摘要 :
In this paper, we aim to introduce and study the (locally strongly convex) equiaffine isoparametric functions on the affine space A(n+1), making the emphasis on their relation with the affine isoparametric hypersurfaces. Motivated...
展开
In this paper, we aim to introduce and study the (locally strongly convex) equiaffine isoparametric functions on the affine space A(n+1), making the emphasis on their relation with the affine isoparametric hypersurfaces. Motivated by the case in the Euclidean space En+1, we first introduce the concept of equiaffine parallel hypersurfaces in A(n+1), and then equivalently re-define the equiaffine isoparametric hypersurfaces to be ones that are among families of equiaffine parallel hypersurfaces in A(n+1) of constant affine mean curvature. As the main result, we prove that an equiaffine isoparametric hypersurface is nothing but exactly a regular level set of some equiaffine isoparametric function.
收起
摘要 :
Li and Zhang (2014) studied affine hypersurfaces of Rn+1 with parallel difference tensor relative to the affine alpha-connection del((alpha)), and characterized the generalized Cayley hypersurfaces by Kn-1 not equal 0 and del K-(a...
展开
Li and Zhang (2014) studied affine hypersurfaces of Rn+1 with parallel difference tensor relative to the affine alpha-connection del((alpha)), and characterized the generalized Cayley hypersurfaces by Kn-1 not equal 0 and del K-(alpha) = 0 for some nonzero constant a, where the affine alpha-connection del((alpha)) of information geometry was introduced on affine hypersurface. In this paper, by a slightly different method we continue to study affine hypersurfaces with del K-(alpha) = 0, if alpha = Owe further assume that the Pick invariant vanishes and affine metric is of constant sectional curvature. It is proved that they are either hyperquadrics or improper affine hypersphere with flat indefinite affine metric, the latter can be locally given as a graph of a polynomial of at most degree n + 1 with constant Hessian determinant. In particular, if the affine metric is definite, Lorentzian, or its negative index is 2, we complete the classification of such hypersurfaces. (C) 2014 Elsevier B.V. All rights reserved.
收起
摘要 :
We study Lorentzian affine hypersurfaces of ?~(n+1) having parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric. As main result, we obtain a complete classification of these hypersurfaces.
摘要 :
In this paper, we complete the classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric.
摘要 :
The classifications of locally strongly convex isotropic equiaffine spheres and isotropic centroaffine hypersurfaces have been completed in the last decade, see [1] and [4]. In this paper, we consider isotropic Calabi hypersurface...
展开
The classifications of locally strongly convex isotropic equiaffine spheres and isotropic centroaffine hypersurfaces have been completed in the last decade, see [1] and [4]. In this paper, we consider isotropic Calabi hypersurfaces (also called graph immersion with Calabi normalization) and obtain a complete classification in the affine space Rn+1. As a corollary, we give a new characterization of elliptic paraboloid: Elliptic paraboloids are the only constant isotropic Calabi hypersurfaces in Rn+1.(c) 2022 Elsevier B.V. All rights reserved.
收起
摘要 :
In this paper, we investigate the locally strongly convex affine hypersurfaces with semi-parallel cubic form relative to the Levi-Civita connection of affine metric. We obtain two results on such hypersurfaces which admit at most ...
展开
In this paper, we investigate the locally strongly convex affine hypersurfaces with semi-parallel cubic form relative to the Levi-Civita connection of affine metric. We obtain two results on such hypersurfaces which admit at most one affine principal curvature of multiplicity one: (1) classify these being not affine hyperspheres; (2) classify these affine hyperspheres with constant scalar curvature. For the latter, by proving the parallelism of their cubic forms we translate the classification into that of affine hypersurfaces with parallel cubic form, which has been completed by Hu-Li-Vrancken (J Differ Geom 87:239-307, 2011).
收起
摘要 :
We extend the concept of Codazzi-equivalence from Riemannianmetrics in [14] to affine connections. Applications to relative hypersurface the-ory show that this concept simplifies the investigation of pairs of hypersurfaceswith par...
展开
We extend the concept of Codazzi-equivalence from Riemannianmetrics in [14] to affine connections. Applications to relative hypersurface the-ory show that this concept simplifies the investigation of pairs of hypersurfaceswith parallel normalization, moreover we get a better understanding of theaffine GauB maps. We give a new proof of Calabi's global affine Minkowskiproblem; see [1, 10].
收起
摘要 :
We study Lorentzian affine hypersurfaces in R~(n+1) with parallel cubic form with respect to the Levi-Civita connection of the affine metric. As main result, a complete classification of such non-degenerate affine hypersurfaces in R~4 is given.
摘要 :
We prove the following: a relative hypersurface with parallel shape operator is either a relative hypersphere, or it is affinely equivalent to an example constructed by Th. Binder. Furthermore, based on Binder's example, we give a...
展开
We prove the following: a relative hypersurface with parallel shape operator is either a relative hypersphere, or it is affinely equivalent to an example constructed by Th. Binder. Furthermore, based on Binder's example, we give another simple and more explicit example; this way we improve the classification and show that it is completely determined by functions k(t) and C(v(i); t) , the latter being solutions of certain Monge-AmpSre equations. Our example geometrically is constructed from a plane curve and a family of relative hyperspheres. In case of an affine sphere with Blaschke geometry we show that our classification can be considered as a construction coming from a plane curve together with a family of improper affine hyperspheres. Especially in R-4 this construction is determined by only three functions of a single variable.
收起
摘要 :
In this paper, we study a whole family of n-dimensional equiaffine homogeneous hypersurfaces with a parameter alpha, constructed by Eastwood and Ezhov (Proc Steklov Inst Math 253:221-224, 2006), called the generalized Cayley hyper...
展开
In this paper, we study a whole family of n-dimensional equiaffine homogeneous hypersurfaces with a parameter alpha, constructed by Eastwood and Ezhov (Proc Steklov Inst Math 253:221-224, 2006), called the generalized Cayley hypersurfaces. By introducing a new parametrization we find that the generalized Cayley hypersurfaces are improper affine hypersphere with flat affine metric and vanishing Pick invariant, whose difference tensor K satisfies del((alpha)) K = 0 and Kn-1 not equal 0, where the affine alpha-connection del((alpha)) of information geometry is first introduced on affine hypersurface for each alpha is an element of R. As main result, we establish a characterization of the generalized Cayley hypersurfaces by the last two properties for some nonzero constant alpha.
收起
原文获取