On k-nullity foliations in Finsler geometry and completeness

TL;DR

This paper introduces and studies k-nullity foliations in Finsler geometry, establishing their properties, involutivity, and conditions for completeness, extending classical nullity concepts from Riemannian to Finsler manifolds.

Contribution

It defines k-nullity foliations in Finsler manifolds, proves their involutivity and geodesic properties, and links their completeness to the manifold's curvature characteristics, extending Riemannian nullity theory.

Findings

k-nullity foliation is involutive when dimension is constant

Maximal integral manifolds are totally geodesic

Completeness of integral manifolds implies manifold completeness

Abstract

Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant then the distribution is involutive and each maximal integral manifold is totally geodesic. Characterization of the k-nullity foliation is given, as well as some results concerning constancy of the flag curvature, and completeness of their integral manifolds, providing completeness of (M, F). The introduced k-nullity space is a natural extension of nullity space in Riemannian geometry, introduced by S. S. Chern and N. H. Kuiper and enlarged to Finsler setting by H. Akbar-Zadeh and contains it as a special case.