projectively flat affine surfaces

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J. Geom. 79 (2004) 31 – 45 0047–2468/04/020031 – 15 © Birkh¨ auser Verlag, Basel, 2004 DOI 10.1007/s00022-003-1674-2 Projectively flat affine surfaces Thomas Binder Abstract. We study non-degenerate affine surfaces in A 3 with a projectively flat induced connection. The curvature of the affine metric ˆ K, the affine mean curvature H , and the Pick invariant J are related by ˆ K = H + J . Depending on the rank of the span of the gradients of these functions, a local classification of three groups is given. The main result is the characterization of the projectively flat but not locally symmetric surfaces as a solution of a system of ODEs. In the final part, we classify projectively flat and locally symmetric affine translation surfaces. Mathematics Subject Classification (2000): Primary 53A15; Secondary 53B05. Key words: Projectively flat induced connection, locally symmetric induced connection, affine Theorema Egregium. 1. Introduction The study of projectively flat affine surfaces is motivated by the following interesting aspects. While in dimension greater than two the projectively flat affine hypersurfaces are exactly the affine hyperspheres, in the surface case there are examples which are not affine spheres. Secondly, local symmetry of a surface implies its projective flatness. As the local symmetry case has been studied by B. Opozda [8], W. Jelonek [2], and other geometers, the difference set attracts our attention. Finally, all classes mentioned are very rich. F. Podest` a [12] studies projectively flat affine surfaces with diagonalizable shape operator; B. Opozda [9] assumes rank one shape operator and studies realizability problems [10]. Projectively flat affine surfaces with non-diagonalizable shape operator were classified completely by W. Jelonek [4]. Another classification under the assumption ˆ K = 0 is due to I. C. Lee and L. Vrancken [6]. It is common to all the examples in these papers that at least one of the functions ˆ K , H , or J is constant. Indeed the following list comprises all examples of projectively flat affine surfaces we know. This list will be extended in Section 5. EXAMPLE 1. (i) H = const. This class consists of affine spheres and affine ruled surfaces only. An affine ruled surface is projectively flat if and only if its affine mean curvature is constant, see [3]. (ii) ˆ K = 0. In this case, [6] gives a complete classification. It contains numerous examples which occur also in other contexts, among them also affine spheres and ruled surfaces. 31

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Page 1: Projectively flat affine surfaces

J. Geom. 79 (2004) 31 – 450047–2468/04/020031 – 15© Birkhauser Verlag, Basel, 2004DOI 10.1007/s00022-003-1674-2

Projectively flat affine surfaces

Thomas Binder

Abstract. We study non-degenerate affine surfaces in A3 with a projectively flat induced connection. Thecurvature of the affine metric K , the affine mean curvature H , and the Pick invariant J are related by K = H +J .Depending on the rank of the span of the gradients of these functions, a local classification of three groups is given.The main result is the characterization of the projectively flat but not locally symmetric surfaces as a solution of asystem of ODEs. In the final part, we classify projectively flat and locally symmetric affine translation surfaces.

Mathematics Subject Classification (2000): Primary 53A15; Secondary 53B05.Key words: Projectively flat induced connection, locally symmetric induced connection, affine Theorema Egregium.

1. Introduction

The study of projectively flat affine surfaces is motivated by the following interesting aspects.While in dimension greater than two the projectively flat affine hypersurfaces are exactlythe affine hyperspheres, in the surface case there are examples which are not affine spheres.Secondly, local symmetry of a surface implies its projective flatness. As the local symmetrycase has been studied by B. Opozda [8], W. Jelonek [2], and other geometers, the differenceset attracts our attention. Finally, all classes mentioned are very rich.

F. Podesta [12] studies projectively flat affine surfaces with diagonalizable shape operator;B. Opozda [9] assumes rank one shape operator and studies realizability problems [10].Projectively flat affine surfaces with non-diagonalizable shape operator were classifiedcompletely by W. Jelonek [4]. Another classification under the assumption K = 0 is due toI. C. Lee and L. Vrancken [6]. It is common to all the examples in these papers that at leastone of the functions K , H , or J is constant. Indeed the following list comprises all examplesof projectively flat affine surfaces we know. This list will be extended in Section 5.

EXAMPLE 1. (i) H = const. This class consists of affine spheres and affine ruledsurfaces only. An affine ruled surface is projectively flat if and only if its affine meancurvature is constant, see [3].

(ii) K = 0. In this case, [6] gives a complete classification. It contains numerousexamples which occur also in other contexts, among them also affine spheres andruled surfaces.

31

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32 Thomas Binder J. Geom.

(iii) Any surface with vanishing J must be a quadric (when the metric is definite) or aruled surface (indefinite metric). Hence the projectively flat surfaces with J = 0 arecontained in (i).

Following the above examples it is natural to look for projectively flat surfaces withK = const �= 0 or J = const �= 0. However, the following theorem states that newexamples do not exist.

THEOREM 1. Let x : M2 → A3 be a projectively flat affine surface.

(i) If K = const, then either K = 0 or x is an affine sphere or a ruled surface.1

(ii) If J = const, then either J = 0 or x is an affine sphere.

In Example 1 there are surfaces which are not locally symmetric. The existence of thesesurfaces was shown in [12]. I. C. Lee [5] gave concrete examples in the class of affinerotation surfaces, his examples are part of the list in [6] (where K = 0). Also, Jelonek’snon-ruled examples with non-diagonalizable shape operator [4] are not locally symmetric;they have flat affine metric, too.

The remaining two theorems treat the case where none of the functions K , H , or J is constanton an open subset. The three gradients form a parallelogram which may degenerate to aline.

THEOREM 2. Let x : M2 → A3 be a projectively flat affine surface and suppose that noneof grad K , grad H , grad J is locally zero. Then

dim span{grad K, grad H, grad J } = 2

implies that x is locally symmetric.

THEOREM 3. Let x : M2 → A3 be a projectively flat affine surface and suppose that noneof grad K , grad H , grad J is locally zero. Moreover, assume

dim span{grad K, grad H, grad J } = 1.

Then x is locally given by a solution of the systems of ODE (30) or (34) below. All thesesurfaces form a five-parameter family.

No rank assertions can be made about locally symmetric surfaces. More precisely, we willidentify a four-parameter subfamily of locally symmetric surfaces under the assumptionsof Theorem 3.

In the last Section 5, we mention a class of affine translation surfaces which is projectivelyflat but not locally symmetric and has non-constant functions K , H , and J . A reference for

1This classification was suggested in [5].

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Vol. 79, 2004 Projectively flat affine surfaces 33

affine translation surfaces is [11]. Projective flatness is preserved under homotheties, so itis sufficient to carry out classifications up to affine (and not unimodular) equivalences.

The results are contained in the author’s dissertation [1].

The author wishes to thank Luc Vrancken for stimulating discussions.

2. Affine surfaces

For a detailed introduction to the subject see e.g. [7] or [13].

An affine surface is a non-degenerate C∞-immersion x : M2 → A3 of a two-dimensionalconnected C∞-manifold given with its Blaschke normal y. A3 denotes real flat affine space.The structure equations of x with respect to y read as follows:

∇udx(v) = dx(∇uv) + h(u, v)y, dy(u) = −dx(Su); u, v ∈ X(M).

Here X(M) denotes the C∞-module of vector fields over M . h is a regular symmetricbilinear form, the affine metric. ∇ is a torsion-free Ricci-symmetric affine connection calledthe induced connection. S designates the shape operator. Its trace 2H := trace S is theaffine mean curvature. We denote the Levi-Civita connection of h by ∇. Define the (1,2)-difference tensor by C(u, v) := ∇uv − ∇uv. The Blaschke normal satisfies the apolaritycondition trace C = 0. Given x, we can compute the Blaschke normal y as follows:

�ij := det(dx(∂1), dx(∂2), ∇∂i

dx(∂j )), h := | det �|− 1

4 �, y := 1

2�x, (1)

where (∂1, ∂2) and � denote a Gauß basis and the Laplacian with respect to h, respectively.The integrability conditions read

R(u, v)w = h(v, w)Su − h(u, w)Sv, (2)

∇h is totally symmetric, (3)

∇S is totally symmetric, (4)

h(Su, v) = h(u, Sv), (5)

where R is the curvature tensor of ∇. The Ricci tensor of ∇ satisfies

Ric(u, v) = 2Hh(u, v) − h(Su, v) for all u, v ∈ X(M). (6)

By 2K := traceh Ric we denote the normed scalar curvature of h. The Pick invariant J

is given by 2J := ‖C‖2, where ‖·‖2 denotes the norm on tensors with respect to h. Theaffine Theorema Egregium is a consequence of the integrability conditions:

K = H + J. (7)

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34 Thomas Binder J. Geom.

An affine sphere is an affine surface with S = H id. An affine sphere is called proper ifH �= 0, and improper if H = 0. It is a classical result that an affine surface is locally ruled ifand only if its affine metric is indefinite and J = 0. A quadric is a surface that can be givenby a quadratic relation of the coordinate functions. It is well-known that affine quadrics arecharacterized by C = 0.

The following lemma is a generalized version of [14], Lemma 2.1. We omit the proofwhich is analogous.

LEMMA 1. Let x : M2 → A3 be an affine surface. Then, for each p ∈ M , there is aneighborhood U of p, such that, if necessary after replacing y by −y, exactly one of thefollowing holds:

CASE 1. Sq is real diagonalizable for each q ∈ U . Then there exists a frame (e1, e2) suchthat

h =(

1 00 ε

)and S =

(λ1 00 λ2

)where ε = ±1.

CASE 2. Sq is complex diagonalizable for each q ∈ U . There exists a frame (e1, e2)

such that

h =(

0 11 0

)and S =

(H −µ

µ H

)where µ �= 0.

CASE 3. Sq is non-diagonalizable over C for each q ∈ U . We can find a frame (e1, e2)

such that

h =(

0 11 0

)and S =

(H 01 H

).

CASE 4. Sp is non-diagonalizable over C, and any open neighborhood of p contains a q

such that Sq is real or complex diagonalizable.

As our considerations are local, we shall make restrictions to suitable subsets withoutmentioning this each time. Any point p ∈ M satisfying Case 4 has arbitrarily closeneighbors with neighborhoods from Cases 1 or 2. Thus, from the local point of view,Case 4 is uninteresting; we will ignore it from now on.

The proof of Lemma 1 is based on the roots of the characteristic equation of S. Thecoefficients of this polynomial are differentiable, but its roots might not necessarily bedifferentiable in points where the discriminant is zero. Since the coefficients of S mightnot be differentiable in an umbilic point p having non-umbilic points in any of its openneighborhoods, we will also exclude such points p from our local considerations.

The choice of the frame defines differentiable functions a and b by [e1, e2] =: ae1 + be2.The Lie bracket for a function ϕ is a synonym for the equation

e1(e2(ϕ)) − e2(e1(ϕ)) = ae1(ϕ) + be2(ϕ).

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Vol. 79, 2004 Projectively flat affine surfaces 35

In the remainder of this section we will restate the integrability conditions (2)–(5) in termsof the frames (e1, e2) defined in Cases 1 and 2 of Lemma 1. We will not study Case 3, sinceany projectively flat affine surface with non-diagonalizable shape operator is either ruledwith constant mean curvature or has K = 0, cf. [4].

CASE 1. We can express the induced connection, the Levi-Civita connection of h, and thecomponents of C as

∇e1e1 = −δe1 + ε(γ − a)e2, ∇e1e1 = −εae2, C(e1, e1) = −δe1 + εγ e2,

∇e1e2 = (γ + a)e1 + δe2, ∇e1e2 = ae1, C(e1, e2) = γ e1 + δe2,

∇e2e1 = γ e1 + (δ − b)e2, ∇e2e1 = −be2,

∇e2e2 = ε(δ + b)e1 − γ e2, ∇e2e2 = εbe1, C(e2, e2) = εδe1 − γ e2.

Let us substitute the eigenvalues of S by 2H = λ1 + λ2 and µ := λ2 − λ1, where µ isdifferentiable due to the restriction made above. The integrability conditions read

K − H = 2(εγ 2 + δ2) = J, (8)

e1

(H + 1

)= e1(λ2) = (b − δ)µ, (9)

e2

(H − 1

)= e2(λ1) = (a + γ )µ, (10)

e1(γ ) + e2(δ) = −3(aδ − bγ ), (11)

e1(δ) − εe2(γ ) = 3(εaγ + bδ) − 1

2µ, (12)

where the Gauß curvature K of h satisfies

K = e1(b) − εe2(a) − εa2 − b2.

The curvature tensor R is given by

R(e1, e2)e1 = −λ2e2, R(e1, e2)e2 = ελ1e1. (13)

CASE 2. For ∇, ∇ and C we obtain

∇e1e1 = −be1 + γ e2, ∇e1e1 = −be1, C(e1, e1) = γ e2,

∇e1e2 = be2, ∇e1e2 = be2, C(e1, e2) = 0,

∇e2e1 = −ae1, ∇e2e1 = −ae1,

∇e2e2 = δe1 + ae2, ∇e2e2 = ae2, C(e2, e2) = δe1. (14)

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36 Thomas Binder J. Geom.

The integrability conditions can be rewritten as

K − H = γ δ = J, (15)

e1(H) + e2(µ) = −µ(2a + γ ), (16)

e1(µ) − e2(H) = µ(2b − δ), (17)

e2(γ ) = −3aγ + µ, (18)

e1(δ) = 3bδ − µ, (19)

whereK = e2(b) − e1(a) + 2ab. (20)

The curvature tensor R satisfies

R(e1, e2)e1 = He1 − µe2, R(e1, e2)e2 = −µe1 − He2. (21)

3. Projectively flat and locally symmetric affine surfaces

Two affine connections ∇, ∇ are said to be projectively equivalent if there is a one-form �

such that∇uv − ∇

uv = �(u)v + �(v)u for all u, v ∈ X(M).

A connection is called projectively flat if it is locally projectively equivalent to a flat connec-tion. Projective equivalence is equivalent to the coincidence of the images of correspondinggeodesics. A connection is called locally symmetric if ∇R = 0. An affine surface is calledprojectively flat or locally symmetric if the induced connection has the respective property.

It is due to H. Weyl that a torsion-free and Ricci-symmetric connection ∇ on a two-dimensional manifold is projectively flat if and only if ∇ Ric is totally symmetric. From [12]it is known that for the induced connection this is equivalent to

traceh ∇S = 0. (22)

For n = 2 local symmetry is the same as ∇ Ric = 0, since we can reconstruct R fromRic. Thus, locally symmetric connections can be considered as “trivial” projectively flatconnections.

Let us now recompute (22) in terms of the local notation in Section 2.

CASE 1. The projective flatness of ∇ is equivalent to

e1(µ) = 2bµ, e2(µ) = −2aµ, (23)

and additionally, we can eliminate λ1, λ2 by rewriting (9), (10) as

e1(H) = −δµ, e2(H) = γµ. (24)

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Vol. 79, 2004 Projectively flat affine surfaces 37

The Lie bracket for µ implies µ(e1(a) + e2(b)) = 0. From now on, we will exclude allaffine spheres, hence

e1(a) = −e2(b), (25)

and the Lie bracket for H gives e1(γ ) + e2(δ) = aδ − bγ . If we compare this with (11)we see that

aδ = bγ. (26)

CASE 2. The projective flatness adds two PDEs for H and µ, so together with (16), (17)we have four PDEs:

e1(H) = −µγ, e1(µ) = 2bµ,

e2(H) = µδ, e2(µ) = −2aµ. (27)

The respective Lie brackets for µ and H imply

e1(a) = −e2(b), (28)

aγ = bδ. (29)

We will now extend [12], Proposition 3.1. For an even more general version without apolarityand diagonalizability assumptions we refer to Proposition 2.8 in [1].

PROPOSITION 1. Let x : M2 → A3 be a projectively flat affine surface with real orcomplex diagonalizable shape operator. Then the induced connection is locally symmetricif and only if det S = const.

Proof. CASE 1. Assume that x is not an affine sphere, i.e. µ �= 0. Using (13) a straightfor-ward calculation yields

(∇e1R)(e1, e2)e1 = −(2Hγ + µa)e1 − (2Hδ + µb)e2,

(∇e1R)(e1, e2)e2 = ε(e1(λ1) − 2λ1δ

)e1 + (2Hγ + µa)e2,

(∇e2R)(e1, e2)e1 = −ε(2Hδ + µb)e1 + (−e2(λ2) + 2λ2γ)e2,

(∇e2R)(e1, e2)e2 = ε(2Hγ + µa)e1 − ε(2Hδ + µb)e2.

Using (23) we can replace ei(λi) above and verify that ∇R = 0 implies

0 = 2Hδ + µb = −µ−1e1(λ1λ2), 0 = 2Hγ + µa = µ−1e2(λ1λ2),

which is equivalent to the assertion.

CASE 2. From (14) and (21) it is straightforward to compute

(∇e1R)(e1, e2)e1 = (e1(H) + µγ )e1 + (−e1(µ) + 2Hγ − 2µb)e2,

(∇e1R)(e1, e2)e2 = (−e1(µ) + 2µb)e1 + (−e1(H) − µγ )e2,

(∇e2R)(e1, e2)e1 = (e2(H) − µδ)e1 + (−e2(µ) − 2µa)e2,

(∇e2R)(e1, e2)e2 = (−e2(µ) − 2Hδ + 2µa)e1 + (−e2(H) + µδ)e2.

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38 Thomas Binder J. Geom.

Using (27), local symmetry for projectively flat surfaces is equivalent to

Hγ = 2µb, Hδ = 2µa.

By taking derivatives it is easy to verify that the latter equation holds if and only ifdet S = H 2 + µ2 = const. �

REMARK 1. Suppose that x is a projectively flat affine surface. From the preceding proofit is obvious that, in our local notation, x is locally symmetric if and only if:

CASE 1. 2Hδ + µb = 0 = 2Hγ + µa,

CASE 2. Hγ − 2µb = 0 = Hδ − 2µa.

4. Proof of the theorems

We continue to pursue the two local cases.

CASE 1. For simplification, we write

si z ={

sin z if ε = +1,sinh z if ε = −1,

and co z ={

cos z if ε = +1,cosh z if ε = −1.

If ε = −1 and J < 0, we can switch the sign of the Blaschke normal and then inter-change e1 and e2. After these steps, we have J > 0 and the invariants match those ofSection 2. again. Let us introduce polar coordinates by γ =: j si φ and δ =: j co φ such thatJ = 2(δ2 + εγ 2) = 2j2. Without loss of generality we may introduce new functions ν andτ by

e1(j) =: τ si φ − 1

4ν co φ and e2(j) =: ετ co φ + 1

4ν si φ.

We can formulate the degeneracy of the parallelogram equivalently as H = const or τ = 0,but H = const is impossible as we demand µ �= 0 and j �= 0 by excluding all affine spheresand ruled surfaces. (26) implies that we can also substitute a = ρj si φ and b = ρj co φ

for a new function ρ. Rewriting (11), (12) gives

e1(φ) = −ετ

jco φ +

(2µ − ν

4j− 3ρj

)si φ,

e2(φ) = ετ

jsi φ + ε

(2µ − ν

4j− 3ρj

)co φ.

We obtain a pair of PDEs for ρ by rewriting (25) and (8) in terms of our polar coordinates.Defining

A(H, j, µ, ρ) := 1

2j(2H + ρµ) − 2j (ρ2 − 1)

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Vol. 79, 2004 Projectively flat affine surfaces 39

we havee1(ρ) = A co φ, e2(ρ) = −A si φ.

By a lengthy calculation we find that the Lie bracket for ρ is

(2H + ρµ)j−2τ = 0.

As 2H = −ρµ means local symmetry (cf. Remark 1), we can restrict to τ = 0 from nowon. In this case, we get a pair of PDEs for ν by evaluating the Lie brackets for j and φ.Abbreviating

B(H, j, µ, ν, ρ) := −12j (H + 2j2) − 1

2j(2µ − ν)(µ − ν) + 2jρ(4µ − ν),

we obtain the linear system

co φe2(ν) + si φe1(ν) = 0, si φe2(ν) − co φe1(ν) = B,

which is solved bye1(ν) = B co φ, e2(ν) = −B si φ.

The Lie bracket for ν gives no further information. At this point, we have twelve PDEs forthe six functions φ, H , j , µ, ν, ρ. All six Lie brackets are satisfied.

We now rotate our frame as follows:

e1 := co φ e1 − ε si φ e2, e1 = co φ e1 + ε si φ e2,

e2 := si φ e1 + co φ e2, e2 = − si φ e1 + co φ e2.

Calculating the Lie bracket of the transformed frame we get

[e1, e2] = 1

4j−1(2µ − ν − 8j2ρ)e2 =: b e2.

Look for a nowhere vanishing function σ such that [e1, σ e2] = 0. It is easy to check thatthis is the case if and only if e1(σ ) = −bσ . Such a function always exists. Hence there areparameters (t, s) with Gauß basis ∂t = e1 and ∂s = σ e2. Rewriting the twelve PDEs in theparameters we observe that

φt = 0 and φs = ε

4σj−1(2µ − ν − 12j2ρ),

while the remaining five functions only depend on t . Thus, we may write ′ instead of ∂t

and obtain a closed system of ODEs which is independent of the choice of σ :

H ′ = −jµ, j ′ = −1

4ν,

µ′ = 2jρµ, ν′ = B(H, j, µ, ν, ρ), ρ′ = A(H, j, µ, ρ). (30)

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40 Thomas Binder J. Geom.

For any choice of five initial values (30) has a unique solution. Note that different choicesof σ correspond to a reparametrization of the s-lines only. We could thus demand σs = 0.

We are about to prove Theorem 1. Its assertion is now plausible because we simply addmore equations to the well-determined system (30). Observe that either assumption ofTheorem 1 implies τ = 0.

(i) K = const implies 0 = K ′ = (H + 2j2)′ = −j (µ + ν), thus µ = −ν. Differenti-ating the latter we get jρ = j

µK + 1

4µj

. Differentiating this again, using the above

formula and the one for ρ′ from (30), we get K = 0.(ii) J = const means j = const, hence ν = 0 and jρ = 3

2jµK + 1

8µj

. Differentiating

this we get that also 0 = K ′ = −µj , which is only possible for affine spheres orruled surfaces.

CASE 2. Note that J = γ δ cannot vanish on an open set as this would contradict (18) or(19). Relation (29) implies that we can write a =: ρδ and b =: ργ for some new function ρ.Without loss of generality we can express

e1(J ) =: τδ − νγ and e2(J ) =: τγ + νδ.

Similarly to Case 1, the degeneracy of the parallelogram can be reformulated as τ = 0.Substituting J = γ δ and using (18), (19) we get

e1(γ ) = τ + γ

δ(µ − ν − 3ργ δ), e2(δ) = τ − δ

γ(µ − ν − 3ργ δ).

From (28) and (15) we get

e1(a) = ab − 1

2(H + γ δ) and e2(b) = −ab + 1

2(H + γ δ). (31)

Using this, we can calculate e1(b) and e2(a) by taking derivatives in aγ = bδ with respectto e1 and e2, respectively. The result is

e1(b) = −5b2 + ρτ + γ

δ

(ρ(2µ − ν) − 1

2(H + γ δ)

)and

e2(a) = 5a2 + ρτ − δ

γ

(ρ(2µ − ν) − 1

2(H + γ δ)

). (32)

Rewriting (31) and (32) in terms of ρ we obtain

e1(ρ) = − 1

2δ(H − 2µρ) − 1

2γ (1 + 4ρ2), e2(ρ) = 1

2γ(H − 2µρ) + 1

2δ(1 + 4ρ2).

By a lengthy calculation we find that the Lie bracket for ρ is

1

2(γ 2 + δ2)γ −2δ−2(H − 2µρ)τ = 0.

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Vol. 79, 2004 Projectively flat affine surfaces 41

As H = 2ρµ means local symmetry (cf. Remark 1), we can restrict to τ = 0 from nowon. In this case, we get a pair of PDEs for ν by evaluating the Lie brackets for γ and δ:

e1(ν) = 3γ (H + γ δ) − δ−1(2µ − ν)(µ − ν) + 2b(4µ − ν) and

e2(ν) = −3δ(H + γ δ) + γ −1(2µ − ν)(µ − ν) − 2a(4µ − ν). (33)

The Lie bracket for ν gives no further information. At this point, we have twelve PDEs forthe six functions γ , δ, H , ρ, µ, and ν. All six Lie brackets are satisfied.

We now transform our frame as follows:

e1 := γ −1 e1 − δ−1 e2, e1 = 1

2γ (e1 + e2),

e2 := γ −1 e1 + δ−1 e2, e2 = 1

2δ(−e1 + e2).

Calculating the Lie bracket of the transformed frame we get

[e1, e2] = 2

γ δ− 2ρ

)e2 =: b e2.

Look for a nowhere vanishing function σ such that [e1, σ e2] = 0. It is easy to check thatthis is the case if and only if e1(σ ) = −bσ . Such a function always exists. Hence there areparameters (t, s) with Gauß basis ∂t = e1 and ∂s = σ e2. When rewriting the PDEs in theparameters it makes sense to substitute γ and δ by J and γ /δ since

(γ /δ)t = 0 and (γ /δ)s = 2σδ−2(2µ − ν − 6ρJ ).

The remaining five functions only depend on t . Thus, we will write ′ instead of ∂t andobtain a system of ODEs which is independent of the choice of σ :

H ′ = −2µ, ρ′ = −4ρ2 − J−1(H − 2ρµ) − 1,

J ′ = −2ν, ν′ = 6(H + J ) − 2J−1(2µ − ν)(µ − ν) + 4ρ(4µ − ν), (34)

µ′ = 4ρµ.

Again, by prescribing initial values for the five functions we obtain a unique solution. Asin Case 1, it is now easy to prove Theorem 1.

(i) K = const implies µ = −ν. Differentiating this relation we get 4ρµ = 2J−1

µ2 − K . Differentiating this again yields K = 0.(ii) J = const means ν = 0, hence 8ρµ = 2J−1µ2 − 3(H + J ). Differentiating this

again implies K = const, which is impossible since µ �= 0 in Case 2.

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42 Thomas Binder J. Geom.

REMARK 2. There are locally symmetric solutions of (30) and (34) since by eliminating H

through the respective assumptions 2H = −µρ or H = 2µρ the order is reduced by one.Roughly speaking, fixing four parameters gives locally symmetric surfaces.

Another assumption that reduces the order by one in Case 1 is that grad H has fixed angleφ = const with respect to the eigendirections of S, i.e. 2µ − ν = 12j2ρ. The analogousorder-reducing assumption for Case 2 is γ /δ = const or 2µ − ν = 6ρJ .

It would be of interest to know more about the systems (30), (34).

5. Examples among the translation surfaces

An affine surface x : M2 → A3 is called a translation surface if, in an appropriate localparametrization, it admits a decomposition into the sum of two planar curves. Up to asuitable unimodular transformation any translation surface can be given by

x : (u1, u2) �→ (α1(u1), α2(u

2), β1(u1) + β2(u

2))T ,

where αi and βi are differentiable functions.

In the following, the freedom in the parametrization will be fixed. We use (1) to calculatethe affine invariants. We easily get �12 = 0 and �ii = α′

j (α′iβ

′′i − β ′

iα′′i ), where i = 1, 2.

In order to obtain a conformal parametrization of h we will assume

β ′′i − β ′

i

α′′i

α′i

= εi, (i = 1, 2)

for some εi = ±1 from now on. Defining

eγ := |α′1α

′2|

12 , we have γi = 1

2

α′′i

α′i

.

In this notation the affine metric can be written as

h = εeγ (ε1du1 ⊗ du1 + ε2du2 ⊗ du2).

The choice of ε = ±1 corresponds to fixing the orientation of the Blaschke normal. Usingthe conformal change formula for the Levi-Civita connection we get

∇i∂i = 1

2γi∂i − 1

2εiεj γj ∂j , ∇1∂2 = ∇2∂1 = 1

2(γ2∂1 + γ1∂2).

The explicit expression

y = ε

2e−γ (ε1α

′′1 , ε2α

′′2 , ε1β

′′1 + ε2β

′′2 )T

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Vol. 79, 2004 Projectively flat affine surfaces 43

for the Blaschke normal permits a straightforward computation of the remaining affineinvariants:

∇i∂i = γi∂i − εiεj γj ∂j , ∇1∂2 = ∇2∂1 = 0,

Sii = −(γ ′i + γ 2

i ), S12 = S21 = γ1γ2,

C(∂i, ∂i) = 1

2γi∂i − 1

2εiεj γj ∂j , C(∂1, ∂2) = C(∂2, ∂1) = −1

2(γ2∂1 + γ1∂2).

The affine mean curvature, Pick invariant, and curvature of h read (use also (7))

H = −ε

2e−γ (ε1(γ

′1 + γ 2

1 ) + ε2(γ′2 + γ 2

2 )), J = ε

2e−γ (ε1γ

21 + ε2γ

22 ),

K = −ε

2e−γ (ε1γ

′1 + ε2γ

′2). (35)

A translation surface is determined up to affine equivalences if we prescribe the function γ

along with the signature vector (ε1, ε2).

Before studying projectively flat affine translation surfaces (as suggested in [12]), we willgive some examples.

EXAMPLE 2 (Quadrics). γ = const, i.e. γ1 = 0 = γ2. Up to translation and affineequivalences we get αi(u

i) = ui and βi(ui) = 1

2εi(ui)2, which leads to the paraboloids

z = ε1x2 + ε2y

2.

EXAMPLE 3. Assume γ is a linear function but γ �= const. Without loss of generality wecan set γ1 = 1

2a1 ∈ R\{0} and γ2 = 12a2 ∈ R. If a2 �= 0, then up to translations and affine

equivalences we get αi(ui) = eaiu

iand βi(u

i) = dieaiu

i − εi

aiui for i = 1, 2. We end up in

z = log x + a log y, a = ε1ε2a21a−2

2 ∈ R\{0}.The case a2 = 0 leads to

z = −ε1 log x + ε2y2.

We will abbreviate these surfaces by D1(a1, a2). They appear in the classification list of [6].Note that the paraboloids from Example 2 correspond to D1(0, 0).

EXAMPLE 4. When γ is a quadratic function, we can write γi(ui) = aiu

i for ai ∈ R,i = 1, 2, after a parameter translation. Assume a1a2 �= 0 in the following. Then α′

i (ui) =

eai(ui )2

and β ′′i − 2aiu

iβ ′i = εi . In the generic case, this cannot be integrated elementarily;

integration involves the error function∫

e−ξ2dξ . We write D2(a1, a2) for this surface.

THEOREM 4. Let x : M2 → A3 be an affine translation surface which is locally sym-metric. Then x(M) is affinely equivalent to an open part of one of the following surfaces:

Page 14: Projectively flat affine surfaces

44 Thomas Binder J. Geom.

(i) (x2 − z)3 = ±y2 (improper affine sphere),(ii) (a) z = ε1x

2 + ε2 cosh((sinh + id)−1y),(b) z = ε1x

2 − ε2 cosh((sinh − id)−1y),(iii) z = ε1x

2 − ε2 cos((sin + id)−1y),(iv) the surfaces D1(a1, a2); a1,2 ∈ R described in Examples 2 and 3.

Proof. The curvature tensor R of the induced connection is given by

R(∂1, ∂2)∂1 = −γ1γ2∂1 + ε1ε2(γ′2 + γ 2

2 )∂2,

R(∂1, ∂2)∂2 = −ε1ε2(γ′1 + γ 2

1 )∂1 + γ1γ2∂2.

Computing the components of ∇R we get

(∇1R)(∂1, ∂2)∂1 = −2γ ′1γ2∂1 − 2ε1ε2γ1γ

′2∂2,

(∇1R)(∂1, ∂2)∂2 = −ε1ε2(γ′′1 + 2γ ′

1γ1)∂1 + 2γ ′1γ2∂2,

(∇2R)(∂1, ∂2)∂1 = −2γ1γ′2∂1 + ε1ε2(γ

′′2 + 2γ2γ

′2)∂2,

(∇2R)(∂1, ∂2)∂2 = 2ε1ε2γ′1γ2∂1 + 2γ1γ

′2∂2.

Obviously, ∇R = 0 if and only if γ ′1 = 0 = γ ′

2 or (γi = 0 and γ ′j = −γ 2

j + c for i �= j andsome c = const). Integration of γ ′

1 = 0 = γ ′2 gives (iv). For the second part, without loss of

generality we can assume i = 1 and j = 2. We obtain the functions of the second parametergiven in the following table, where we have applied numerous affine transformations andlinear reparametrizations.

α2(u2) = (u2)3

c = 0 γ2(u2) = 1

u2β2(u

2) = − ε22 (u2)2 (i)

c > 0 and α2(u2) = sinh(u2) + u2

|γ2(0)| <√

cγ2(u

2) = √c tanh u2

β2(u2) = ε2

4ccosh u2 (ii.a)

c > 0 and α2(u2) = sinh(u2) − u2

|γ2(0)| >√

cγ2(u

2) = √c coth u2

β2(u2) = − ε2

4ccosh u2 (ii.b)

α2(u2) = sin(u2) + u2

c < 0 γ2(u2) = −√|c| tan u2

β2(u2) = − ε2

4|c| cos u2 (iii)

The case c > 0 and |γ2(0)| = √c gives Class (iv) again. �

THEOREM 5. Let x : M2 → A3 be an affine translation surface which is projectively flat.Then x(M) is locally symmetric or it is affinely equivalent to an open part of D2(ε1a, ε2a),where a ∈ R\{0}.

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Vol. 79, 2004 Projectively flat affine surfaces 45

Proof. Expressing the total symmetry of ∇ Ric in terms of γ yields

(∇1 Ric)(∂2, ∂1) = −2γ ′1γ2 = −ε1ε2(2γ ′

2γ2 + γ ′′2 ) = (∇2 Ric)(∂1, ∂1),

(∇1 Ric)(∂2, ∂2) = −ε1ε2(2γ ′1γ1 + γ ′′

1 ) = −2γ1γ′2 = (∇2 Ric)(∂1, ∂2).

Since γi depends on ui only, γ1γ′′2 = 0 = γ ′′

1 γ2 is a necessary condition (take derivatives).Hence it suffices to consider the cases (γ1 = 0 and γ ′

2 = −γ 22 + c) and (γ1(u

1) = ε1au1 +b �= 0 and γ2(u

2) = ε2au2 + c, where a, b, c ∈ R). All cases were treated in Theorem 4except the second case when a �= 0. Here we get Example 4. �

REMARK 3. The surfaces D2(ε1a, ε2a) are the only affine translation surfaces which areprojectively flat but not locally symmetric. The classification of these surfaces was proposedin [12]. From (35) it is obvious that none of K , H , J is locally constant. Hence Theorem 2implies that

dim span{grad K, grad H, grad J } = 1.

Moreover, since D2(ε1a, ε2a) has real diagonalizable shape operators, it represents a sub-family of the solutions of the system of ODEs (30).

References

[1] T. Binder, Two Codazzi problems for relative surfaces, PhD thesis, Fak. II, Inst. f. Math., TU Berlin, 2002,Shaker Verlag, Aachen, 2002.

[2] W. Jelonek, Affine locally symmetric surfaces, Geom. Dedicata 44 (1992) 189–221.[3] W. Jelonek, Characterization of affine ruled surfaces, Glasg. Math. J. 39 (1997) 17–20.[4] W. Jelonek, Two families of affine projectively flat surfaces, J. Geom. 58 (1997) 117–122.[5] I.C. Lee, Projectively flat affine surfaces that are not locally symmetric, Proc. Am. Math. Soc. 123 (1995)

237–246.[6] I.C. Lee and L. Vrancken, Projectively flat affine surfaces with flat affine metric, J. Geom. 70 (2001) 85–100.[7] K. Nomizu and T. Sasaki, Affine Differential Geometry, Cambridge University Press, 1994.[8] B. Opozda, Locally symmetric connections on surfaces, Result. Math. 20 (1991) 725–743.[9] B. Opozda, A class of projectively flat surfaces, Math. Z. 219 (1995) 77–92.

[10] B. Opozda, On the realizability of projectively flat connections on surfaces, J. Geom. 70 (2001) 133–138.[11] H. Pabel, Translationsflachen in der aquiaffinen Differentialgeometrie, J. Geom. 40 (1991) 148–164.[12] F. Podesta, Projectively flat surfaces in A3, Proc. Am. Math. Soc. 119 (1993) 255–260.[13] U. Simon, A. Schwenk-Schellschmidt and H. Viesel, Introduction to the affine differential geometry of

hypersurfaces, Lecture Notes, Science University of Tokyo, 1991. Distribution: TU Berlin.[14] L. Vrancken, Affine surfaces with constant affine curvatures, Geom. Dedicata 33 (1990) 177–194.

Thomas BinderUniversity of LubeckInstitute for Neuro- and BioinformaticsSeelandstr. 1aD-23569 LubeckGermanye-mail: [email protected]

Received 11 November 2002.