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Minimal total absolute curvature for equiaffine immersions

Yuta Yamauchi

Abstract

Koike (2001) defined the Lipschitz--Killing curvature and established a Chern--Lashof type inequality for equiaffine immersions of arbitrary codimensions. In this paper, we study the equality case. We prove that the total absolute curvature of an $n$-dimensional equiaffine immersion is equal to $2$ if and only if the image is a convex hypersurface embedded in an $(n+1)$-dimensional affine subspace.

Minimal total absolute curvature for equiaffine immersions

Abstract

Koike (2001) defined the Lipschitz--Killing curvature and established a Chern--Lashof type inequality for equiaffine immersions of arbitrary codimensions. In this paper, we study the equality case. We prove that the total absolute curvature of an -dimensional equiaffine immersion is equal to if and only if the image is a convex hypersurface embedded in an -dimensional affine subspace.

Paper Structure

This paper contains 6 sections, 8 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:main']}
  4. Reduction of codimension
  5. Minimal total absolute curvature and convexity
  6. Examples

Key Result

Theorem A

Let $(M,\nabla,\theta)$ be an oriented compact $n$-dimensional manifold $M$ with an equiaffine structure $(\nabla, \theta)$, and let $f:(M,\nabla,\theta) \to ({\boldsymbol R}^{n+r},\tilde{\nabla}, \omega)$ be an equiaffine immersion into an $(n+r)$-dimensional affine space ${\boldsymbol R}^{n+r}$ wi

Figures (3)

  • Figure 1: Tangent hyperplanes at $f(p_n)$ and $f(q)$ (the proof of $(1)$ of Lemma \ref{['lem:nu_critical_point']}).
  • Figure 2: The correspondence between sequences $\{\phi_n\}$ and $\{p_n\}$ (the proof of $(2)$ of Lemma \ref{['lem:nu_critical_point']}).
  • Figure 3: The convex surface $\Sigma$ in Example \ref{['ex:semidefinite']} and the image of degenerate locus ${\rm Deg} (\alpha_\xi)$ (blue curves). The affine fundamental form of $\Sigma$ is positive semi-definite

Theorems & Definitions (21)

  • Theorem A
  • Definition 2.1: Koi1
  • Definition 2.2
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 11 more