Centroaffine Hypersurfaces Research Articles

On conformally flat centroaffine hypersurfaces with semi-parallel cubic form

In this paper, we study locally strongly convex centroaffine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of centroaffine metric. As the main result, we classify such hypersurfaces being not of flat centroaffine metric. In particular, 2,3-dimensional locally strongly convex centroaffine hypersurfaces with semi-parallel cubic form are completely determined.

Classification of isotropic Calabi hypersurfaces

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.

Lightlike pseudo-isotropic centroaffine Lorentzian hypersurfaces of dimension 3

In this paper, we classify locally lightlike pseudo-isotropic but not pseudo-isotropic Lorentzian centroaffine hypersurfaces in R4. We prove that they have a structure of warped product manifold and they are all part of hyperquadrics in R4.

Classification of locally strongly convex isotropic centroaffine hypersurfaces

In this paper, we establish a complete classification of locally strongly convex isotropic centroaffine hypersurfaces in the (n+1)-dimensional affine space Rn+1.

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

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field (sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\widetilde{J}$-tangent affine hyperspheres with the property that at least one null-direction of the second fundamental form coincides with either $\mathcal{D}^+$ or $\mathcal{D}^-$. The main purpose of this paper is to give a full local classification of the above mentioned hypersurfaces. In particular, we prove that every nondegenerate centro-affine hypersurface of dimension $3$ with a $\widetilde{J}$-tangent centro-affine vector field which has two null-directions $\mathcal{D}^+$ and $\mathcal{D}^-$ must be both an affine hypersphere and a hyperquadric. Some examples of these hypersurfaces are also given.

An Optimal Inequality on Locally Strongly Convex Centroaffine Hypersurfaces

In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in $\mathbb{R}^{n+1}$ involving the norm of the covariant derivatives of both the difference tensor $K$ and the Tchebychev vector field $T$. Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in [4], we can completely classify the hypersurfaces which realize the equality case of the inequality.

$\delta^{\sharp}(2,2)$-Ideal Centroaffine Hypersurfaces of Dimension $5$

The notion of an ideal submanifold was introduced by Chen at the end of the last century. A survey of recent results in this area can be found in his book [9]. Recently, in [10], an optimal collection of Chen's inequalities was obtained for Lagrangian submanifolds in complex space forms. As shown in [2], these inequalities have an immediate counterpart in centroaffine differential geometry. Centroaffine hypersurfaces realising the equality in one of these inequalities are called ideal centroaffine hypersurfaces. So far, most results in this area have only been related with $3$- and $4$-dimensional $\delta^{\sharp}(2)$-ideal centroaffine hypersurfaces. The purpose of this paper is to classify $\delta^{\sharp}(2,2)$-ideal hypersurfaces of dimension~$5$ in centroaffine differential geometry.

Classification of the Locally Strongly Convex Centroaffine Hypersurfaces with Parallel Cubic Form

In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the complete classification of locally strongly convex equiaffine hypersurfaces with parallel cubic form due to Hu, Li and Vrancken [12].

Three-dimensional locally homogeneous nondegenerate centroaffine hypersurfaces with nondiagonalizable Tchebychev operator

Liu and Wang (Beitrage Algebra Geom 35:109–117, 1994) determined homogeneous nondegenerate centroaffine surfaces in the 3-dimensional affine space. However, n-dimensional homogeneous or locally homogeneous nondegenerate centroaffine hypersurfaces for \(n\ge 3\) have not been completely classified yet. In this paper, we determine 3-dimensional locally homogeneous nondegenerate centroaffine hypersurfaces with nondiagonalizable Tchebychev operator.

Completeness of hyperbolic centroaffine hypersurfaces

This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity conditions on the boundary of the convex cone generated by the hypersurface. The main result is that completeness holds for hyperbolic components of level sets of homogeneous cubic polynomials. This implies that every such component defines a complete quaternionic K\"ahler manifold of negative scalar curvature.

Three dimensional locally homogeneous nondegenerate centroaffine hypersurfaces with null Tchebychev vector field

Liu and Wang (Beitrage Algebra Geom 35:109–117, 1994) determined homogeneous nondegenerate centroaffine surfaces in the 3-dimensional affine space. However, n-dimensional homogeneous or locally homogeneous nondegenerate centroaffine hypersurfaces for \(n\ge 3\) have not been completely classified yet. In this paper,we determine 3-dimensional locally homogeneous nondegenerate centroaffine hypersurfaces with null Tchebychev vector field.

Realization of Robertson–Walker spacetimes as affine hypersurfaces

Due to the growing interest in embeddings of spacetimes in higher dimensional spaces, we consider a special type of embedding. We prove that Robertson–Walker spacetimes can be embedded as centroaffine hypersurfaces and graph hypersurfaces in some affine spaces in such a way that the induced relative metrics are exactly the Lorentzian metrics on the Robertson–Walker spacetimes. Such realizations allow us to view Robertson–Walker spacetimes and their submanifolds as affine submanifolds in a natural way. Consequently, our realizations make it possible to apply the tools of affine differential geometry to study Robertson–Walker spacetimes and their submanifolds.

Geometry of Affine Warped Product Hypersurfaces

The purpose of this article is the study of warped product manifolds which can be realized either as centroaffine or graph hypersurfaces in some affine space. First, we show that there exist many such realizations. Then we establish general optimal inequalities in terms of the warping function and the Tchebychev vector field for such affine hypersurfaces. We also investigate warped product affine hypersurfaces which verify the equality case of the inequalities. Several applications are also presented.

An Optimal Inequality and Extremal Classes of Affine Spheres in Centroaffine Geometry

A hypersurface f : M → Rn+1 in an affine (n+1)-space is called centroaffine if its position vector is always transversal to f*(TM) in Rn+1. In this paper, we establish a general optimal inequality for definite centroaffine hypersurfaces in Rn+1 involving the Tchebychev vector field. We also completely classify the hypersurfaces which verify the equality case of the inequality.

δ-invariants and their applications to centroaffine geometry

We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in R n + 1 . The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in R n + 1 , n ⩾ 3 , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article.

Conformally Flat Semi-Riemannian Manifolds with Nilpotent Ricci Operators and Affine Differential Geometry

We investigate the conformally flat semi-Riemannian manifolds withnilpotent Ricci operators. We construct a lot of complete orhomogeneous, conformally flat semi-Riemannian manifolds with nilpotentRicci operators. In this construction, we show interesting relationsbetween the semi-Riemannian geometry and the affine differentialgeometry of centro-affine hypersurfaces.

The Centroaffine Volume of Generalized Geodesic Balls Under Inversion at the Sphere

On an analytic Riemannian manifold (M,g), several authors have studied the Taylor expansion for the volume of geodesic balls under the exponential mapping. In the foregoing paper [1] we studied a more general structure (M,D,g), where D is a torsion-free and Ricci-symmetric connection. We calculated the Taylor expansion up to order (n+4) for the volume of what we called a generalized geodesic ball under the exponential mapping in case that all metric notions are Riemannian, while the exponential mapping is induced from the connection D. For the structure % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $(M,D,{\cal G})$ the coefficients of the Taylor expansion are much more complicated than in the Riemannian case. It is one of the main objectives of the present paper to study centroaffine hypersurfaces in Euclidean space, their geometric invariants which appear in the very complicated coefficient of order (n+4), and their behaviour under polarization (inversion at the unit sphere). Our results complement applications in the foregoing paper [1], where mainly the coefficients up to order (n+2) and geometric consequences have been studied.