J-Tangent Affine Hyperspheres

In this paper we study J-tangent affine hyperspheres. Under some additional conditions we give a local characterization of 3-dimensional J-tangent affine hyperspheres.


Introduction
Centro-affine real hypersurfaces with a J-tangent transversal vector field were first studied by Cruceanu in [1]. He proved that such hypersurfaces f : M 2n+1 → C n+1 can be locally expressed in the form f (x 1 , . . . , x 2n , z) = Jg(x 1 , . . . , x 2n ) cos z + g(x 1 , . . . , x 2n ) sin z, where g is some smooth function defined on an open subset of R 2n . He also showed that if the induced almost contact structure is Sasakian then a hypersurface must be a hyperquadric. The latter result was generalized in [3] to arbitrary hypersurfaces with J-tangent transversal vector field.
Since the class of centro-affine hypersurfaces with a J-tangent transversal vector field is quite large, the question arises whether there are affine hyperspheres with a J-tangent Blaschke normal field. A nontrivial 3-dimensional example was provided in [4]. The main purpose of this paper is to give a local characterization of 3-dimensional J-tangent affine hyperspheres with involutive contact distribution D.
In Sect. 2 we briefly recall basic formulas of affine diferential geometry and recall the notion of an affine hypersphere.
In Sect. 3 we recall the notion of a J-tangent transversal vector field, a definition of the induced almost contact structure as well as some results obtained in [3]. Section 4 contains the main results of this paper. We prove that there are no improper J-tangent affine hyperspheres and we give a local representation of 3-dimensional J-tangent affine hyperspheres under additional condition that the contact distribution is involutive.

Preliminaries
We briefly recall the basic formulas of affine differential geometry. For more details, we refer to [2]. Let f : M → R n+1 be an orientable connected differentiable n-dimensional hypersurface immersed in the affine space R n+1 equipped with its usual flat connection D. Then for any transversal vector field C we have and where X, Y are vector fields tangent to M . It is known that ∇ is a torsion-free connection, h is a symmetric bilinear form on M , called the second fundamental form, S is a tensor of type (1, 1), called the shape operator, and τ is a 1-form, called the transversal connection form. We assume that h is nondegenerate so that h defines a semi-Riemannian metric on M . If h is nondegenerate, then we say that the hypersurface or the hypersurface immersion is nondegenerate. In this paper we assume that f is always nondegenerate. We have the following Theorem 2.1 ([2], Fundamental equations). For an arbitrary transversal vector field C the induced connection ∇, the second fundamental form h, the shape operator S, and the 1-form τ satisfy the following equations: For a hypersurface immersion f : M → R n+1 a transversal vector field C is said to be equiaffine (resp. locally equiaffine) if τ = 0 (resp. dτ = 0).
When f is nondegenerate, there exists a canonical transversal vector field C, called the affine normal (or the Blaschke normal field). The affine normal is uniquely determined up to sign by the following conditions: the metric volume form ω h of h is ∇-parallel and coincides with the induced volume form Θ, where The affine immersion f with the Blaschke normal field C is called a Blaschke hypersurface. In this case fundamental equations can be rewritten as follows For a Blaschke hypersurface f , we have the following fundamental equations: If λ = 0 f is called an improper affine hypersphere, if λ = 0 f is called a proper affine hypersphere.
For simplicity we shall omit f * in front of vector fields in most cases.

Induced Almost Contact Structures
Let dim M = 2n + 1 and f : M → R 2n+2 be a nondegenerate affine hypersurface. We always assume that R 2m C m is endowed with the standard complex structure J. In particular, if m = n + 1 we have Let C be a transversal vector field on M . We say that We also define a distribution D on M as the biggest J invariant distribution on M , that is, for every x ∈ M . We use the notation X ∈ D for vectors as well as for D-fields. We say that the distribution D is nondegenerate if h is nondegenerate on D.
Recall that a (2n + 1)-dimensional manifold M is said to have an almost contact structure if there exist on M a tensor field ϕ of type (1,1), a vector field ξ and a 1-form η which satisfy Let f : M → R 2n+2 be a nondegenerate hypersurface with a J-tangent transversal vector field C. Then we can define a vector field ξ, a 1-form η and a tensor field ϕ of type (1,1) as follows: It is easy to see that (ϕ, ξ, η) is an almost contact structure on M . This structure is called the almost contact structure on M induced by C (or simply induced almost contact structure).
For an induced almost contact structure we have the following theorem for every X, Y ∈ X (M ).

J -Tangent Affine Hyperspheres
An affine hypersphere with a transversal J-tangent Blaschke normal field we call a J-tangent affine hypersphere. It is obvious that the standard hypersphere S 2n+1 (r) in R 2n+2 is a J-tangent affine hypersphere, since it is an affine hypersphere and the affine normal field is orthogonal to it. The next example shows that there are also other J-tangent affine hyperspheres: ). Let us consider the affine immersion f defined as follows: with the transversal vector field Thus, h is not positive definite.
As an immediate consequence of Theorem 3.1 we have the following: for some λ > 0.
We split the proof of Theorem 4.2 into several lemmas.  Proof. First observe that h is nondegenerate on D, it means that for every x ∈ M h x is nondegenerate on D x . We will prove it by contradiction, namely, suppose there exists x ∈ M such that h x is degenerate on D x . Now, we can find w ∈ D x , w = 0 such that h x (v, w) = 0 for every v ∈ D x . From formula (3.11) we also have h x (ξ x , w) = 0. Since every vector t ∈ T x M can be expressed in the form for all t ∈ T x M . We have that h x is nondegenerate on T x M so it follows that w = 0, which contradicts the assumption. Now we show that for every x ∈ M we can find a D-field Z such that Then, for any v, w ∈ D x we have: where λ is some positive constant.