The total absolute curvature of submanifolds with singularities

TL;DR

This paper extends the Chern-Lashof theorem to singular submanifolds called frontals, establishing a lower bound on total absolute curvature related to Betti numbers and characterizing convexity in special cases.

Contribution

It generalizes classical curvature bounds to singular frontals and provides conditions under which these frontals are convex domains.

Findings

Total absolute curvature is at least the sum of Betti numbers.

Equality in total absolute curvature implies the frontal is a convex domain.

Singularities of the first kind lead to convexity when curvature equals 2.

Abstract

In this paper, we give a generalization of the Chern-Lashof theorem for submanifolds with singularities called frontals in Euclidean space. We prove that, for an $n$-dimensional admissible compact frontal in $(n+r)$-dimensional Euclidean space $\boldsymbol{R}^{n+r}$, its total absolute curvature is greater than or equal to the sum of the Betti numbers. Furthermore, if the total absolute curvature is equal to $2$, and all singularities are of the first kind, then the image of the frontal coincides with a closed convex domain of an affine $n$-dimensional subspace of $\boldsymbol{R}^{n+r}$.