Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 2, 437-458.

A New Intrinsic Curvature Invariant for Centroaffine Hypersurfaces

Christine Scharlach, Udo Simon Leopold Verstraelen, Luc Vrancken

Fachbereich Mathematik, MA 8-3, Technische Universität Berlin Straße des 17. Juni 136, D - 10623 Berlin, Germany Departement Wiskunde, Katholieke Universiteit Leuven Celestijnenlaan 200 B, B - 3001 Leuven, Belgium

Abstract. We consider centroaffine hypersurfaces M^{n} and introduce a new intrinsic curvature invariant of the centroaffine metric. For n > 2 we derive an inequality between this invariant and the square norm of the Tchebychev field. We study the class of hypersurfaces which realize the equality and obtain a wide class of examples and a partial classification. Furthermore we discuss our results under polarization of the hypersurface and finally relate them to recent asymptotic spectral results on second order Laplace type operators in affine differential geometry. MSC 1991: 53A15, 58G25 Key words and phrases. centroaffine hypersurfaces, curvature invariant, proper affine hyperspheres, polar hypersurfaces, asymptotic spectral geometry of Laplace type operators

1. Introduction

In this paper we study definite centroaffine hypersurfaces M^{n} in R^{n+1}, n ≥ 2. For such hypersurfaces, the centroaffine normal induces a definite symmetric bilinear form h on M^{n}. Therefore we can consider (M^{n}, h) as a Riemannian manifold and we can define the invariant ϵ_{M} by

ϵ_{M}(p) = \frac{n(n−1)}{2} κ̂(p) − (sup K̂)(p),

where κ̂(p) is the normalized scalar curvature, K̂(π) is the sectional curvature of a plane section π in T_{p}M and

(sup K̂)(p) := sup{K̂(π) | π a plane section in T_{p}M}.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1996 Heldermann Verlag Since ϵ_{M} = 0 for n = 2, a study of ϵ_{M} is only of interest for n ≥ 3. A similar invariant, replacing supremum by infimum, was first introduced by B. Y. Chen (see [2]) in order to study submanifolds of real space forms. For further results see also [3], [4], [8] and [5].

The paper is organized as follows. In Section 2, we recall the basic facts about centroaffine hypersurfaces and introduce the necessary notations. We also introduce some notions about distributions on Riemannian manifolds which we will need in Section 5. In Section 3, we derive an inequality relating the invariant ϵ_{M} with the length of the Tchebychev vector field T. For a positive definite centroaffine hypersurface M^{n} in R^{n+1} we show that

ϵ_{M} ≥ −\frac{n^{2}(n−2)}{2(n−1)}h(T^{♯}, T^{♯}) + \frac{1}{2}(n + 1)(n − 2).  (1.1)

Then we continue with investigating those centroaffine hypersurfaces which realize the equality in (1.1). Clearly, if M is two-dimensional, the equality is trivially satisfied. If the dimension of M^{n} is greater than two, then hypersurfaces satisfying the equality can be characterized by a special property of the difference tensor K from (2.5) as shown in Section 3. In particular, for these hypersurfaces the Tchebychev vector field vanishes identically, which means they are proper affine hyperspheres in the sense of Blaschke.

In Section 4, we define on such hypersurfaces a distribution D by

D_{p} = {v ∈ T_{p}M|K(v, w) = 0 ∀w ∈ T_{p}M}.

We show that the distribution D is differentiable (on an open and dense subset) and that (on that subset) D is integrable and autoparallel. We use these results in Section 5 to obtain classification theorems under the additional assumption that either M^{n} has constant scalar curvature (Thm. 5.3) or the distribution D^{⊥} is integrable (Thm. 5.8). In particular, we obtain a wide class of examples related to equiaffine elliptic spheres in R^{3}, realizing the equality in (1.1). Similar results can be obtained for negative definite centroaffine hypersurfaces and for positive definite hypersurfaces in relative geometry. These will be mentioned throughout the paper. Some of the extensions need subtle proofs which will appear in [15]. Finally some remarks about polarization and asymptotic spectral invariants will be made in Section 6 and 7.

2. Preliminaries

First, we recall the basic facts about centroaffine hypersurfaces. For more details and proofs see [14, Chapter 6]. Let M^{n} be a n-dimensional C^{∞}-manifold and let

f : M^{n} → R^{n+1}

be a non-degenerate hypersurface immersion whose position vector is nowhere tangent. Then f can be regarded as a transversal field along itself and we call ξ = −f the centroaffine normal. Following Nomizu, we call f together with this normalisation a centroaffine hypersurface. We get the centroaffine structure equations:

D_{X}f_{∗}(Y) = f_{∗}(∇_{X}Y) + h(X, Y)(−f),  (2.1)
D_{X}ξ = −f_{∗}(X).  (2.2)

Here D denotes the canonical flat connection of R^{n+1}, ∇ is a torsionfree connection on M^{n}, called the induced centroaffine connection, and h is a non-degenerate symmetric (0,2)- tensor field, called the centroaffine metric. The corresponding integrability conditions (equations of Gauss (2.3) and Codazzi (2.4)) read

R(X, Y)Z = h(Y, Z)X − h(X, Z)Y,  (2.3)
(∇_{X}h)(Y, Z) = (∇_{Y} h)(X, Z).  (2.4)

If we define C(X, Y, Z) = (∇_{X}h)(Y, Z), we get a totally symmetric (0,3)-tensor field, called the cubic form.

From now on we restrict our attention to so called definite centroaffine hypersurfaces; that means h is definite. In order to obtain a more unified approach, we take for negative definite hypersurfaces ξ = f as affine normal, so that the corresponding second fundamental form h again becomes positive definite; (2.1), (2.2) and (2.3) then read

D_{X}f_{∗}(Y) = f_{∗}(∇_{X}Y) + h(X, Y)f,  (2.1′)
D_{X}ξ = f_{∗}(X),  (2.2′)
R(X, Y)Z = h(Y, Z)X − h(X, Z)Y.  (2.3′)

Denote by ˆ∇ the Levi-Civita connection of h, by R̂, R̂ic and κ̂ the curvature tensor, the Ricci tensor and the normalized scalar curvature of ˆ∇, respectively. The difference tensor K, which is defined by

K_{X}Y = K(X, Y) = ∇_{X}Y − ˆ∇_{X}Y,  (2.5)

is a symmetric (1,2)-tensor field related to the cubic form by C(X, Y, Z) = −2h(K_{X}Y, Z). Hence, for every X, K_{X} is selfadjoint with respect to h. Furthermore, we define the Tchebychev form T, the Tchebychev vector field T^{♯} and the Pick invariant J by

nT (X) = trace K_{X},  (2.6)
h(T^{♯}, X) = T (X),  (2.7)
4n(n − 1)J = 4h(K, K) = h(C, C).  (2.8)

It is also well known that in centroaffine geometry

h(K_{X}Y, Z) = h(Y, K_{X}Z),  (2.9)
R̂(X, Y)Z = K_{Y} K_{X}Z − K_{X}K_{Y} Z + ε(h(Y, Z)X − h(X, Z)Y),  (2.10)
( ˆ∇K)(X, Y, Z) = ( ˆ∇K)(Y, X, Z),  (2.11)
κ̂ = J + ε − \frac{n}{n−1}h(T^{♯}, T^{♯}),  (2.12)

where ε = 1 (resp. ε = −1) if M is positive definite (resp. negative definite).

Next, we recall some definitions and properties of distributions on Riemannian manifolds. Details can be found e.g. in [11]. Let (M^{n}, h) be a Riemannian manifold with Levi Civita connection ˆ∇. Let E define a distribution on M^{n}. Denote by E^{⊥} its orthogonal distribution. Then E is called parallel if ˆ∇_{X}Y ∈ E for all vectorfields X tangent to M^{n} and Y ∈ E; it is called autoparallel if ˆ∇_{X}Y ∈ E for all X, Y ∈ E; it is called totally umbilical if there exists a vector H ∈ E^{⊥} such that h( ˆ∇_{X}Y, Z) = h(X, Y)h(H, Z) for all X, Y ∈ E and for all Z ∈ E^{⊥}, in this case H is called the mean curvature normal of the distribution E. We call E spherical if it is totally umbilical and its mean curvature normal H satisfies h( ˆ∇_{X}H, Z) = 0 for all X ∈ E and Z ∈ E^{⊥}. If E is autoparallel, totally umbilical or spherical, then E is involutive and all the leaves of the foliation of M^{n} induced by E are totally geodesic, totally umbilical or spherical respectively.

3. A Centroaffine Inequality and its Applications

Unless otherwise mentioned, M^{n} will denote a positive definite centroaffine hypersurface in R^{n+1}. Following the approach of Chen (see [2]), we will derive an inequality involving ϵ_{M}. We will need the following algebraic inequality from [2]:

Lemma 3.1. Let a_{1}, …, a_{n}, c be n + 1 real numbers, n ≥ 2, such that

(∑_{i=1}^{n} a_{i})^{2} = (n − 1)(∑_{i=1}^{n} a_{i}^{2} + c).  (3.1)

Then 2a_{1}a_{2} ≥ c, with equality holding if and only if a_{1} + a_{2} = a_{3} = … = a_{n}.

Theorem 3.2. Let M^{n} be an n-dimensional positive definite centroaffine hypersurface in R^{n+1}. Then

ϵ_{M} ≥ −\frac{n^{2}(n−2)}{2(n−1)}h(T^{♯}, T^{♯}) + \frac{1}{2}(n + 1)(n − 2).

Proof. Define the centroaffine invariant

δ := n(n − 1)κ̂ + \frac{n^{2}(n−2)}{n−1}h(T^{♯}, T^{♯}) − (n + 1)(n − 2).  (3.2)

Then (2.12) and (3.2) imply that

n^{2}h(T^{♯}, T^{♯}) = (n − 1) (h(K, K) − (δ − 2)) .  (3.3)

Let π ⊂ T_{p}M be a plane section. We choose an h-orthonormal frame {e_{1}, …, e_{n}} such that π = span{e_{1}, e_{2}} and an h-orthonormal frame {u_{1}, …, u_{n}} such that u_{1} is in the direction of the Tchebychev vector field T^{♯}. In the special case that T^{♯} = 0, we choose an arbitrary h-orthonormal frame {u_{1}, …, u_{n}}. We introduce the notation K_{ij}^{k} := h(K_{u_{k}}e_{i}, e_{j}); then (3.3) yields

(∑_{i=1}^{n} K_{ii}^{1})^{2} = (n − 1) (∑_{i=1}^{n}(K_{ii}^{1})^{2} + ∑_{i≠j}^{n}(K_{ij}^{1})^{2} + ∑_{r=2}^{n} ∑_{i,j=1}^{n}(K_{ij}^{r})^{2} − δ + 2) .  (3.4)

Applying now Lemma 3.1 gives

2K_{11}^{1}K_{22}^{1} ≥ ∑_{i≠j}^{n}(K_{ij}^{1})^{2} + ∑_{r=2}^{n} ∑_{i,j=1}^{n}(K_{ij}^{r})^{2} − δ + 2.  (3.5)

From (2.10), we have

K̂(π) = h(R̂(e_{1}, e_{2})e_{2}, e_{1}) = 1 − h(K(e_{1}, e_{1}), K(e_{2}, e_{2})) + h(K(e_{1}, e_{2}), K(e_{1}, e_{2})).

Therefore,

K̂(π) = 1 − K_{11}^{1}K_{22}^{1} − ∑_{r=2}^{n} K_{11}^{r}K_{22}^{r} + ∑_{r=1}^{n}(K_{12}^{r})^{2},

and we deduce from (3.5) that

K̂(π) ≤ 1 − \frac{1}{2} (∑_{i≠j}(K_{ij}^{1})^{2} + ∑_{r=2}^{n} ∑_{i,j=1}^{n}(K_{ij}^{r})^{2} − δ + 2) − ∑_{r=2}^{n} K_{11}^{r}K_{22}^{r} + ∑_{r=1}^{n}(K_{12}^{r})^{2}
= −\frac{1}{2} ∑_{r=2}^{n}(K_{11}^{r} + K_{22}^{r})^{2} − ∑_{r=1}^{n} ∑_{j=3}^{n} ((K_{1j}^{r})^{2} + (K_{2j}^{r})^{2})
− \frac{1}{2} ∑_{i≠j>2}^{n}(K_{ij}^{1})^{2} − \frac{1}{2} ∑_{r=2}^{n} ∑_{i,j>2}^{n}(K_{ij}^{r})^{2} + \frac{1}{2}δ  (3.6)
≤ \frac{1}{2}δ.

Combining (3.6) with (3.2) gives

K̂(π) ≤ \frac{n(n−1)}{2} κ̂ + \frac{n^{2}(n−2)}{2(n−1)}h(T^{♯}, T^{♯}) − \frac{1}{2}(n + 1)(n − 2).  (3.7)

Hence the assertion follows from the definition of ϵ_{M}.

Lemma 3.3. Let M^{n} be an n-dimensional positive definite centroaffine hypersurface in R^{n+1}. Then, M^{n} realizes the equality in (1.1) at a point p of M^{n} if and only if there exist tangent bases {e_{1}, …, e_{n}} and {u_{1}, …, u_{n}} of T_{p}M such that the linear maps K_{u_{i}} take, with respect to the basis {e_{1}, …, e_{n}}, the following form:

K_{u_{1}} = ( \begin{matrix}a & 0 & 0 & … & 0 \\ 0 & b & 0 & … & 0 \\ 0 & 0 & μ & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & μ \end{matrix} ) , K_{u_{r}} = ( \begin{matrix}K_{11}^{r} & K_{12}^{r} & 0 & … & 0 \\ K_{12}^{r} & −K_{11}^{r} & 0 & … & 0 \\ 0 & 0 & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 0 \end{matrix} ) , (r > 1),

where a + b = μ.

Proof. If M^{n} satisfies the equality in (1.1), then the inequalities in (3.5) and (3.6) become equalities. Therefore, we get that, for i, j > 2, r ≥ 2:

K_{1j}^{1} = K_{2j}^{1} = 0,
K_{ij}^{1} = 0,
K_{1j}^{r} = K_{2j}^{r} = K_{ij}^{r} = 0,
K_{11}^{2} + K_{22}^{2} = … = K_{11}^{n} + K_{22}^{n} = 0.

Since K_{u_{1}}: π → π is a symmetric linear operator, we can still choose e_{1} and e_{2} which diagonalize K_{u_{1}}, i.e. K_{12}^{1} = 0. Moreover, by applying Lemma 3.1, we have that K_{11}^{1} + K_{22}^{1} = K_{33}^{1} = … = K_{nn}^{1}.

From now on, we assume that M^{n} is a centroaffine hypersurface which satisfies the equality in (1.1). The following lemmas extract the information of Lemma 3.3 in a form independent of the choice of the basis. They will be used to show that the Tchebychev vector field T^{♯} vanishes identically; that means that the hypersurface is a proper affine hypersphere.

Lemma 3.4. Let M^{n} be an n-dimensional positive definite centroaffine hypersurface in R^{n+1} satisfying the equality in (1.1). If y is a tangent vector at a point p of M^{n}, then the symmetric linear operator K_{y} has at most three eigenvalues and the dimension of one of the eigenspaces is at least (n − 2). Moreover, the following cases can occur:

(1) K_{y} = 0;

(2) K_{y} has a non-zero real eigenvalue α with a 2-dimensional eigenspace and an eigenvalue −2α with an (n − 2)-dimensional eigenspace;

(3) K_{y} has zero as an eigenvalue with a 1-dimensional eigenspace and a non-zero real eigenvalue α with an (n − 1)-dimensional eigenspace;

(4) K_{y} has a non-zero eigenvalue α with a 1-dimensional eigenspace, a non-zero eigenvalue β with a 1-dimensional eigenspace and an eigenvalue α + β with an (n − 2)- dimensional eigenspace.

Lemma 3.5. Let M^{n} be as in Lemma 3.4, let p ∈ M and let y be a tangent vector at p. Suppose that K_{y} satisfies case 4 of Lemma 3.4. Let v be an eigenvector belonging to one of the 1-dimensional eigenspaces and denote the (n − 2)-dimensional eigenspace by W. Then (1) V = {K_{u}v|u ∈ T_{p}M} is a vector space containing v. The dimension of V is at most

two and for all u′ ∈ T_{p}M, K_{u′}V ⊂ V,

(2) for all u′ ∈ T_{p}M, there exists a real number κ(u′) such that K_{u′}w = κ(u′)w for all w ∈ W.

Recall that the equiaffine support function ρ satisfies T = \frac{n+2}{2n}d log ρ (see (5.1), (5.1.3) and (6.3.2) in [14]). Now, we can prove the following theorem.

Theorem 3.6. Let M^{n} be a positive definite centroaffine hypersurface in R^{n+1}, n ≥ 3. If M^{n} realizes the equality in (1.1) at a point p of M^{n}, then the Tchebychev vector field T^{♯} vanishes at p and the equiaffine support function is stationary at p. Moreover, if M^{n} realizes the equality for every p ∈ M^{n}, then M^{n} is a proper affine hypersphere centered at the origin.

Proof. Our proof is based on a method from [9] which was extended in [16]. Let us assume that the equality holds at p ∈ M and that the dimension n > 2. If for all tangent vectors u, v and w, h(K(u, v), w) = 0, there is nothing to prove. So we assume that this is not the case. We define a function f_{p} : U_{p}M → R by f_{p}(v) = h(K(v, v), v), where U_{p}M = {v ∈ T_{p}M|h(v, v) = 1}. Since U_{p}M is a compact set, there exists a vector u in U_{p}M such that f_{p} attains an absolute maximum at u. Then f_{p}(u) > 0 and h(K(u, u), w) = 0, for all w orthogonal to u. So u is an eigenvector of the symmetric operator K_{u} (cp. (2.5)). We put e_{1} = u and choose e_{2}, …, e_{n} in such a way that {e_{1}, …, e_{n}} is an orthonormal basis of T_{p}M and e_{i} is an eigenvector of K_{e_{1}} with eigenvalue λ_{i}. Since f_{p} attains an absolute maximum in e_{1}, the function f_{i} : R → R, i ≥ 2, defined by f_{i}(t) = f(cos te_{1} + sin te_{i}) has a relative maximum at t = 0. So, by computing f′′_{i}(0), we get λ_{1} − 2λ_{i} ≥ 0. Since λ_{1} > 0, this implies λ_{1} ≠ λ_{i} for all i ≥ 2. Now we have K_{e_{i}}e_{1} = K_{e_{1}}e_{i} = λ_{i}e_{i}, so by Lemma 3.5 with y = v = e_{1}, we may assume λ_{3} = … = λ_{n} = 0. From Lemma 3.4, we get λ_{1} + λ_{2} = 0. So, we have for i ≥ 3:

K_{e_{1}}e_{1} = λ_{1}e_{1},
K_{e_{1}}e_{2} = −λ_{1}e_{2},
K_{e_{1}}e_{i} = 0.

Since K_{e_{2}}e_{1} = −λ_{1}e_{2}, we also get from Lemma 3.5 (with y = v = e_{1}) that the space spanned by e_{1} and e_{2} is invariant under all K_{u′}, where u′ is an arbitrary tangent vector. Using Lemma 3.4 and Lemma 3.5, we can write for i ≥ 3:

K_{e_{2}}e_{1} = −λ_{1}e_{2},
K_{e_{2}}e_{2} = −λ_{1}e_{1} + αe_{2},
K_{e_{2}}e_{i} = αe_{i}.

However, since K_{e_{3}}e_{2} = K_{e_{2}}e_{3} = αe_{3}, we get that α = 0.

Consider now K_{e_{k}}, with k ≥ 3. Since 0 = K_{e_{k}}e_{1} = K_{e_{k}}e_{2}, the space spanned by {e_{1}, e_{2}} is an eigenspace with eigenvalue 0 and span{e_{3}, …, e_{n}} also is an eigenspace. Lemma 3.4 now implies that K_{e_{k}} = 0. Finally, we obtain nT (e_{i}) = trace K_{e_{i}} = 0 for all i.

The following lemma is an immediate consequence of the proof of Theorem 3.6 and (3.2).

Lemma 3.7. Let M^{n}, n ≥ 3, be a positive definite centroaffine hypersurface satisfying the equality in (1.1) and let p ∈ M. Then, there exist an h-orthonormal basis {e_{1}, …, e_{n}} of T_{p}M and a number λ such that for i, j ≥ 3:

\begin{aligned}K(e_{1}, e_{1}) & = λe_{1}, \\ K(e_{2}, e_{2}) & = −λe_{1}, \end{aligned}  \begin{aligned}K(e_{1}, e_{2}) & = −λe_{2}, \\ K(e_{2}, e_{i}) & = 0, \end{aligned}  \begin{aligned}K(e_{1}, e_{i}) & = 0, \\ K(e_{i}, e_{j}) & = 0. \end{aligned}

The function λ(= λ_{1}) is a continuous positive function determined by −4λ^{2} = n(n − 1) − n(n − 1)κ̂.

In the case that M^{n} is a negative definite centroaffine hypersurface, taking f as affine normal and following exactly the same approach, we can prove:

Theorem 3.8. Let M^{n} be a negative definite centroaffine hypersurface in R^{n+1}, n ≥ 3. Then

ϵ_{M} ≥ −\frac{n^{2}(n−2)}{2(n−1)}h(T^{♯}, T^{♯}) − \frac{1}{2}(n + 1)(n − 2).  (3.8)

If M^{n} realizes the equality at a point p of M^{n}, then the Tchebychev vector field T^{♯} vanishes at p and the equiaffine support function is stationary at p. Moreover, if M^{n} realizes the equality for every p ∈ M^{n}, then M^{n} is a proper affine hypersphere centered at the origin.

Also, in the case that f: M → R^{n+1} is a (positive definite) hypersurface with relative affine normal ξ, in the sense of [14], a similar inequality can be derived. We denote by H the mean curvature and introduce a curvature like tensor by

R̃(X, Y)Z = \frac{1}{2}(R(X, Y)Z + R̄(X, Y)Z),

where R̄ is the curvature tensor associated with the conjugate connection ¯∇. We denote by K̃(π) the sectional curvature of a plane π with respect to this curvature tensor. Then,

sup(K̂(π) − K̃(π)) ≤ \frac{1}{2}n(n − 1)(κ̂ − H) + \frac{n^{2}(n−2)}{2(n−1)}h(T^{♯}, T^{♯}).  (3.9)

Similar computations as in Theorem 3.6 and Lemma 3.7 show that if M^{n} satisfies the equality in (3.9), then the Tchebychev vector field has to vanish identically (hence the relative normal is a constant multiple of the equiaffine normal) and at each point there exists an h-orthonormal basis {e_{1}, …, e_{n}} such that for i, j ≥ 3:

\begin{aligned}K(e_{1}, e_{1}) & = λe_{1}, \\ K(e_{2}, e_{2}) & = −λe_{1}, \end{aligned}  \begin{aligned}K(e_{1}, e_{2}) & = −λe_{2}, \\ K(e_{2}, e_{i}) & = 0, \end{aligned}  \begin{aligned}K(e_{1}, e_{i}) & = 0, \\ K(e_{i}, e_{j}) & = 0. \end{aligned}

4. A Differentiable Choice of a Basis

Throughout this section, we will assume that M^{n} is a centroaffine positive definite hypersurface satisfying the equality in (1.1). We define on such a hypersurface a distribution D by

D_{p} = {v ∈ T_{p}M|K(v, w) = 0 ∀w ∈ T_{p}M}.

Denote by V = {p ∈ M|K_{p} ≠ 0}. It is clear that V is an open subset. Denote by W the interior of M\V. It is well known that either W = ∅ or W is an open part of a quadric. Then, on V, D defines at each point p an (n − 2)-dimensional linear subset of T_{p}M and λ varies continuously. Let p ∈ V and let {e_{1}, …, e_{n}} be an h-orthonormal basis of T_{p}M satisfying Lemma 3.7. In the following we show that we can extend this basis to local vector fields {E_{1}, …, E_{n}} which satisfy Lemma 3.7 in a neighborhood of the point p of V. In order to do so, we again consider the function f_{q}(v) = h(K(v, v), v), q ∈ M (see the proof of Theorem 3.6).

Lemma 4.1. Let q ∈ M and let s_{q} be a critical value of f_{q}. Then s_{q} ∈ {−λ(q), 0, λ(q)}.

Proof. Let s_{q} be a critical value. Then there exists a v ∈ U_{p}M such that f_{q}(v) = s_{q} and h(K(v, v), w) = 0 for all w orthogonal to v. We take an orthonormal basis {e_{1}, …, e_{n}} satisfying Lemma 3.7. If necessary by changing {e_{3}, …, e_{n}}, we may assume that v = α_{1}e_{1} + α_{2}e_{2} + α_{3}e_{3}, α_{1}^{2} + α_{2}^{2} + α_{3}^{2} = 1. So K(v, v) = λ(q)(α_{1}^{2} − α_{2}^{2})e_{1} − 2λ(q)α_{1}α_{2}e_{2}. Then s_{q} = h(K(v, v), v) is a critical value of f_{q} if and only if the following two equations are satisfied:

\begin{aligned}0 & = h(K(v, v), α_{2}e_{1} − α_{1}e_{2}) = λ(q)(α_{1}^{2} − α_{2}^{2})α_{2} + 2λ(q)α_{1}^{2}α_{2} = λ(q)(3α_{1}^{2} − α_{2}^{2})α_{2}, \\ 0 & = h(K(v, v), α_{3}e_{1} − α_{1}e_{3}) = λ(q)(α_{1}^{2} − α_{2}^{2})α_{3}. \end{aligned}

First we assume α_{2} = 0. The second equation becomes α_{1}^{2}α_{3} = 0. Since α_{1}^{2} + α_{2}^{2} + α_{3}^{2} = 1, we get that either v = ±e_{1} or v = ±e_{3}. In the first case h(K(v, v), v) = ±λ(q), while in the second case h(K(v, v), v) = 0. If α_{2} ≠ 0, the first equation implies that α_{2} = ϵ\sqrt{3}α_{1}, ϵ = ±1. Then the second equation gives that α_{3} = 0 and therefore h(K(v, v), v) = ±λ(q). This completes the proof of the lemma.

Proposition 4.2. Let M^{n} be a positive definite centroaffine hypersurface in R^{n+1} which realizes the equality in (1.1). Let V be the set of points at which the difference tensor does not vanish identically. Then, there exist local h-orthonormal vector fields E_{1}, …, E_{n} in a neighborhood of p ∈ V such that for i, j ≥ 3:

\begin{aligned}K(E_{1}, E_{1}) & = λE_{1}, \\ K(E_{1}, E_{2}) & = −λE_{2}, \\ K(E_{2}, E_{2}) & = −λE_{1}, \end{aligned}  \begin{aligned}K(E_{1}, E_{i}) & = 0, \\ K(E_{2}, E_{i}) & = 0, \\ K(E_{i}, E_{j}) & = 0. \end{aligned}

Proof. We use a similar construction technique as in [16]. Let p ∈ V. First, we show that if f_{p} attains an absolute maximum in v, we can extend v differentiably to a vector field V on a neighborhood U of p such that h(V, V) = 1 and, at every point q of U, f_{q} attains an absolute maximum in V (q). In order to do this, let U_{1}, …, U_{n} be an arbitrary local differentiable orthonormal basis on a neighborhood U of p such that U_{i}(p) = e_{i}, with e_{1} = v, satisfying Lemma 3.7. Then we define a function γ: R^{n} × U ⟶ R^{n} by γ(a_{1}, …, a_{n}, q) = (b_{1}, …, b_{n}), where for k = 1, 2, …, n:

b_{k} = ∑_{i,j=1}^{n} a_{i}a_{j}h(K(U_{i}, U_{j}), U_{k}) − λ(p)a_{k}.

Using the fact that {U_{1}(p), …, U_{n}(p)} satisfies Lemma 3.7, we get

\frac{∂b_{k}}{∂a_{m}} (1, 0, …, 0, p) = 2h(K(U_{1}(p), U_{k}(p)), U_{m}(p)) − λ(p)δ_{km}
= 0, if k ≠ m,
= λ(p), if k = m = 1,
= −3λ(p), if k = m = 2,
= −λ(p), if k = m > 2.

The implicit function theorem shows that there exist differentiable functions a_{1}, …, a_{n} defined on a neighborhood of p such that the local vector field V defined by

V = a_{1}U_{1} + a_{2}U_{2} + … + a_{n}U_{n}

satisfies V (p) = v and K(V, V) = λ(p)V. Therefore the vector field W = V/\sqrt{h(V, V)} satisfies K(W, W) = (λ(p)/\sqrt{h(V, V)})W. Hence, on a neighborhood of p, f_{q} has a critical point W (q). Therefore λ(p)/\sqrt{h(V, V)} is a critical value, and hence equal to λ(q), 0 or −λ(q). The latter cases are impossible, so λ(q) = λ(p)/\sqrt{h(V, V)} and W is the desired vector field.

We now take V = E_{1} and look at K_{E_{1}}. We know that E_{1} is an eigenvector of K_{E_{1}} with eigenvalue λ. Since λ ≠ 0, Lemma 3.7 implies that K_{E_{1}} has two more eigenspaces. The first one is one-dimensional and corresponds to the eigenvalue −λ and the second one is (n − 2)-dimensional and corresponds to the eigenvalue 0. Since these three eigenvalues are different and have constant multiplicities on a neighborhood of p, a result of [11] allows us to find local orthonormal vector fields {E_{2}, …, E_{n}} such that for i > 2:

\begin{aligned}K_{E_{1}}E_{2} & = −λE_{2}, \\ K_{E_{1}}E_{i} & = 0. \end{aligned}

Combining this with Lemma 3.7 completes the proof of the lemma.

Since on V, D = span{E_{3}, …, E_{n}}, it immediately follows that D is differentiable on an open and dense subset of M^{n}, and thus determines a distribution on that set. Let p ∈ V. We take the local orthonormal basis {E_{1}, …, E_{n}} constructed in Lemma 4.2. We introduce local functions γ̂_{ij}^{k} by γ̂_{ij}^{k} := h( ˆ∇_{E_{i}}E_{j}, E_{k}). Obviously, γ̂_{ij}^{k} + γ̂_{ik}^{j} = 0. Then, we have the following lemma.

Lemma 4.3. For i, j > 2 we have

γ̂_{11}^{i} − γ̂_{22}^{i} = 0,  (4.1)
γ̂_{12}^{i} + γ̂_{21}^{i} = 0,  (4.2)
γ̂_{ij}^{1} = γ̂_{ij}^{2} = 0,  (4.3)
γ̂_{i1}^{2} = −\frac{1}{3}(γ̂_{12}^{i}).  (4.4)

Moreover, the function λ satisfies the following system of differential equations:

E_{1}(λ) = −3λγ̂_{21}^{2},  (4.5)
E_{2}(λ) = 3λγ̂_{11}^{2},  (4.6)
E_{i}(λ) = −λγ̂_{1i}^{1}.  (4.7)

Proof. We write ( ˆ∇K)(E_{i}, E_{j}, E_{k}) = ( ˆ∇_{E_{i}}K)(E_{j}, E_{k}). Since, for i, j > 2,

( ˆ∇K)(E_{1}, E_{i}, E_{j}) = ˆ∇_{E_{1}}K(E_{i}, E_{j}) − K( ˆ∇_{E_{1}}E_{i}, E_{j}) − K(E_{i}, ˆ∇_{E_{1}}E_{j}) = 0,
( ˆ∇K)(E_{i}, E_{1}, E_{j}) = ˆ∇_{E_{i}}K(E_{1}, E_{j}) − K( ˆ∇_{E_{i}}E_{1}, E_{j}) − K(E_{1}, ˆ∇_{E_{i}}E_{j})
= −K(E_{1}, ˆ∇_{E_{i}}E_{j}),

Codazzi’s equation (2.4) yields γ̂_{ij}^{1} = γ̂_{ij}^{2} = 0. This proves (4.3). Next we compute

( ˆ∇K)(E_{i}, E_{1}, E_{1}) = ˆ∇_{E_{i}}K(E_{1}, E_{1}) − 2K( ˆ∇_{E_{i}}E_{1}, E_{1})
= E_{i}(λ)E_{1} + λ ˆ∇_{E_{i}}E_{1} + 2h( ˆ∇_{E_{i}}E_{1}, E_{2})λE_{2},
( ˆ∇K)(E_{1}, E_{i}, E_{1}) = ˆ∇_{E_{1}}K(E_{i}, E_{1}) − K( ˆ∇_{E_{1}}E_{i}, E_{1}) − K(E_{i}, ˆ∇_{E_{1}}E_{1})
= −h( ˆ∇_{E_{1}}E_{i}, E_{1})λE_{1} + h( ˆ∇_{E_{1}}E_{i}, E_{2})λE_{2}.

Since h( ˆ∇_{E_{i}}E_{j}, E_{1}) = 0, for i, j > 2, by comparing components, we get

\begin{matrix}E_{i}(λ) = −λh( ˆ∇_{E_{1}}E_{i}, E_{1}) = −λγ̂_{1i}^{1}, \\ 3h( ˆ∇_{E_{i}}E_{1}, E_{2}) = −h( ˆ∇_{E_{1}}E_{2}, E_{i}). \end{matrix}

This proves (4.4) and (4.7). Similarly, from ( ˆ∇K)(E_{i}, E_{2}, E_{2}) = ( ˆ∇K)(E_{2}, E_{i}, E_{2}), we obtain (4.1): γ̂_{22}^{i} = γ̂_{11}^{i}. Finally, we have

( ˆ∇K)(E_{2}, E_{1}, E_{1}) = ˆ∇_{E_{2}}K(E_{1}, E_{1}) − 2K( ˆ∇_{E_{2}}E_{1}, E_{1})
= E_{2}(λ)E_{1} + λ ˆ∇_{E_{2}}E_{1} + 2h( ˆ∇_{E_{2}}E_{1}, E_{2})λE_{2},
( ˆ∇K)(E_{1}, E_{2}, E_{1}) = ˆ∇_{E_{1}}K(E_{2}, E_{1}) − K( ˆ∇_{E_{1}}E_{2}, E_{1}) − K(E_{2}, ˆ∇_{E_{1}}E_{1})

= −E_{1}(λ)E_{2} − λ ˆ∇_{E_{1}}E_{2} − h( ˆ∇_{E_{1}}E_{2}, E_{1})λE_{1} + h(∇_{E_{1}}E_{1}, E_{2})λE_{1}.

So, by comparing components, we get

h( ˆ∇_{E_{2}}E_{1} + ˆ∇_{E_{1}}E_{2}, E_{i}) = 0,
E_{2}(λ) = 3λγ̂_{11}^{2},
E_{1}(λ) = −3λγ̂_{21}^{2}.

This completes the proof of the lemma.

Clearly, Lemma 4.3 implies that the distribution D is integrable and autoparallel.

5. Some Classification Theorems

The inequality (1.1) gives rise to a global centroaffine curvature invariant for hyperovaloids and a characterization of hyperellipsoids in terms of this global invariant.

Proposition 5.1. Consider a centroaffine hyperovaloid f: M^{n} → R^{n+1} of dimension n ≥ 3 with normalization as in (2.1). Then,

∫ (ϵ_{M} + \frac{n^{2}(n−2)}{2(n−1)}h(T^{♯}, T^{♯}))ω(h) ≥ \frac{1}{2}(n + 1)(n − 2) vol(M, h),

where ω(h) denotes the Riemannian volume form of (M^{n}, h) and vol(M, h) the total volume. Moreover, we have equality if and only if the hyperovaloid is an hyperellipsoid.

Proof. We only need to discuss the equality. Theorem 3.6 gives an affine hypersphere and finally an hyperellipsoid from the theorem of Blaschke and Deicke ([10], Theorem 2.4.7).

Remark 5.2. According to Sections 3 and 4 in [13], h and T^{♯} form a complete system of centroaffine invariants on hyperovaloids; that means a centroaffine hyperovaloid can be uniquely determined from {h, T^{♯}}. The global centroaffine invariant on the left hand side depends only on this system.

Theorem 5.3. Let M^{n}, n ≥ 3, be a positive definite centroaffine hypersurface realizing the equality in (1.1). If M^{n} has constant scalar curvature, then M^{n} is an open part of an hyperellipsoid centered at the origin.

Proof. Let M^{n} be a centroaffine hypersurface with constant scalar curvature satisfying (1.1). Then, it follows from Lemma 3.7 that either K vanishes identically on M^{n} or λ is a strictly positive constant. In the first case, it is well known (see e.g. [14], Section 7.1.1) that M^{n} is affinely equivalent with an open part of an hyperellipsoid centered at the origin.

Therefore, we may assume that λ is a positive constant. Lemma 4.3 then implies that, for i, j > 2:

γ̂_{11}^{i} − γ̂_{22}^{i} = 0,  (5.1)
γ̂_{12}^{i} + γ̂_{21}^{i} = 0,  (5.2)
γ̂_{ij}^{1} = γ̂_{ij}^{2} = 0,  (5.3)
γ̂_{i1}^{2} = −\frac{1}{3}(γ̂_{12}^{i}),  (5.4)
0 = −3λγ̂_{21}^{2},  (5.5)
0 = 3λγ̂_{11}^{2},  (5.6)
0 = −λγ̂_{1i}^{1}.  (5.7)

Combining (5.1), (5.5), (5.6) and (5.7) we immediately get that

ˆ∇_{E_{1}}E_{1} = ˆ∇_{E_{2}}E_{2} = 0.

Clearly, we can still assume that {E_{3}, …, E_{n}} are chosen in such a way that

ˆ∇_{E_{1}}E_{2} = bE_{3},  (5.8)

where b is a locally defined function. Then, (5.2) implies

ˆ∇_{E_{2}}E_{1} = −bE_{3}.  (5.9)

If n > 3, then by (2.10) we have 1 = h(R̂(E_{1}, E_{4})E_{4}, E_{1}). Since by (5.3) resp. (5.7) and (5.8), ˆ∇_{E_{4}}E_{4} and ˆ∇_{E_{1}}E_{4} are orthogonal to E_{1} and E_{2}, this becomes

1 = h( ˆ∇_{ˆ∇_{E_{4}} E_{1}}E_{4}, E_{1}) = 0,

which is a contradiction. Thus K ≡ 0. So, n = 3, and ˆ∇ is given by

\begin{aligned}ˆ∇_{E_{1}}E_{1} & = 0, \\ ˆ∇_{E_{1}}E_{2} & = bE_{3}, \\ ˆ∇_{E_{2}}E_{1} & = −bE_{3}, \end{aligned}  \begin{aligned}ˆ∇_{E_{2}}E_{2} & = 0, \\ ˆ∇_{E_{1}}E_{3} & = −bE_{2}, \\ ˆ∇_{E_{3}}E_{1} & = −\frac{1}{3}bE_{2} \end{aligned}  \begin{aligned}ˆ∇_{E_{3}}E_{3} & = 0, \\ ˆ∇_{E_{2}}E_{3} & = bE_{1}, \\ ˆ∇_{E_{3}}E_{2} & = \frac{1}{3}bE_{1}. \end{aligned}

Using again (2.10) we get

1 = h(R̂(E_{1}, E_{3})E_{3}, E_{1}) = b^{2}.  (5.10)

Hence b = ±1. To obtain now the desired contradiction, we use again (2.10):

1 + 2λ^{2} = h(R̂(E_{1}, E_{2})E_{2}, E_{1}) = −\frac{5}{3}.

This completes the proof of the theorem.

From now on, we will concentrate on classifying those hypersurfaces for which the distribution D^{⊥} is integrable. Clearly, we may restrict ourselves to the open set V consisting of the points p with K_{p} ≠ 0. Since this distribution is locally spanned by the vector fields E_{1} and E_{2}, its integrability is equivalent to h( ˆ∇_{E_{1}}E_{2} − ˆ∇_{E_{2}}E_{1}, E_{i}) = 0, for i ≥ 3. Combining this with Lemma 4.3 gives us, for i > 2:

γ̂_{12}^{i} = γ̂_{21}^{i} = 0.  (5.11)

If we suppose that D^{⊥} is also autoparallel on some open set, then both D and D^{⊥} are parallel distributions and therefore h(R̂(E_{3}, E_{1})E_{1}, E_{3}) = 0. On the other hand, (2.10) implies that h(R̂(E_{3}, E_{1})E_{1}, E_{3}) = 1, which gives us a contradiction.

Therefore we can define an open dense set U of V as the set of points where D^{⊥} is not autoparallel. On U, we can define a unique (up to sign) vector field E_{3} (which belongs to D) such that by (4.1):

\begin{aligned}ˆ∇_{E_{1}}E_{1} & = a_{1}E_{2} + a_{2}E_{3}, \\ ˆ∇_{E_{2}}E_{2} & = −a_{3}E_{1} + a_{2}E_{3}, \end{aligned}

where a_{1}, a_{2}, a_{3} are locally defined functions with a_{2} ≠ 0. Our assumption (5.11) together with Lemma 4.3 implies that for i ≥ 3:

ˆ∇_{E_{1}}E_{2} = −a_{1}E_{1},  (5.12)
ˆ∇_{E_{2}}E_{1} = a_{3}E_{2},  (5.13)
ˆ∇_{E_{i}}E_{1} = ˆ∇_{E_{i}}E_{2} = 0.  (5.14)

We now define 3 local distributions H_{0}, H_{1} and H_{2} by

H_{0} = span {E_{3}}, H_{1} = span {E_{1}, E_{2}}, H_{2} = span {E_{4}, …, E_{n}}.

In order to simplify our notation, we introduce different ranges of indices. From now on, p, q ∈ {1, 2} and i, j, k ∈ {4, …, n}.

Lemma 5.4. The distribution H_{0} is autoparallel and its integral curves are ellipses centered at the origin in R^{n+1}.

Proof. By Lemma 4.3, we already know that ˆ∇_{E_{3}}E_{3} has no H_{1}-component. In order to prove that ˆ∇_{E_{3}}E_{3} has no H_{2}-component, we use (2.10). For i ≥ 4, we have

0 = h(R̂(E_{1}, E_{3})E_{1}, E_{i})
= h( ˆ∇_{E_{1}} ˆ∇_{E_{3}}E_{1} − ˆ∇_{E_{3}} ˆ∇_{E_{1}}E_{1} − ˆ∇_{ˆ∇_{E_{1}} E_{3}− ˆ∇_{E_{3}} E_{1}}E_{1}, E_{i})
= −h( ˆ∇_{E_{3}}(a_{1}E_{2} + a_{2}E_{3}), E_{i}) = −a_{2}h( ˆ∇_{E_{3}}E_{3}, E_{i}).

Since a_{2} ≠ 0, this shows that ˆ∇_{E_{3}}E_{3} has no H_{2}-component and thus ˆ∇_{E_{3}}E_{3} = 0.

Let γ be an integral curve of the vector field E_{3}. Since, we have that D_{E_{3}}E_{3} = −f (cf. (2.1) and (2.5)), the curve γ satisfies the differential equation γ′′ = −γ. Hence γ is an ellipse centered at the origin.

Lemma 5.5. The distribution H_{1} is spherical and the distribution H_{0}⊕H_{1} is autoparallel.

Proof. For p, q ∈ {1, 2}, we know that

ˆ∇_{E_{p}}E_{q} − a_{2}δ_{pq}E_{3} ∈ H_{1},  (5.15)

showing that H_{1} is totally umbilical with mean curvature normal η = a_{2}E_{3}. So, in order to show that H_{1} is spherical, it is sufficient to prove that the following two conditions are satisfied:

E_{1}(a_{2}) = E_{2}(a_{2}) = 0,  (5.16)
h( ˆ∇_{E_{1}}E_{3}, E_{i}) = h( ˆ∇_{E_{2}}E_{3}, E_{i}) = 0, i ≥ 4.  (5.17)

In order to prove (5.16) and (5.17), we use again the equation (2.10). Modulo the distribution H_{1}, we have

0 ≡ R̂(E_{1}, E_{2})E_{1}
≡ ˆ∇_{E_{1}} ˆ∇_{E_{2}}E_{1} − ˆ∇_{E_{2}} ˆ∇_{E_{1}}E_{1} − ˆ∇_{ˆ∇_{E_{1}} E_{2}− ˆ∇_{E_{2}} E_{1}}E_{1}
≡ ˆ∇_{E_{1}}(a_{3}E_{2}) − ˆ∇_{E_{2}}(a_{1}E_{2} + a_{2}E_{3}) − ˆ∇_{−a_{1}E_{1}−a_{3}E_{2}}E_{1}
≡ −a_{1}a_{2}E_{3} − E_{2}(a_{2})E_{3} − a_{2} ˆ∇_{E_{2}}E_{3} + a_{1}a_{2}E_{3},
≡ −E_{2}(a_{2})E_{3} − a_{2} ˆ∇_{E_{2}}E_{3} (mod H_{1}),

which shows that E_{2}(a_{2}) = 0 and h( ˆ∇_{E_{2}}E_{3}, E_{i}) = 0 for i ≥ 4. The other equations follow in a similar manner by computing R̂(E_{1}, E_{2})E_{2}. This shows that H_{1} is spherical.

The fact that H_{0} ⊕H_{1} is autoparallel now follows immediately from (5.14), (5.15) and (5.17).

Lemma 5.6. The distribution H_{2} is spherical and the distribution H_{0}⊕H_{2} is autoparallel.

Proof. First, we notice that the distribution H_{0} ⊕ H_{2} = D is autoparallel. To show that H_{2} is spherical, we first compute ˆ∇_{E_{i}}E_{j} modulo H_{2}. It follows from Lemma 4.3 that there exist local functions α_{ij} such that

ˆ∇_{E_{i}}E_{j} ≡ α_{ij}E_{3} (mod H_{2}),

for i, j ≥ 4. From (2.10), we then get

δ_{ij} = h(R̂(E_{i}, E_{1})E_{1}, E_{j})
= h( ˆ∇_{E_{i}} ˆ∇_{E_{1}}E_{1} − ˆ∇_{E_{1}} ˆ∇_{E_{i}}E_{1} − ˆ∇_{ˆ∇_{E_{i}} E_{1}− ˆ∇_{E_{1}} E_{i}}E_{1}, E_{j})
= h( ˆ∇_{E_{i}}(a_{1}E_{2} + a_{2}E_{3}) + ˆ∇_{ˆ∇_{E_{1}} E_{i}}E_{1}, E_{j})
= a_{2}h( ˆ∇_{E_{i}}E_{3}, E_{j}),

from which we deduce α_{ij} = −δ_{ij}1/a_{2}.

Since ˆ∇_{E_{i}}E_{j} ∈ H_{0} ⊕ H_{2}, we see that H_{2} is spherical if and only if a_{2} is constant along H_{2}. We have, for i ≥ 4,

0 = h(R̂(E_{i}, E_{1})E_{1}, E_{3})
= h( ˆ∇_{E_{i}} ˆ∇_{E_{1}}E_{1} − ˆ∇_{E_{1}} ˆ∇_{E_{i}}E_{1} − ˆ∇_{ˆ∇_{E_{i}} E_{1}− ˆ∇_{E_{1}} E_{i}}E_{1}, E_{3})
= h( ˆ∇_{E_{i}}(a_{1}E_{2} + a_{2}E_{3}) + ˆ∇_{ˆ∇_{E_{1}} E_{i}}E_{1}, E_{3}) = E_{i}(a_{2}).

Thus a_{2} is constant along H_{2} and H_{2} is spherical.

Lemma 5.7. The function a_{2} satisfies the following system of differential equations:

\begin{matrix}E_{1}(a_{2}) = E_{2}(a_{2}) = E_{i}(a_{2}) = 0, i ≥ 4, \\ E_{3}(a_{2}) = 1 + a_{2}^{2}. \end{matrix}

Proof. The first statement is clear from Lemma 5.5 and Lemma 5.6. In order to show the second statement, we use again (2.10). We have

1 = h(R̂(E_{3}, E_{1})E_{1}, E_{3})
= h( ˆ∇_{E_{3}} ˆ∇_{E_{1}}E_{1} − ˆ∇_{E_{1}} ˆ∇_{E_{3}}E_{1} − ˆ∇_{ˆ∇_{E_{3}} E_{1}− ˆ∇_{E_{1}} E_{3}}E_{1}, E_{3})
= h( ˆ∇_{E_{3}}(a_{1}E_{2} + a_{2}E_{3}) + ˆ∇_{ˆ∇_{E_{1}} E_{3}}E_{1}, E_{3}) = E_{3}(a_{2}) − a_{2}^{2},

which completes the proof.

Theorem 5.8. Let M^{n}, n ≥ 3, be a simply connected positive definite centroaffine hypersurface satisfying the equality in (1.1). Suppose moreover that the distribution D^{⊥} is integrable. Then M^{n} is locally centroaffinely equivalent to the hypersurface

f(t, y, x) = cos t (ψ(y), 0, …, 0) + sin t (0, 0, 0, ϕ(x)),

where the map x ↦ ϕ(x) describes an hyperellipsoid in R^{n−2} centered at the origin and the map y ↦ ψ(y) describes a proper affine sphere in R^{3}.

Proof. Our considerations are of local character. Here, we use the notation M^{n} for an appropriate local neighborhood. Using Nölker’s result [11] and using the formalism of [6], the above lemmas imply that M^{n} is locally isometric to a warped product manifold, say M^{n} = (−ϵ, ϵ) ×_{ρ_{1}} M_{1} ×_{ρ_{2}} M_{2}, where the warping functions ρ_{1} and ρ_{2} are respectively determined by

\begin{matrix}a_{2} = −E_{3}(log ρ_{1}), \\ (−\frac{1}{a_{2}} ) = −E_{3}(log ρ_{2}), \end{matrix}

with the initial conditions that ρ_{1}(0) = ρ_{2}(0) = 1. So, if we denote the variable on (−ϵ, ϵ) by t, with \frac{∂}{∂t} = E_{3}, we get that

\begin{matrix}a_{2} = −\frac{∂}{∂t}(log ρ_{1}), \\ \frac{1}{a_{2}} = \frac{∂}{∂t}(log ρ_{2}). \end{matrix}

Since, by the previous lemma,

\frac{∂a_{2}}{∂t} = 1 + a_{2}^{2},

we get that the function a_{2}, which only depends on t, is given by

a(t) = tan (t − s),

where s is an arbitrary constant. By translating the variable t, we may assume that s = 0. It now follows that

\begin{aligned}ρ_{1}(t) & = D_{1} cos t, \\ ρ_{2}(t) & = D_{2} sin t, \end{aligned}

where D_{1} and D_{2} are determined by the initial conditions.

We now denote by u and v local coordinates on M_{1} and by x a point on M_{2}. We identify M^{n} with its image under f and define a map ψ from M^{n} into R^{n+1} by

ψ(t, u, v, x) = cos t f(t, u, v, x) − sin t E_{3}(t, u, v, x).  (5.18)

Then, for i ≥ 4,

ψ_{∗}(E_{3}) = − sin t f(t, u, v, x) + cos t E_{3}(t, u, v, x)
− cos t E_{3}(t, u, v, x) − sin t (−f(t, u, v, x)) = 0.
ψ_{∗}(E_{i}) = cos t E_{i} − sin t \frac{1}{a_{2}(t)}E_{i} = 0.

Hence ψ actually reduces to a well-defined map from M_{1} into R^{n+1}. We now put ψ(u, v) = ψ(t, u, v, x). We compute

ψ_{∗}(E_{1}) = (cos t + sin t a_{2}(t))E_{1} = \frac{1}{cos t}E_{1},
ψ_{∗}(E_{2}) = (cos t + sin t a_{2}(t))E_{2} = \frac{1}{cos t}E_{2}.

A straightforward computation shows that

D_{E_{1}}ψ_{∗}(E_{1}) = \frac{1}{cos t}(λE_{1} + a_{1}E_{2} + a_{2}E_{3} − f)
= \frac{1}{cos t}(λE_{1} + a_{1}E_{2}) − \frac{1}{cos^{2} t}(cos t f − sin t E_{3})
= \frac{1}{cos t}(λE_{1} + a_{1}E_{2}) − \frac{1}{cos^{2} t}ψ.

Similarly we get that

D_{E_{1}}ψ_{∗}(E_{2}) = \frac{1}{cos t}(−a_{1}E_{1} − λE_{2}),
D_{E_{2}}ψ_{∗}(E_{1}) = \frac{1}{cos t}(a_{3} − λ)E_{2},
D_{E_{2}}ψ_{∗}(E_{2}) = \frac{1}{cos t}(−(a_{3} + λ)E_{1}) − \frac{1}{cos^{2} t}ψ.

Using the previous equations, it follows immediately that the image of ψ is contained in a three-dimensional linear subspace A_{1} (through the origin) and that ψ: M_{1} → A_{1} is a positive definite centroaffine immersion with vanishing Tchebychev vector field.

Finally, we define still another map from ϕ: M → R^{n+1} by

ϕ(t, u, v, x) = \frac{1}{sin t} (f(t, u, v, x) − cos t ψ(u, v)) .  (5.19)

Then, using (5.18), we have

ϕ_{∗}(E_{1}) = \frac{1}{sin t} (E_{1} − cos t ψ_{∗}(E_{1})) = 0,
ϕ_{∗}(E_{2}) = \frac{1}{sin t} (E_{2} − cos t ψ_{∗}(E_{2})) = 0,
ϕ_{∗}(E_{3}) = −\frac{1}{sin^{2} t} cos t (f − cos t ψ)
+ \frac{1}{sin t} (E_{3} + sin t ψ)
= \frac{1}{sin^{2} t}(− cos t f + sin t E_{3} + ψ) = 0.

Therefore ϕ actually defines a mapping from M_{2} in R^{n+1}. If we denote by ∇_{X}^{∗}Y the H_{2}-component of ˆ∇_{X}Y, we find that for i, j ≥ 4,

ϕ_{∗}(E_{i}) = \frac{1}{sin t}E_{i},

and

D_{E_{i}}ϕ_{∗}(E_{j}) = \frac{1}{sin t}D_{E_{i}}E_{j}
= \frac{1}{sin t}(K(E_{i}, E_{J}) + ˆ∇_{E_{i}}E_{j} − δ_{ij}f)
= \frac{1}{sin t}(∇_{E_{i}}^{∗}E_{j}) + δ_{ij} \frac{1}{sin t}(−\frac{cos t}{sin t}E_{3} − f)
= \frac{1}{sin t}(∇_{E_{i}}^{∗}E_{j}) + δ_{ij} \frac{1}{sin t}(−\frac{cos t}{sin^{2} t}(cost f − ψ) − f)
= \frac{1}{sin t}∇_{E_{i}}^{∗}E_{j} − \frac{1}{sin^{2} t}δ_{ij}ϕ.

So we notice that ϕ(M_{2}) is contained in an (n − 2)-dimensional affine space containing the origin and that ϕ is a tranversal vector field to the immersion. So, if we take −ϕ as centroaffine normal and denote the induced connection by ∇^{ϕ} and the induced centroaffine metric by h^{ϕ}, we notice that ϕ is positive definite and

\begin{matrix}h^{ϕ}(E_{i}, E_{j}) = \frac{1}{sin^{2} t}δ_{ij} = \frac{1}{sin^{2} t}h(E_{i}, E_{j}), \\ ∇_{E_{i}}^{ϕ}E_{j} = ∇_{E_{i}}^{∗}E_{j}. \end{matrix}

Since ˆ∇ is the Levi Civita connection for h, a straightforward computation shows that ∇^{ϕ} is the Levi Civita connection for h^{ϕ}. Hence the cubic form vanishes and M_{2} is centroaffine equivalent with an hyperellipsoid centered at the origin in an (n − 2)-dimensional linear subspace A_{2} (cp. [14], Section 7.1.1). Since M^{n} is linearly full we have that A_{1} and A_{2} are complementary linear subspaces. Hence it follows from (5.19) that there exists a centroaffine transformation such that

f(t, y, x) = cos t (ψ(y), 0, …, 0) + sin t (0, 0, 0, ϕ(x)).

Also, in the negative definite case, similar results can be obtained. By slightly adapting the proof of Theorem 5.3 it is possible to show that

Theorem 5.9. Let M^{n}, n ≥ 3, be a negative definite centroaffine hypersurface realizing the equality in (3.8). If M^{n} has constant scalar curvature, then M^{n} is an open part of a hyperboloid centered at the origin.

Adapting the proof of Theorem 5.8 to the negative definite case is subtle (cp. [15]). However, it is still possible to show that M^{n} is centroaffinely equivalent with

(1) a quadric or

(2) with a hypersurface

f_{1}(t, y, x) = cosh t (ψ(y), 0, …, 0) + sinh t (0, 0, 0, ϕ(x)),

where the map x ↦ ϕ(x) describes an ellipsoid in R^{n−2} centered at the origin and the map y ↦ ψ(y) describes a negative definite proper affine sphere in R^{3} or

(3) with a hypersurface

f_{2}(t, y, x) = − sinh t (ψ(y), 0, …, 0) + cosh t (0, 0, 0, ϕ(x)),

where the map x ↦ ϕ(x) describes a two-sheeted hyperboloid in R^{n−2} centered at the origin and the map y ↦ ψ(y) describes a positive definite proper affine sphere in R^{3} or

(4) with a hypersurface

f_{3}(t, y, x) = e^{−t}(1, e^{2t} + ϕ(y) + \frac{1}{2} ∑_{i=1}^{n−3} x_{i}^{2}, y, x_{1}, …, x_{n−3}),

where (y, ϕ(y)), ϕ: N^{2} → R, defines an equiaffine improper affine sphere with constant normal vector field (0, 0, 1). This last example can also be seen as a semi-projective composition of improper affine spheres (in the sense of [7]).

6. Remarks on Polarization

Polarity of hypersurfaces is one of the classical concepts of Euclidean hypersurface theory. In [12] it was studied under affine view points: from this, polarity of a non-degenerate pair of hypersurfaces exactly describes the relation between a centroaffine hypersurface and its centroaffine Gauss conormal immersion. We briefly recall this situation.

Let V := R^{n+1}, V^{∗} be its dual space and ⟨ , ⟩: V^{∗} × V → R be the canonical scalar product. Then, the centroaffine hypersurfaces f: M → V and F: M → V^{∗} define a polar pair if

⟨F, f⟩ = −1,  ⟨F, f_{∗}⟩ = 0,  ⟨F_{∗}, f⟩ = 0.  (6.1)

The duality of the situation is obvious: F is the Gauss conormal immersion of f and vice versa. Polarization in the classical terminology means the interchange of f and F, when V is equipped with a Euclidean structure and V and V^{∗} can be identified; in this situation (6.1) can be described in Euclidean terms. In our situation polarization means the interchange of f and F. The Gauss structure equation for F, corresponding to (2.1), read

D_{X}F_{∗}(Y) = F_{∗}( ¯∇_{X}Y) + h̄(X, Y)(−F),  (6.2)

while the normalisation in (2.1’) implies

D_{X}F_{∗}(Y) = F_{∗}( ¯∇_{X}Y) + h̄(X, Y)F.  (6.2′)

Proposition 7.2.1 in [12] contains a list of affine invariants of f and F which are also invariant under polarization. From this list, we only need the following:

Lemma 6.1. For a polar pair with n ≥ 2, we have the following centroaffine relations:

(1) h̄ = h,

(2) h(T̄^{♯}, T̄^{♯}) = h(T^{♯}, T^{♯}),

(3) the connections ∇ and ¯∇ interchange under polarization.

Recall from Remark 5.2 that {h, T^{♯}} define a complete system of invariants which globally determine a hyperovaloid. Theorem 3.6 (resp. Theorem 3.8) and Lemma 6.1 imply for the centroaffine invariants ϵ_{M} and δ (see (3.2)):

Corollary 6.2. For n ≥ 3, we have:

(1) the local centroaffine invariants ϵ_{M}, δ and ϵ_{M} + \frac{n^{2}(n−2)}{2(n−1)}h(T^{♯}, T^{♯}) are also invariant under polarization;

(2) the inequality

\frac{1}{vol(M,h)} ∫ (ϵ_{M} + \frac{n^{2}(n−2)}{2(n−1)}h(T^{♯}, T^{♯}))ω(h) ≥ ε\frac{1}{2}(n + 1)(n − 2)

gives a lower bound, depending only on the dimension n, for the global polarization invariant on the left-hand side; here ε = 1 (ε = −1) corresponding to the normalization in (2.1) (and (2.1’), respectively);

(3) if we have equality in (2) then f and F both are proper affine hyperspheres (of elliptic type if ε = 1, of hyperbolic type if ε = −1).

7. Spectral relation

Recently [1] studied asymptotic spectral invariants of some second order differential operators of Laplace type which are of importance in affine hypersurface theory. We recall the definitions and main results and refer to [1] for details. We restrict to compact M^{n} with ∂M^{n} = ∅ and thus to the case that h is positive definite and f(M^{n}) is a hyperovaloid which is normalized by ξ = −f (see (2.1)).

Consider the induced connection ∇ from (2.1), its Ricci tensor Ric(∇) and its Hessian

(Hess_{∇}f)(V, W) := V (W f) − df(∇_{V} W).  (7.1)

We define the second order operator D by

Df := D(h, ∇)f := trace_{h}(Hess_{∇}f + \frac{1}{n−1} Ric(∇)f) = trace_{h}Hess_{∇}f + nf.  (7.2)

Here, we used that Ric(∇) = (n−1)h (see [14], Section 6.3.3.2). Let {λ_{v}} be the eigenvalues of the elliptic operator D where each eigenvalue is repeated according to its multiplicity and where λ_{1} ≤ λ_{2} ≤ . . .. If t > 0, then exp(−tD) is trace class on L^{2}(M) and the series

Tr_{L^{2}}(exp(−tD)) = ∑_{v} exp(−tλ_{v})

converges uniformly on compact subsets of (0, ∞). As t ↘ 0, there is an asymptotic series of the form

Tr_{L^{2}}(exp(−tD)) ∼ ∑_{k≥0} a_{k}(D)t\frac{1}{2}(k − n).

The coefficients a_{k}(D) are spectral invariants which are locally computable in terms of (M, h, ∇) as follows: there exist local invariants a_{k}(p, D), p ∈ M, depending both on h and ∇ such that

a_{k}(D) = ∫ a_{k}(p, D)ω(h).  (7.3)

Analogously, we can consider the conormal connection ¯∇ from (6.2) and the differential operator D̄ := D(h, ¯∇). From (1.22) and Theorem 6.1.3 in [1] we have (observe that the centroaffine Weingarten operator is the identity):

Lemma 7.1. Let f: M → R^{n+1} be a centroaffine hyperovaloid, n ≥ 2. Then, we have the following relations for asymptotic spectral invariants of D and D̄:

(1) a_{2k+1}(D) = 0 = a_{2k+1}(D̄), for k ∈ N, (2) a_{0}(D) = a_{0}(D̄) = (4π)^{− \frac{n}{2}} vol(M, h),

(3) a_{2}(D) = a_{2}(D̄) = (4π)^{− \frac{n}{2}} ∫ (\frac{1}{6}n(n − 1)κ̂ + n − \frac{1}{4}n^{2}h(T^{♯}, T^{♯}))ω(h).

As a consequence we get relations between intrinsic invariants of (M, h) and the asymptotic spectrum.

Corollary 7.2. Let f be a centroaffine hyperovaloid as above, and n ≥ 3. Then we have the following inequality between global intrinsic invariants of (M, h) and invariants of the asymptotic spectrum:

(1) for D̃ = D or D̃ = D̄, we have

(n − 1) ∫ (ϵ_{M} + \frac{1}{3}n(n − 2)κ̂)ω(h) ≥ (n − 2)(4π)^{\frac{n}{2}} (\frac{1}{2}(n^{2} − 4n − 1)a_{0}(D̃) + 2a_{2}(D̃)),

(2) equality in (1) holds if and only if f is an ellipsoid centered at the origin;

(3) both sides of the inequality in (1) are invariant under polarization, and in case of equality the polar hyperovaloids are centered ellipsoids.

Proof. The inequality in (1) can be easily verified from Proposition 5.1 and Theorem 7.1. Equality yields if and only if equality yields in Proposition 5.1. Finally, the polarization statement follows from Lemma 6.1, as h̄ = h and D and D̄ are interchanged.

8. Acknowledgements

The geometry groups at K.U. Leuven and T.U. Berlin participate in the GADGET program of the EU. The first author was supported by the DFG-Forschungsstipendium Scha 698/1-1. The first, the second and the fourth author were partially supported by the DFG-project Si 163/4-1,2. The second, the third and the fourth author were partially supported by the grant OT/TBA/95/9 of the Research Council of the K.U. Leuven. The forth author is a Senior Research Assistant of the National Fund for Scientific Research (Belgium).

References

[1] N. Bokan, P. B. Gilkey and U. Simon: Applications of spectral geometry to affine and projective geometry. Contributions to Algebra and Geometry 35 (1994), 283–314.

[2] B. Y. Chen: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60 (1993), 568–578.

[3] B.-Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken: Two equivariant totally real immersions into the nearly Kähler 6-sphere and their characterization. Japan. J. Math. 21 (1995), 207–222.

[4] B.Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken: Characterizing a class of totally real submanifolds of S^{6}(1) by their sectional curvatures. Tôhoku Math. J. 47 (1995), 185–198.

[5] B. Y Chen and L. Vrancken: Minimal hypersurfaces in S^{n+1}(1) and H^{n+1}(−1). Proc. Amer. Math.Soc. 123 (1995), 3851-3854.

[6] F. Dillen and S. Nölker: Semi-parallelity, multi-rotation surfaces and the helix-proper-ty. J. Reine Angew. Math. 435 (1993), 33–63.

[7] F. Dillen and L. Vrancken: Calabi Type composition of affine spheres. Diff. Geom. and its Appl. 4 (1994), 303–328.

[8] F. Dillen and L. Vrancken: Totally real submanifolds in S^{6} satisfying Chen’s equality. Trans. Amer. Math. Soc. 348 (1996), 1633–1646.

[9] N. Ejiri: Totally real minimal immersions of n-dimensional real space forms into ndimensional complex space forms. Proc. Amer. Math. Soc. 84 (1982), 243–246.

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

[11] S. Nölker: Isometric immersions of warped products. Diff. Geom. and its Appl. 6 (1996), 1–30.

[12] V. Oliker and U. Simon: Affine geometry and polar hypersurfaces. In: Analysis and Geometry: Trends in Research and Teaching (B. Fuchssteiner and W. A. J. Luxemburg, eds.), BI-Mannheim-Zürich, 1992, 87–112.

[13] R. Schneider: Zur affinen Differentialgeometrie im Großen I. Math. Z. 101 (1967), 375–406.

[14] U. Simon, A Schwenk-Schellschmidt and H. Viesel: Introduction to the affine differential geometry of hypersurfaces. Science University of Tokyo, 1991.

[15] Ch. Scharlach and L. Vrancken: A curvature invariant for centroaffine hypersurfaces, Part II. Geometry and Topology of Submanifolds, VIII, ed. by F. Dillen et al., World Scientific, Singapore, 1996, 341–350.

[16] L. Vrancken, A. M. Li and U. Simon: Affine spheres with constant affine sectional curvature. Math. Z. 206 (1991), 651–658.

Received May 13, 1996