Minimal total absolute curvature for equiaffine immersions

TL;DR

This paper characterizes when the total absolute curvature of an equiaffine immersion reaches its minimal value, showing it occurs precisely for convex hypersurfaces in an affine subspace.

Contribution

It proves that the total absolute curvature equals 2 if and only if the immersion is a convex hypersurface embedded in a lower-dimensional affine space.

Findings

Total absolute curvature equals 2 for convex hypersurfaces.

Characterization of equality case in Lipschitz--Killing curvature inequality.

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.