Affine Hyperspheres Research Articles

Affine hypersurfaces with the Ricci semi-symmetry

The purpose of the present paper is to prove that either a proper affine hypersphere or an affine cylindrical (See Introduction) is the only nondegenerate affine hypersurface of affine space with torsion free affine connection which satisfies the Ricci semi-symmetry and to study the equivalence of semi-symmetric and Ricci semi-symmetric in the case of affine hypersurfaces

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.

Stability of the Graph of a Convex Function

Let x: M → A n+1 be the graph of some strongly convex function x n+1= ƒ( x1,…,xn) defined on a domain Ω ⊂ A n in a real affine space. We consider the relative metric G, defined by \( G=\sum{\partial^{2}f\over\partial x_{i}\partial x_{j}}dx_{i}dx_{j}\).In this paper, we calculate the second variation of the area integral with respect to the relative metric G. We prove that the parabolic affine hyperspheres are stable.

Proper affine hyperspheres which fiber over projective special Kähler manifolds

Proper affine hyperspheres which fiber over projective special Kähler manifolds

Yang–Mills theory and conjugate connections

We develop a new Yang–Mills theory for connections D in a vector bundle E with bundle metric h, over a Riemannian manifold by dropping the customary assumption Dh=0. We apply this theory to Einstein–Weyl geometry (cf. M.F. Atiyah, et al., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London 362 (1978) 425–461, and H. Pedersen, et al., Einstein–Weyl deformations and submanifolds, Internat. J. Math. 7 (1996) 705–719) and to affine differential geometry (cf. F. Dillen, et al., Conjugate connections and Radon's theorem in affine differential geometry, Monatshefts für Mathematik 109 (1990) 221–235). We show that a Weyl structure ( D, g) on a 4-dimensional manifold is a minimizer of the functional (D,g)↦ 1 2 ∫ M‖R D‖ 2v g if and only if ∗R D=±R D ∗ , where D ∗ is conjugate to D. Moreover, we show that the induced connection on an affine hypersphere M is a Yang–Mills connection if and only if M is a quadratic affine hypersurface.

Three Dimensional Affine Hyperspheres Generated by Two Dimensional Partial Differential Equations

It is well-known that locally strongly convex affine hyperspheres can be determinedas solutions of differential equations of Monge-Ampère type. In this paper we study in particular the 3-dimensional case and we assume that the hypersphere admits a Killing vector field (with respect to the affine metric) whose integral curves are geodesics with respect to both the induced affine connection and the Levi-Civita connection of the affine metric. We show that besides the already known examples, such hyperspheres can be constructed starting from the 2-dimensional Poisson equation, the 2-dimensional sine-Gordon equation or the 2-dimensional cosh-Gordon equation.

Realisation of special Kähler manifolds as parabolic spheres

We prove that any simply connected special Kähler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. We also show that a classical result of Calabi and Pogorelov on parabolic spheres implies Lu’s theorem on complete special Kähler manifolds with a positive definite metric.

The magid-ryan conjecture for 4-dimensional affine spheres

In this paper we prove the Magid-Ryan conjecture for 4-dimensional affine hyperspheres in R5. This conjecture states that every affine hypersphere with non-zero Pick invariant and constant sectional curvature is affinely equivalent with either (x12 ±x22)(x32 ±x42...(x2m−12 ±x2m2) = 1 or (x12 ±x22(x32 ±x42)...(x2m−12 ±x2m2)x2m+1 = 1 where the dimensionn satisfiesn = 2m orn =2m + 1. This conjecture was proved in [11] in case the metric is positive definite and in [2] in case the metric is Lorentzian.

Conformal structure in affine geometry: Complete tchebychev hypersurfaces

We give a conformal classification of affine-complete centroaffine Tchebychev hypersurfaces recently introduced by Liu and Wang. This classification is based on partial differential equations known from conformal Riemannian geometry. Moreover we investigate Tchebychev hyperovaloids and generalize the classical theorem of Blaschke and Deicke on affine hyperspheres.

On locally symmetric affine hypersurfaces

1. Introduction. In the present paper, we investigate 3-dimensional. locally strongly convex affine hyperspheres M in N, ~. We will also assume that the Levi Civita connection ~' of the affine metric h is locally symmetric. This problem is related to the following two questions proposed at the conference on affine differential geometry at Oberwolfach ([10]) in 1986:

Affine 3-spheres with constant affine curvature

We classify the affine hyperspheres in R 4 {R^4} which have constant curvature in the affine metric h h and whose Pick invariant is nonzero. In particular, the metric h h must be flat.

Some theorems in affine differential geometry

In this paper we prove that an affine hypersphere with scalar curvature zero in a unimodular affine space of dimensionn+1 must be contained either in an elliptic paraboloid or in an affine image of the hypersurfacex1x2...xn+1=const. We prove also that an affine complete, affine maximal surface is an elliptic paraboloid if its affine normals omit 4 or more directions in general position.

Locally symmetric affine hypersurfaces

The nondegenerate quadratic hypersurfaces and the improper affine hyperspheres are the only nondegenerate hypersurfaces of dimension greater than two of an affine space which are affine locally symmetric with respect to their induced connection.

Locally Symmetric Affine Hypersurfaces

The nondegenerate quadratic hypersurfaces and the improper affine hyperspheres are the only nondegenerate hypersurfaces of dimension greater than two of an affine space which are affine locally symmetric with respect to their induced connection.

Hyperbolic affine hyperspheres

A locally strongly convex hypersurface in the affine space Rn + 1 is called an affine hypersphere if the affine normals (§ 1) through each point of the hypersurface either all intersect at one point, called its center, or else are all mutually parallel. It is called elliptic, parabolic or hyperbolic according to whether the center is, respectively, on the concave side of the hypersurface, at infinity or on the convex side. This class of hypersurfaces was first studied systematically by W. Blaschke ([1]) in the frame of affine geometry. In his paper [3] E. Calabi redefined it and proposed a problem of determining all complete hyperbolic affine hyperspheres and raised a conjecture that these hypersurfaces are asymptotic to the boundary of a convex cone and every non-degenerate cone V determines a hyperbolic affine hypersphere, asymptotic to the boundary of V, uniquely by the value of its mean curvature.

Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens.

Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens.