On cylindricity of submanifolds of nonnegative Ricci curvature in a Minkowski space

TL;DR

This paper studies the geometric structure of Finsler submanifolds with nonnegative Ricci curvature in Minkowski spaces, proving they are cylindrical under certain conditions and providing a counterexample related to convexity.

Contribution

It establishes conditions under which submanifolds are cylindrical and presents a novel example of a non-convex surface with positive flag curvature in Minkowski space.

Findings

Submanifolds with nonnegative Ricci curvature are cylindrical under specific inertia conditions.

A non-convex surface with positive flag curvature exists in three-dimensional Minkowski space.

The paper extends understanding of curvature and shape in Finsler geometry.

Abstract

We consider Finsler submanifolds $M^n$ of nonnegative Ricci curvature in a Minkowski space $\mathbb{M}^{n+p}$ which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition on the inertia of the pencil of the second fundamental forms. We give an example of a surface of positive flag curvature in a three-dimensional Minkowski space which is not locally convex.