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 $n$-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.