J-Tangent Affine Hyperspheres

Abstract

Abstract. 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. Mathematics Subject Classification (2010). 53A15, 53D15.Keywords. Affine hypersurface, almost contact structure, affine hypersphere. IntroductionCentro-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. PreliminariesWe 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 haveandwhere 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 The Eqs. (2.3), (2.4), (2.5), and (2.6) are called the equations of Gauss, Codazzi for h, Codazzi for S and Ricci, respectively.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, th...

On 3 -dimensional e J -tangent centro-affine hypersurfaces and e J -tangent affine hyperspheres with some null-directions

“…Based on the above result we provide a local classification of all 3 -dimensional nondegenerate centro-affine hypersurfaces with a J -tangent transversal centro-affine vector field as well as J -tangent affine hyperspheres with the null-direction D + or D − . Moreover, in this case distribution D = D + ⊕ D − is not involutive, so affine hyperspheres are completely different from those studied in [11]. In particular, we give explicit examples of such hyperspheres.…”

“…First recall from [11] that an affine hypersphere with a transversal J -tangent Blaschke field is called a J -tangent affine hypersphere. Moreover, in [11] we proved that every J -tangent affine hypersphere is proper. Now we shall prove classification theorems for affine hyperspheres with null-directions.…”

“…In [11] the author studied affine hypersurfaces with a J -tangent transversal vector field and gave a local classification of J -tangent affine hyperspheres with an involutive distribution D .…”

“…In this paper we study real affine hypersurfaces f : M 3 → R 4 ∼ = C 2 of the paracomplex space C 2 with a J -tangent transversal vector field C and an induced almost paracontact structure (φ, ξ, η). In [11] the author showed that when C is centro-affine (not necessarily Blaschke) then f can be locally expressed in the form: f (x 1 , . .…”