PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 64 (78), 1998, 133–145

SPECTRAL INVARIANTS OF AFFINE HYPERSURFACES

Neda Bokan^{1}, Peter Gilkey^{2} and Udo Simon

Communicated by Mileva Prvanović

Abstract. Let M be a smooth compact manifold of dimension m with smooth, possibly empty, boundary ∂M. If g is a Riemannian metric on M and if ∇ is an affine connection, let D = D(g, ∇) be the trace of the normalized Hessian; if ∂M is empty, then we impose Dirichlet boundary conditions. The structures (g, ∇) arise naturally in the context of affine differential geometry and we give geometric conditions which ensure that D is formally self-adjoint in this setting. We study the asymptotics of the heat equation trace; we have that a_{m}(D) is an affine invariant. We use the asymptotics of the heat equation to study the affine geometry of affine hypersurfaces.

§0 Introduction

Let M be a smooth compact manifold of dimension m ≥ 2 with smooth, possibly empty, boundary ∂M. Let ∇ be a Ricci symmetric, torsion free connection on the tangent bundle of M. Let g be a Riemannian metric on M. Let D be the trace of the normalized Hessian defined by g and ∇; see §1.3 for details. If the boundary of M is non-empty, we impose Dirichlet boundary conditions; it is also possible to use suitable modified Neumann boundary conditions. Let a_{n}(D) be the coefficients in the asymptotic expansion of the heat trace, see §1.4 for details. In [2], we showed that if two connections ∇ and ˜∇ are projectively equivalent and if two metrics g and g̃ are conformally equivalent, then a_{m}(D(g, ∇)) = a_{m}(D(g̃, ˜∇)).

Here is a brief outline to the paper. In §1, we shall present a brief review of results from [2] and [3] which we shall need. In §2, we review affine differential geometry. We define the metric g and the two torsion free Ricci symmetric tensors 1∇ and ^{2}∇ which are associated to a relative normalization {x, X, y} of a nondegenerate hypersurface. The operators D(g, ∇) for ∇ = ^{1}∇ or for ∇ = ^{2}∇ need not be self-adjoint. In §3, we study conditions on the hypersurface being studied to ensure that these operators are self-adjoint. In §4, we use the invariants of the heat equation to define invariants of affine differential geometry. In §5, we study the spectral geometry of the Gauss map.

AMS Subject Classification (1991): 53A15, 58G25 1Research partially supported by SFS project #04M03 2Research partially supported by the NSF (USA)

Key words: Operators of Laplace type, the Hessian, hypersurface immersed in affine space, the heat equation

§1 Heat equation asymptotics

1.1 Notational conventions. We adopt the following notational conventions. Let Greek indices ν and μ range from 1 through m and index local coordinate frames for the tangent and cotangent bundles of M; let Greek indices α and β range from 1 through m − 1 and index local coordinate frames for the tangent and cotangent bundles of the boundary. Let Roman indices i and j range from 1 through m and index local orthonormal frames for the tangent and cotangent bundles of M; let Roman indices a and b range from 1 through m − 1 and index local orthonormal frames for the tangent and cotangent bundles of ∂M. We shall assume ∂_{m} is perpendicular to the boundary; for the moment we do not assume that it is a unit normal vector field. We adopt the Einstein convention and sum over repeated indices. We shall assume the coordinates are chosen near the boundary so that g_{αm} ≡ 0; this normalization is preserved by conformal rescaling. Let Γ_{∇} and Γ_{g} be the Christoffel symbols of the connection ∇ and of the Levi-Civita connection determined by g;

∇_{∂_{ν}} ∂_{μ} = Γ_{∇,νμ}^{σ}∂_{σ} and ^{g}∇_{∂_{ν}} ∂_{μ} = Γ_{g,νμ}^{σ}∂_{σ}.

The difference Θ of these two connections is tensorial. Since the two connections are torsion free we have

Θ_{νμ}^{σ} := Γ_{∇,νμ}^{σ} − Γ_{g,νμ}^{σ} satisfies Θ_{νμ}^{σ} = Θ_{μν}^{σ}.

Let L be the second fundamental form along the boundary of the metric g;

L_{αβ} = (^{g}∇_{∂_{α}}∂_{β}, ∂_{m}) = −\frac{1}{2}∂_{m}g_{αβ}.

We impose Dirichlet boundary conditions on all operators henceforth;

domain(D) = {f ∈ C^{∞}(M) : f|_{∂M} = 0}.

1.2 Projective equivalence. Let T M, T M^{∗} and S^{2}M ⊂ T^{∗}M ⊗ T^{∗}M be the tangent, cotangent and symmetric 2 cotensor bundles over M. We say that two metrics g̃ and g are conformally equivalent if there exists a smooth function ψ on M so that g̃ = e^{2ψ}g. We say that two connections ∇ and ˜∇ are projectively equivalent if there exists a smooth closed 1-form

θ = θ( ˜∇, ∇) so that:
( ˜∇_{u} − ∇_{u})v = θ(u)v + θ(v)u.

We note that two connections are projectively equivalent if and only if their unparametrized geodesics coincide. If ∇ is a torsion free connection on T M, let

\begin{matrix}R & _{∇}(u, v) : w → (∇_{u}∇_{v} − ∇_{v}∇_{u} − ∇_{[u,v]})w, and \\ ρ & _{∇}(u, w) := − tr(v → R_{∇}((u, v)w) \end{matrix}

be the full curvature tensor and the Ricci tensor of the connection ∇. A connection is said to be Ricci symmetric if ρ_{∇}(u, w) = ρ_{∇}(w, u) for all tangent vectors u and w; we restrict to torsion free Ricci symmetric connections henceforth.

1.3 The Hessian. The Hessian H_{∇} is a second order operator from the space of smooth functions on M to the space of smooth symmetric 2 tensors on M which is defined by the equation:

(H_{∇}f)(u, v) := u(v(f)) − ∇_{u}v(f).

If ω = ω_{νμ}dx^{ν} ∘dx^{μ} is a symmetric 2 tensor, let tr_{g} ω := g^{νμ}ω_{νμ} be the contraction of ω. We contract the Hessian and normalize by adding a suitable multiple of the Ricci tensor to define a second order operator D = D(g, ∇) of Laplace type on C^{∞}(M):

Df := − tr_{g}{H_{∇}(f) + (m − 1)^{−1}fρ_{∇}}.

Although D need not be self-adjoint in general, it satisfies an important transformation rule. Let g̃ = e^{2ψ}g be a metric which is conformally equivalent to g and let ˜∇ be a connection which is projectively equivalent to ∇. Choose a local primitive ϕ so dϕ = θ( ˜∇, ∇). We refer to [2, Lemma 2.1] for the proof of the following identity:

D(g̃, ˜∇) = e^{−2ψ+ϕ}D(g, ∇)e^{−ϕ}.

1.4 Heat equation. The fundamental solution u(x; t) = e^{−tD}ϕ(x) of the heat equation satisfies the equations:

(∂_{t} + D)u(x; t) = 0, u(x; 0) = ϕ(x), and u(y; t) = 0 for y ∈ ∂M.

The operator e^{−tD} for t > 0 is trace class on L^{2}(M). As t ↓ 0, there is an asymptotic series of the form:

tr_{L^{2}}(e^{−tD}) ∼ ∑_{n≥0} a_{n}(D)t^{(n−m)/2}.

The coefficients a_{n}(D) are locally computable invariants which will comprise the focal point of our discussion. Let dx = dx(g) and dy = dy(g) be the Riemannian measures on the interior of M and on the boundary of M. We refer to [2, 6] for the proof of the following result:

1.5 Theorem.

(1) Let D be an operator of Laplace type. There exist local invariants a_{n}(x, D) defined for x ∈ M and a_{n}^{bd}(y, D) defined for y ∈ ∂M so that we have a_{n}(D) = ∫_{M} a_{n}(x, D)dx + ∫_{∂M} a_{n}^{bd}(y, D)dy. If n is odd, then the interior invariants a_{n}(x, D) vanish.

(2) Let g̃ and g be conformally equivalent metrics. Let ˜∇ and ∇ be projectively equivalent torsion free Ricci symmetric connections. Then we have that a_{m}(D(g, ∇)) = a_{m}(D(g̃, ˜∇)).

To describe the local formulae for the invariants of Theorem 1.5, it is convenient at this point to express the operator D invariantly. We refer to [2], [6] for the proof of the following assertion.

1.6 Lemma. Let D = D(g, ∇). Let Θ = ∇ −^{g}∇.

(1) There exists a unique connection ∇_{D} on C^{∞}(M) and a unique function E ∈ C^{∞}(M) so that D = −(tr_{g}(∇_{D}^{2}) + E).

(2) The connection 1 form ω_{D} of ∇_{D} is given by ω_{D,δ} := −\frac{1}{2}g_{νδ}g^{μσ}Θ_{νσ}^{ν}.

(3) We have E := \frac{1}{(m−1)}g^{νμ}ρ_{∇,νμ} − g^{νμ}(∂_{μ}ω_{D,ν} + ω_{D,ν}ω_{D,μ} − ω_{D,σ}Γ_{g,νμ}^{σ}).

Let e_{m} be the inward unit normal vector field on the boundary of M. Let R_{ijkl}, ρ_{ij} := R_{ikkj}, and τ := ρ_{ii} be the curvature tensor, the Ricci tensor, and the scalar curvature of the Levi-Civita connection. Let ^{D}Ω be the curvature tensor of the connection ^{D}∇. We refer to [3, Theorems 1.1 and 1.2] for the proof of the following Theorem:

1.7 Theorem. Let M be a manifold with smooth boundary. Adopt the notation of Lemma 1.6.

(1) a_{0}(D) = (4π)^{−m/2} ∫_{M} dx.

(2) a_{1}(D) = −\frac{1}{4}(4π)^{−(m−1)/2} ∫_{∂M} dy.

(3) a_{2}(D) = \frac{1}{6}(4π)^{−m/2}\{∫_{M}(6E + τ)dx + ∫_{∂M} 2L_{aa}dy\}.

(4) a_{3}(D) = −\frac{1}{384}(4π)^{−(m−1)/2} ∫_{∂M}(96E + 16τ − 8ρ_{mm} + 7L_{aa}L_{bb}

−10L_{ab}L_{ab})dy.

(5) a_{4}(D) = \frac{1}{360}(4π)^{−m/2}\{∫_{M}(60τE + 180E^{2} + 30Ω^{2} + 5τ^{2} − 2ρ^{2} + 2R^{2})dx

+ ∫_{∂M}(−180e_{m}(E) − 30e_{m}(τ) + 120EL_{aa} + 20τL_{aa} − 4ρ_{mm}L_{bb}

−12R_{ambm}L_{ab} + 4R_{abcb}L_{ac} + \frac{40}{21}L_{aa}L_{bb}L_{cc} − \frac{88}{7}L_{ab}L_{ab}L_{cc}
+\frac{320}{21}L_{ab}L_{bc}L_{ca})dy\}.

We note that information concerning the invariants a_{5} is available; see [4] for details.

§2 Operators defined by Affine Differential Geometry

2.1 Affine differential geometry of nondegenerate hypersurfaces. We refer to [1], [5], [8], [9], [11], and [13] for further details concerning this material. Let A be a real affine space which is modeled on a vector space V of dimension m+1, and let V^{∗} be the dual space. The tangent space and cotangent space at a point a ∈ A are modeled on the vector space V and the dual vector space V^{∗}, i.e. T_{a}A = V and T_{a}^{∗}A = V^{∗}. Let ⟨·, ·⟩ : V^{∗} × V → ℝ be the natural pairing between V^{∗} and V. Let x be a smooth immersion of M into A. Let

C(M)_{P} = {X ∈ V^{∗} : ⟨X, dx(v)⟩ = 0, ∀v ∈ T_{P} M^{m}}

be the conormal space at a point P ∈ M; let C(M) be the conormal line bundle over M. We assume that C(M) is trivial and choose a non-vanishing conormal field X on M. We say that the hypersurface x(M) is regular if and only if there exists a conormal field X such that rank(X, dX) = m + 1; if this condition is satisfied for one non-vanishing conormal field, it is satisfied for every non-vanishing conormal field so this notion is affinely invariant. We assume x(M) is regular henceforth; this implies that X is an immersion from M to V^{∗} such that the position field X is transversal to X(M). Define y = y(X) : M → V by the conditions ⟨X, y⟩ = 1 and ⟨dX, y⟩ = 0. The triple (x, X, y) is called a hypersurface with relative normalization. Note that y need not be an immersion.

The relative structure equations given below contain the fundamental geometric quantities of relative hypersurface theory; two connections ^{1}∇ and ^{2}∇, the relative shape (Weingarten) operator B, and two symmetric forms g and B̂. Let A∇ be the flat affine connection on A. We have:

dy(v) = −dx(B(v)), (Weingarten equation) A∇_{w}dx(v) = dx(^{1}∇_{w}v) + g(v, w)y, (Gauss equation)

A∇_{w}dX(v) = dX(^{2}∇_{w}v) − B̂(v, w)X. (Gauss equation)

We shall assume that the metric g is positive definite henceforth; this means that the immersed hypersurface x(M) is locally strongly convex. We will also assume M to be compact; if the boundary of M is empty and if M is simply connected, then M is a hyperovaloid. The relative shape operator B is self-adjoint with respect to g and is related to the Weingarten form B̂ by the identity:

B̂(v, w) = g(B(v), w) = g(v, B(w)).

We define a (1,2) difference tensor A, a totally symmetric relative cubic form Â, and the Tchebychev form T̂ as follows:

A := \frac{1}{2}(^{1}∇ −^{2}∇), Â(v, w, z) := g(A(v, w), z), and T̂(z) := \frac{1}{m} tr_{g}(A(z, ·)).

Let ‘;’ denote multiple covariant differentiation with respect to the Levi-Civita connection ^{g}∇. The Tchebychev tensor T̂ has the symmetry property T̂_{i;j} = T̂_{j;i}; see [12] for further details. Both the induced connection ^{1}∇ and the conormal connection ^{2}∇ are torsion free Ricci symmetric connections on T M. They are conjugate relative to g, i.e.

\frac{1}{2}(^{1}∇ +^{2}∇) = ^{g}∇.

We call the triple {^{1}∇, g,^{2}∇} a conjugate triple. Let A be the difference tensor defined above. We then have ^{1}∇ = ^{g}∇ + A and ^{2}∇ = ^{g}∇ − A. Let H := m^{−1}B_{ii} be the normalized mean curvature and recall the notation for the Ricci tensor from section §1.2. We have:

\begin{matrix}ρ_{^{1}∇} = mHg − B, ρ_{^{2}∇} = (m − 1)B, and \\ tr_{g}(ρ_{^{1}∇}) = tr_{g}(ρ_{^{2}∇}) = m(m − 1)H. \end{matrix}

2.2 Definition.

(1) The relative support function ϱ with respect to x_{0} ∈ V is given by ϱ = −⟨X, x − x_{0}⟩. If b ∈ V, define a generalized spherical function F := ⟨X, b⟩.

(2) A relative normalization is said to be equiaffine if the Tchebychev form T̂ vanishes. A nondegenerate hypersurface with equiaffine normalization is called a Blaschke hypersurface. We denote the support function of this geometry by ϱ(e).

(3) Consider a non degenerate hypersurface x : M ⟶ V such that its position vector is transversal. Then y(c) := −x is called the centroaffine normal. Following Nomizu we call such a hypersurface together with its centroaffine normalization {X(c), y(c)} a centroaffine hypersurface. The associated geometry of {x, X(c), y(c)} is invariant under the group GL(n + 1, ℝ). Then

B̂ := B̂(c) = g(c) = g, mH(c) = m, and T̂(c) = \frac{m + 2}{2m}d lg |ϱ(e)|.

Recall that x is a proper affine sphere with center at O ∈ V if and only if T̂(c) ≡ 0. For a locally strongly convex hypersurface we choose the orientation such that ϱ(e) > 0.

2.3 Operators of Laplace type defined by relative normalizations. The connections ^{1}∇ and ^{2}∇ determined by a relative normalization (x, X, y) are torsion free Ricci symmetric connections. We assume the associated metric g is positive definite and use the construction described in §1.3 to define operators of Laplace type ^{1}D and ^{2}D. These operators and their spectra are not affine invariants of the embedding x since they depend on the relative normalization chosen. However, Theorem 1.5 shows that the coefficients a_{m} where m := dim(M) are affine invariants. To study these invariants, we recall some notations and results from [2].

2.4 Lemma. Let ϵ_{1} = 1 and let ϵ_{2} = −1. We adopt the notation of Lemma 1.6.

(1) We have Ω(^{r}D) = 0, Θ(^{r}D) := ∇ − ^{g}∇ = ϵ_{r}A, ω(^{r}D) = −\frac{1}{2}ϵ_{r}mT̂, and E(^{r}D) = mH − \frac{1}{4}m^{2}|T̂|_{g}^{2} + \frac{1}{2}ϵ_{r}mT̂_{i;i}; here r = 1, 2 and i = 1,…, m.

(2) Let^{g}D be determined by the Levi-Civita connection associated to the metric g. Then Ω(^{g}D) = 0, Θ(^{g}D) = 0, ω(^{g}D) = 0, and E(^{g}D) = \frac{1}{m−1}τ_{g}.

We use Lemma 2.4 and Theorem 1.7 to determine the heat equation asymptotics in this setting:

2.5 Theorem. Let M be a manifold with smooth boundary. Let {x, X, y} be a relative normalization of a regular embedding x. Assume the associated quadratic form g is positive definite so M is strictly convex.

(1) a_{0}(^{r}D) = (4π)^{−m/2}vol(M).

(2) a_{1}(^{r}D) = −\frac{1}{4}(π)^{−(m−1)/2}vol(∂M).

(3) a_{2}(^{r}D) = (4π)^{−m/2} ∫_{M}\{\frac{1}{6}τ_{g} + mH − \frac{1}{4}m^{2}|T̂|_{g}^{2} + \frac{1}{2}ϵ_{r}mT̂_{i;i}\}dx

+\frac{1}{6}(4π)^{−m/2} ∫_{∂M} 2L_{aa}dy.

(4) a_{3}(^{r}D) = −\frac{1}{384}(4π)^{−(m−1)/2} ∫_{∂M}\{96(mH − \frac{1}{4}m^{2}|T̂|_{g}^{2} + \frac{1}{2}ϵ_{r}mT̂_{i;i}) +16τ_{g} + 8R_{g,amam} + 7L_{aa}L_{bb} − 10L_{ab}L_{ab}}dy.

(5) a_{4}(^{r}D) = (4π)^{−m/2} \frac{1}{360} ∫_{M}\{60τ_{g}(mH − \frac{1}{4}m^{2}|T̂|_{g}^{2} + \frac{1}{2}ϵ_{r}mT̂_{i;i})

+180(mH − \frac{1}{4}m^{2}|T̂|_{g}^{2} + \frac{1}{2}ϵ_{r}mT̂_{i;i})^{2}
+60(mH − \frac{1}{4}m^{2}|T̂|_{g}^{2} + \frac{1}{2}ϵ_{r}mT̂_{i;i})_{;jj}
+12τ_{g;kk} + 5τ_{g}^{2} − 2|ρ_{g}|_{g}^{2} + 2|R_{g}|_{g}^{2}}dx
+\frac{1}{360}(4π)^{−m/2} ∫_{∂M}\{−120(mH − \frac{1}{4}m^{2}|T̂|_{g}^{2} + \frac{1}{2}ϵ_{r}mT̂_{i;i})_{;m}
−18τ_{g;m} + 120(mH − \frac{1}{4}m^{2}|T̂|_{g}^{2} + \frac{1}{2}ϵ_{r}mT̂_{i;i})L_{aa}

+20τ_{g}L_{aa} +4R_{g,amam}L_{bb} −12R_{g,ambm}L_{ab} +4R_{g,abcb}L_{ac} +24L_{aa:bb}

+\frac{40}{21}L_{aa}L_{bb}L_{cc} − \frac{88}{7}L_{ab}L_{ab}L_{cc} + \frac{320}{21}L_{ab}L_{bc}L_{ca}dy\}.

§3 Affine geometries where the operators ^{1}D and ^{2}D are self-adjoint

We begin our study with the following result:

3.1 Theorem. Let M be a manifold with smooth boundary. Let {x, X, y} be a relative normalization of a regular embedding x. Assume the associated quadratic form g is positive definite so M is locally strictly convex.

(1) The operator ^{1}D +^{2}D is self adjoint.

(2) Let x be a hyperovaloid and let {x, X, y} be a relative normalization. The

following assertions are equivalent: 2-a) The operator ^{1}D is self-adjoint. 2-b) The operator ^{2}D is self-adjoint. 2-c) We have ^{1}D = ^{2}D. 2-d) The Tchebychev tensor T vanishes identically.

(3) The Tchebychev tensor T vanishes identically if and only if the relative normalization {x, X, y} is equiaffine.

(4) We have the identity: ∫_{M}(f ·^{1}Df̃− f̃·^{1}Df) = ∫_{M}(f̃·^{2}Df − f ·^{2}Df̃).

Proof. Let Δ_{g} be the scalar Laplacian defined by the metric g. Since ^{1}∇ = ^{g}∇+A, since ^{2}∇ = ^{g}∇ − A, and since T = tr_{g} A, we see that

1Df = Δ_{g}f − mT^{r}f_{;r} + mHf and ^{2}Df = Δ_{g}f + mT^{r}f_{;r} + mHf.

Thus ^{1}D + ^{2}D = 2Δ_{g} + 2mHf. Since 2mH is a term of order zero and 2Δ_{g} is self-adjoint, the first assertion follows. Let Vf := T^{r}f_{;r}; this operator is selfadjoint if and only if the tensor T vanishes. Assertion (2) now follows. We refer to Proposition 4.13 in [8] for the proof of assertion (3). Assertion (4) follows since the operator ^{1}D +^{2}D is self-adjoint. □

We use similar techniques to prove the next result

3.2 Theorem.

(1) Let {x, X, y} be the Euclidean normalization. Then the following assertions

are equivalent:

1-a) The Gauss-Kronecker curvature K = K_{n} is constant. 1-b) We have ^{1}D = ^{2}D. 1-c) The operator ^{1}D or the operator ^{2}D is self-adjoint.

(2) Let x be a compact centroaffine hypersurface with nonempty boundary.

The following assertions are equivalent: 2-a) We have ^{1}D = ^{2}D. 2-b) We have that ^{1}D or ^{2}D are self-adjoint. 2-c) We have that x is a proper affine sphere.

(3) Let x be a compact centroaffine hypersurface without boundary. Then the

following assertions are equivalent: 3-a) We have ^{1}D = ^{2}D. 3-b) We have that ^{1}D or ^{2}D is self-adjoint. 3-c) We have that x is a hyperellipsoid.

Proof. For a hypersurface with Euclidean normalization non-degeneracy means that

K = K_{n} ≠ 0 and T = −\frac{1}{2n}d lg |K|;

see [13], (6.1.2.1) for details. Thus if K is constant and non-zero, we have T vanishes identically, ^{1}D = ^{2}D, and these operators are self-adjoint. On the other hand, if K ≠ 0 and if one of the other conditions is satisfied, then necessarily K is constant. This proves the first assertion.

If the hypothesis of (2) are satisfied, one can see that T (c) ≡ 0 so that therefore x is a proper affine sphere.

If x is compact without boundary a proof like that in case (2) together with the well known result of Blaschke and Deicke [8, p. 121] imply the third assertion. □

§4 Spectral invariants of affine geometry

In this section we use the heat equation asymptotics of the operators ^{r}D (r = 1, 2) and ^{g}D on hypersurfaces M with non-empty boundary immersed in an affine space A to study their geometry.

Recall that if x is a compact, locally strongly convex Blaschke hypersurface, then ^{1}D = ^{2}D. We use Theorem 2.5 to establish the following Lemma:

4.1 Lemma. Let x be a compact, locally strongly convex Blaschke hypersurface with boundary. Then we have

(1) a_{0}(D) = (4π)^{−m/2}vol(M).

(2) a_{1}(D) = −\frac{1}{4}(4π)^{−(m−1)/2}vol(∂M).

(3) a_{2}(D) = \frac{1}{6}(4π)^{−m/2}\{∫_{M}(τ_{g} + 6mH)dx + 2 ∫_{∂M} L_{aa}dy\}.

(4) a_{3}(D) = −\frac{1}{384}(4π)^{−(m−1)/2} ∫_{∂M}\{96mH + 16τ_{g} + 8R_{g,amam} + 7L_{aa}L_{bb}

−10L_{ab}L_{ab}}dy.

(5) a_{4}(D) = (360)^{−1}(4π)^{−m/2}{∫_{M}(60mHτ_{g} + 180m^{2}H^{2} + 60mH_{;jj} + 12τ_{g;kk} +5τ_{g}^{2} −2|ρ_{g}|_{g}^{2} +2|R_{g}|_{g}^{2})dx+∫_{∂M}(−120mH_{;m} −18τ_{g;m} +120mHL_{aa}

+20τ_{g}L_{aa} + 4R_{g,amam}L_{bb} − 12R_{g,ambm}L_{ab} + 4R_{g,abcb}L_{ac}
+\frac{40}{21}L_{aa}L_{bb}L_{cc} − \frac{88}{7}L_{ab}L_{ab}L_{cc} + \frac{320}{21}L_{ab}L_{bc}L_{ca})dy\}.

We can use this Lemma to draw the following conclusion:

4.2 Theorem. Let x be a compact, locally strongly convex Blaschke hypersurface with boundary. Then:

(1) Let (x, X, y) be a relative normalization. Let c(m) := 384(4π)^{(m−1)/2}. 1-a) If m < 5, then c(m){a_{3}(D) − a_{3}(^{g}D)} ≤ ∫_{∂M} 96mJ.

1-b) If m = 5, then c(m){a_{3}(D) − a_{3}(^{g}D)} = ∫_{∂M} 96mJ.
1-c) If m > 5, then c(m){a_{3}(D) − a_{3}(^{g}D)} ≥ ∫_{∂M} 96mJ.

(2) Equality in assertions (1-a) and (1-c) holds if and only if the normalization is equiaffine.

(3) If the normalization is equiaffine, then a_{3}(D) − a_{3}(^{g}D) ≥ 0. Equality holds if and only if ∫_{∂M} J = 0.

Proof. For the operator ^{g}D, we have Θ = ω = Ω = 0 and E_{D} = \frac{1}{m−1}τ_{g}. Thus we may use the formulas given previously to see that

\begin{matrix}a_{3}(D) = −c(m) ∫_{∂M}\{\frac{16(5+m)}{m−1}τ_{g} + 8R_{g,amam} + 7L_{aa}L_{bb} − 10L_{ab}L_{ab}\} \\ c(m)(a_{3}(D) − a_{3}(^{g}D) = ∫_{∂M}\{\frac{96}{m−1}τ_{g} − 96(mH − \frac{1}{4}m^{2}|T̂|^{2})\} \end{matrix}

We use the Theorema egregium in relative geometry (see [13], 4.12.2.2) to see that

κ = J + H − \frac{m}{m−1}|T̂|^{2} i.e. τ_{g} = m(m − 1)(J + H) − m^{2}|T̂|^{2}.

We combine these two displays to see that

c(m)(a_{3}(D) − a_{3}(^{g}D)) = 96m ∫_{∂M}\{J + \frac{m(m−5)}{4(m−1)}|T̂|^{2}\}.

As the metric g is positive definite, we have that J ≥ 0 and |T̂|^{2} ≥ 0; the first assertion holds. Recall that T̂ = 0 characterizes an equiaffine normalization. The second assertion now follows as J ≥ 0. □

4.3 Corollary. Let x be a compact, locally strongly convex Blaschke hypersurface with boundary. Assume that m ≥ 3, that the affine mean curvature H is constant on M, and that a_{3}(D) = a_{3}(^{g}D). We may then conclude that x(M) lies on a quadric.

Proof. The previous result shows that ∫_{∂M} J = 0. The result now follows from Theorem 3.1.6.5 in [8]. □

4.4 Remark. One can prove an analogous result assuming H_{1} := H ≠ 0 and the quotient \frac{H_{2}}{H_{1}} is a non-zero constant on M where H_{r} is the r-th (r = 1,…, m) normed elementary symmetric function of the affine principal curvatures (apply 3.1.6.8 in [8]).

4.5 Volumes of convex bodies. Let M be a compact Blaschke hyperovaloid without boundary. If f, f^{#} are smooth functions on M, then Theorem 3.1 implies the following integral formula for the operator D = ^{1}D = ^{2}D:

(1) ∫ D(ff^{#}) = ∫ {fDf^{#} + ⟨gradf, gradf^{#}⟩} = ∫ {f^{#}Df + ⟨gradf, gradf^{#}⟩}.

Let ϱ and F be as defined in §2.2. We then have Dϱ = ^{2}Dϱ = m and DF = ^{2}DF = 0 on M. We refer to [13, §4.13] for details. Recall that, for a hyperovaloid and for any choice of basepoint x_{0} ∈ V, the volume of the convex body K enclosed by M is given by

vol(K) = \frac{1}{m+1} ∫_{M} ϱ.

4.6 Theorem. Let x : M ⟶ A be a Blaschke hyperovaloid. Then

(1) We have m ∫ Hϱ^{2} = m(m + 1)vol(K) + ∫ ∥gradϱ∥^{2}.

(2) We have ∫ Hϱ^{2} ≥ (m + 1)vol(K). Furthermore, equality holds if and only if x is a hyperellipsoid.

Proof. Let 1 ∈ C^{∞}(M) denote the constant function. Then D1 = mH. Note D is self-adjoint and the boundary of M is empty. We use equation (1) and the observations made in §4.5 to see that

m ∫_{M} ϱ^{2}H = ∫_{M} ϱ^{2}D1 = ∫_{M} D(ϱ^{2}) = ∫_{M}(ϱDϱ + ∥gradϱ∥^{2})
= ∫_{M}(mϱ + ∥gradϱ∥^{2}) = m(m + 1)vol(K) + ∫_{M} ∥gradϱ∥^{2}.

Assume that equality holds in the second assertion. Then ϱ = const. We apply Lemma 7.2.4 of [13] to see that x is an affine sphere. We use a Theorem of Blaschke and Deicke (see Theorem 2.4.7 [7]) to see x is a hyperellipsoid. □

Recall the affine isoperimetric inequality for Blaschke hyperovaloids ([8], p. 237). Denote by σ_{m+1} the volume of the unit ball in Euclidean (m + 1)-space. Let Area := ∫ 1 be the affine area of the hyperovaloid. We then have the inequality

(Area)^{m+2} ≤ {(m + 1)σ_{m+1}}^{2} · {(m + 1)vol(K)}^{m}.

Equality holds exactly for hyperellipsoids. We can now establish the result:

4.7 Corollary. Let x be a Blaschke hyperovaloid.

(1) We have (Area)^{m+2} ≤ {(m + 1)σ_{m+1}}^{2} · {∫ Hϱ^{2}}^{m}; equality holds if and only if x is a hyperellipsoid.

(2) We have {∫ Hϱ}^{m+2} ≤ {(m + 1)σ_{m+1}}^{2} · {∫ Hϱ^{2}}^{m}; equality holds if and only if x is a hyperellipsoid.

(3) Assume that the affine Weingarten operator has maximal rank. This implies that the m-th curvature function, the affine Gauss-Kronecker curvature, is nonzero. Then ∫ \frac{H_{m−1}}{H_{m}} ≤ ∫ Hϱ^{2} ; equality holds exactly for hyperellipsoids.

Proof. The first assertion is obvious from the previous discussion; the second assertion follows from the affine Minkowski formula ∫ 1 = ∫ Hϱ ([8, p. 165]) and the third from a related formula ∫ \frac{H_{m−1}}{H_{m}} = ∫ ϱ (l.c., p.169). For the discussion of equality compare the proof of Theorem 4.6. □

We now turn our attention to centroaffine normalizations. The following formula are immediate from our previous calculations.

4.8 Lemma. Let x have centroaffine normalization. We have

(1) a_{0}(^{1}D) = (4π)^{−m/2}vol(M).

(2) a_{1}(^{1}D) = −\frac{1}{4}(4π)^{−(m−1)/2}vol(∂M).

(3) a_{2}(^{1}D) = (4π)^{−m/2}\{∫_{M}(\frac{1}{6}τ_{g} − \frac{1}{4}m^{2}|T̂|^{2} + \frac{1}{2}mT̂_{i;i})dx + 2 ∫_{∂M} L_{aa}dy\}

+a_{0}(^{1}D).

(4) a_{3}(^{1}D) = −\frac{1}{384}(4π)^{−(m−1)/2} ∫_{∂M}\{−24m^{2}|T̂|^{2} + 16τ_{g} + 8R_{g,amam} + 7L_{aa}L_{bb}

−10L_{ab}L_{ab}}dy + a_{1}(^{1}D).

§5 The geometry of affine Gauss maps

In this section we consider a Blaschke hypersurface {x, X, y} and its two affine Gauss maps X : M ⟶ V^{∗} and y : M ⟶ V, see [13, §4.6]. Then X is an immersion with transversal position vector (also denoted by X), while y is an immersion if and only if the equiaffine Weingarten operator B satisfies rank(B) = m. In the latter case, both Gauss maps define centroaffine hypersurfaces in the sense of §2.2 above, i.e. X̌ := −X and y̌ := −y are their centroaffine normals, respectively.

If rank(B) = m the Gauss structure equations take the form

\begin{aligned}¯∇_{v}dX(w) & = dX(^{2}∇_{v}w) + h(X̌)(v, w)X̌ \\ ¯∇_{v}dy(w) & = dy(^{1} ˇ∇_{v}w) + h(y̌)(v, w)y̌. \end{aligned}

The centroaffine metrics of both hypersurfaces X and y coincide. This means that h(X̌) = B̂ = h(y̌) and we have

5.1 Lemma.

(1) We have that {^{1} ˇ∇, B̂,^{2}∇} is a conjugate triple. The connection ^{1} ˇ∇ is torsion-free and Ricci symmetric and satisfies the relation given in [10, section 5]: ^{1} ˇ∇_{u}v = B^{−1}(^{1}∇_{u}(Bv)).

(2) We have that ^{1} ˇ∇ is the induced connection of the Gauss map y, and that 2∇ is the associated conormal connection.

(3) We have that ^{2}∇ is the induced connection of the conormal Gauss map X, and that ^{1} ˇ∇ is the associated conormal connection.

5.2 Lemma. Let {x, X, y} be a Blaschke hypersurface with rank(B) = m. Then we have the following statements are equivalent:

(1) The equiaffine Gauss-Kronecker curvature H_{m} = H_{m}(e) = det(B) is a nonzero constant.

(2) The map y defines a proper affine sphere.

(3) The map X defines a proper affine sphere.

(4) The Tchebychev field Ť = Ť(y) vanishes.

Proof. Lemma 5.2 follows from the relation 2mŤ = d lg |H_{m}(e)|. This follows from Lemma 5.1 and the result T (e) ≡ 0; see [7, p. 182] for further details. □

For a Blaschke hypersurface with rank(B) = m and associated conjugate triple {^{1} ˇ∇, B̂,^{2}∇} we have associated operators ^{1}Ď, ^{2}D and B̂D according to the definitions in section 1.3 above. The following result is proved analogously with previously established results:

5.3 Theorem. Let x be a Blaschke hyperovaloid with rank(B) = m. Then

(1) We have that the operators ^{1}Ď and ^{2}Ď = ^{2}D satisfy the global conjugacy relation: ∫_{M}(f ·^{1}Ďf̃− f̃·^{1}Ďf) = ∫_{M}(f̃·^{2}Ďf − f ·^{2}Ďf̃).

(2) The following assertions are equivalent:

2-a) The operator ^{1}Ď is self-adjoint.

2-b) The operator ^{2}D = ^{2}Ď is self-adjoint. 2-c) The immersion x defines a hyperellipsoid.

Proof. Since the centroaffine normalization is a relative normalization we apply Theorem 3.1 to establish the first assertion. We have that the operator ^{r}Ď is selfadjoint if and only if the map y defines a hyperellipsoid. We use Lemma 5.2 to see that this implies H_{m}(e) = const. We use [8, Theorem 3.1.26] to conclude x is a hyperellipsoid and show 2-a) or 2-b) implies 2-c); the converse is immediate. □

5.4 Theorem. Let x : M ⟶ A be a locally strongly convex Blaschke hypersurface with boundary. Then the following assertions are equivalent.

(1) We have H_{m}(e) = const ≠ 0. (3) We have the operator ^{1}Ď is self-adjoint.

(2) We have ^{1}Ď = ^{2}Ď. (4) We have the operator ^{2}Ď is self-adjoint.

Proof. Use the relationship 2mŤ = d lg |H_{m}(e)| given above; the proof then is analogous to the proof of Theorem 3.2 (1). □

References

[1] W. Blaschke, Vorlesungen über Differentialgeometrie; II. Affine Differentialgeometrie, Springer-Verlag, Berlin, 1923.

[2] N. Bokan, P.B. Gilkey and U. Simon, Applications of spectral geometry to affine and projective geometry, Beiträge zur Algebra und Geometrie 35 (1994), 283–314.

[3] T. Branson and P.B. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. in PDE 15 (1990), 245–272.

[4] T. Branson, P. Gilkey, and D. Vassilevich, The asymptotics of the Laplacian on a manifold with boundary II, Suppl. fasc. 2, Boll. Unione Mat. Ital. 11-B (1997), 39–67.

[5] E. Calabi, Hypersurfaces with maximal affinely invariant area, Amer. J. Math. 104 (1982), 91–126.

[6] P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index theorem (2^{nd} edition), ISBN 0-8493-7874-4, CRC Press, Boca Raton, Florida, 1994.

[7] A.M. Li, U. Simon and B. Chen, A two step Monge-Ampère procedure for solving a fourth order PDE for affine hypersurfaces with constant curvature, J. Reine Ang. Math. 487 (1997), 179–200.

[8] A.M. Li, U. Simon and G. Zhao, Global affine differential geometry of hypersurfaces, De Gruyter, Berlin and New York, 1993.

[9] K. Nomizu and T. Sasaki, Affine differential geometry, Cambridge Univ. Press, 1993.

[10] K. Nomizu and U. Simon, Notes on conjugate connections, Geom.& Top. of Submanifolds IV (eds.: F. Dillen et al.) World Scientific Singapore (1992), 152–172.

[11] P.A. and A.P. Schirokow, Affine Differentialgeometrie, Teubner Verlag, Leipzig, 1962.

[12] U. Simon, Connections and conformal structure in affine differential geometry, Differential geometry and its Applications, Proc. of the Conf. August 24–30, (1986), Brno, Czechoslovakia, 315–327.

[13] U. Simon, A. Schwenk-Schellschmidt and H. Viesel, Introduction to the affine differential geometry of hypersurfaces, Lecture Notes Science University Tokyo, 1991, ISBN 3798315299.

Neda Bokan  Peter Gilkey  Udo Simon
Matematički fakultet  Mathematics Department  Technische Universität Berlin
Studentski trg 16  University of Oregon  Strasse des 17. Juni 135
11001 Beograd, p.p. 550  Eugene Oregon 97403  D-10623 Berlin
Yugoslavia  USA  Deutschland

neda@matf.bg.ac.yu       gilkey@darkwing.uoregon.edu    simon@math.tu-berlin.de

(Received 25 05 1998)