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Conformally flat affine hypersurfaces with semi-parallel cubic form

Huiyang Xu, Cece Li

Abstract

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

Conformally flat affine hypersurfaces with semi-parallel cubic form

Abstract

In this paper, we study locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of affine metric. As the main result, we classify such hypersurfaces being not of flat affine metric. In particular, -dimensional locally strongly convex affine hypersurfaces with semi-parallel cubic form are completely determined.
Paper Structure (4 sections, 12 theorems, 80 equations)

This paper contains 4 sections, 12 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conformally flat affine hypersurfaces with $\hat{R}\cdot C=0$
  4. Proof of Theorem \ref{['thm:1.3']}

Key Result

Theorem 1.1

Let $M$ be an $n$-dimensional ($n\geq2$) locally strongly convex affine hypersurface in $\mathbb{R}^{n+1}$ with $\hat{\nabla} C=0$. Then $M$ is either a hyperquadric (i.e., $C=0$) or a hyperbolic affine hypersphere with $C\not=0$; in the latter case either

Theorems & Definitions (27)

  • Theorem 1.1: cf. HLV
  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1: cf. Theorem 1.1 and Corollary 2.1 of CH
  • ...and 17 more