The classification of 3-dimensional Lorentzian affine hypersurfaces with parallel cubic form

Hu, Zejun; Li, Cece

doi:10.1016/j.difgeo.2011.03.005

Cited by 18 publications

(20 citation statements)

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“…This normalization induces an affine connection ∇ and a semi-Riemannian metric h on M , called the Berwald-Blaschke metric. Let∇ denote the Levi-Civita connection of this metric h. Then the difference tensor K is introduced by K(X, Y ) := K X Y = ∇ X Y −∇ X Y , which is related to the cubic form C = ∇h by On the other hand, the classification of general non-degenerate affine hypersurfaces with∇C = 0 is known only up to dimension n = 3 [3,7,13]. To state these results, we introduce the coordinates (x 1 , .…”

Section: Introductionmentioning

confidence: 99%

Lorentzian Affine Hypersurfaces with Parallel Cubic Form

et al. 2011

Results. Math.

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Section: Introductionmentioning

confidence: 99%

Lorentzian Affine Hypersurfaces with Parallel Cubic Form

et al. 2011

Results. Math.

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“…Moreover the affine metric of Calabi product is flat if and only if both components have a flat affine metric. Now using (5.10) and results from [16,14,15] one can obtain some classification results for J -tangent affine hyperspheres with the parallel cubic form. For example, when dim M = 5, we have the following Corollary 5.3.…”

Section: Some Applicationsmentioning

confidence: 99%

J-Tangent Affine Hyperspheres

Szancer

2014

Results. Math.

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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...

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“…As for the general nondegenerate case, there also have been some interesting partial classification results, see for example the series of published papers by Z.J. Hu etc: [10], [11] and [12]. In this direction, a very recent development is the preprint article [8] in which the author aimed at a complete classification of nondegenerate centroaffine hypersurfaces with parallel Fubini-Pick form.…”

Section: Introductionmentioning

confidence: 99%

On the correspondence between symmetric equiaffine hyperspheres and the minimal symmetric Lagrangian submanifolds

Li¹

2014

Sci. Sin.-Math.

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In this paper, a correspondence via duality is established between the set of locally strongly convex symmetric equiaffine hyperspheres and the set of minimal symmetric Lagrangian submanifolds in a certain complex space form. By using this correspondence theorem, we are able to provide an alternative proof of the classification theorem for the locally strongly convex equiaffine hypersurfaces with parallel Fubini-Pick forms, which has been established recently by Z.J. Hu etc in a totally different way.

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The classification of 3-dimensional Lorentzian affine hypersurfaces with parallel cubic form

Cited by 18 publications

References 17 publications

Lorentzian Affine Hypersurfaces with Parallel Cubic Form

Lorentzian Affine Hypersurfaces with Parallel Cubic Form

J-Tangent Affine Hyperspheres

On the correspondence between symmetric equiaffine hyperspheres and the minimal symmetric Lagrangian submanifolds

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