A compactness theorem in Finsler geometry

TL;DR

This paper proves a compactness theorem for complete Finsler manifolds based on integral conditions of a curvature invariant along geodesics emanating orthogonally from a submanifold.

Contribution

It establishes a new criterion for compactness in Finsler geometry using integral bounds of Ric_k curvature along specific geodesics.

Findings

If the integral of Ric_k along geodesics is positive, then the manifold is compact.

The result generalizes classical compactness theorems by involving a curvature invariant.

Provides a new link between curvature integrals and topological properties in Finsler geometry.

Abstract

Let (M.F) be a complete Finsler manifold and P be a minimal and compact submanifold of M. Ric_k(x), x in M is a differential invariant that interpolates between the flag curvature and the Ricci curvature. We prove that if on any geodesic c(t) emanating orthogonally from P we have \int_{0}^{\infty}\mathbf{Ric}_{k}(t)>0, then M is compact.