2 = Kθ3/2

12

• ∇I = 0?


• ∇µ =

i iXωi ⊗ ωi



• ∇I = θ2 ⊗K − θ3 ⊗ J

∇I = 0 ⇔ α = Jθ2/2 = Kθ3/2

12

• ⇔ α = −d logµ/2.

• Riemannian volume form vg

• ∇vg = −(2k +2)(d logµ)vg

• µ−(2k+2)vg invariant volume form

11

THE CONNECTION

• Levi-Civita connection ∇ holonomy Sp(k) · Sp(1)


• 1-form α

• ∇ZY = ∇ZY + α(Z)Y + α(Y )Z − α(IY )IZ − α(IZ)IY

−α(JY )JZ − α(JZ)JY − α(KY )KZ − α(KZ)KY

• holonomy SL(k,H) · U(1)

11

DIMENSION 4

• SL(1,H) · U(1) = U(2)

• ∇ = Levi-Civita connection of µ−2g

• self-dual Einstein ∼ scalar-flat Kahler

K.P.Tod, The SU(∞)-Toda field equation and special four-dimensional metrics, in “Geometry and physics (Aarhus, 1995)”,Dekker, 1997, 317–312

12



1

µ(µ1I + µ2J + µ3K)


F.Battaglia, Circle actions and Morse theory on quaternion-

Kahler manifolds, J.Lond.Math.Soc 59 (1999) 345 – 358.

S-S.Chern, On Riemannian manifolds of four dimensions,

Bull. Amer. Math. Soc. 51 (1945) 964–971.

A.Derdzinski, Self-dual Kahler manifolds and Einstein mani-

folds of dimension four, Comp. Math. 49 (1983) 405-433.

11

EXAMPLE HP1 = S4

gS4 =1

(1 + ρ2 + σ2)2(dρ2 + ρ2dϕ2 + dσ2 + σ2dθ2)

• X = ∂/∂θ X = σ2dθ/(1 + ρ2 + σ2)2 = u2dθ

• u = σ/(1 + ρ2 + σ2), v = (ρ2 + σ2 − 1)/ρ

gS4 =1

1− 4u2du2 + u2dθ2 +

1− 4u2

(v2 + 4)2dv2 +

1− 4u2

v2 + 4dϕ2

13

• µ = (1− 4u2)1/2

g =1

(1− 4u2)2du2+

1

(1− 4u2)u2dθ2+

1

(v2 + 4)2dv2+

1

v2 + 4dϕ2

• H2 × S2 scalar curvature −4+ 4 = 0

(u = (tanh2x)/2 and v = 2tan y)

14

• µ = (1− 4u2)1/2

g =1

(1− 4u2)2du2+

1

(1− 4u2)u2dθ2+

1

(v2 + 4)2dv2+

1

v2 + 4dϕ2

• H2 × S2 scalar curvature −4+ 4 = 0

(u = (tanh2x)/2 and v = 2tan y)

• ..... on S4 minus the circle ρ = 0,σ = 1

14

COMPACT QUATERNION KAHLER MANIFOLDS

• Wolf spaces G/K symmetric

• Sp(n+1)/Sp(n) · Sp(1), SU(n+2)/S(U(n)× U(2)),

SO(n+4)/SO(n) · SO(4)

• E6/SU(6) · SU(2), E7/Spin(12) · Sp(1), E8/E7 · Sp(1)

• F4/Sp(3) · Sp(1), G2/SO(4)

15

NEXT LECTURE...

• Hyperkahler manifolds

• Hyperholomorphic bundles

• Moduli space examples

2

differential geometry and the …lott/chernlecture1.pdfdifferential geometry and the quaternions...

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