Nullity distributions associated with Chern connection

TL;DR

This paper investigates the nullity distributions of curvature tensors related to the Chern connection in Finsler geometry, proving properties about their integrability and providing counterexamples and a class of Landsberg spaces with singularities.

Contribution

It analyzes the nullity distributions of curvature tensors in Finsler geometry, establishing their properties and providing counterexamples and new classes of spaces.

Findings

Nullity distribution $ abla_{R^*}$ is not the kernel of $ar{R}$.

Nullity distribution $ abla_{P^*}$ is not fully integrable.

A class of non-Berwaldian Landsberg spaces with singularities is constructed.

Abstract

The nullity distributions of the two curvature tensors \, $\overast{R}$ and $\overast{P}$ of the Chern connection of a Finsler manifold are investigated. The completeness of the nullity foliation associated with the nullity distribution $\N_{R^\ast}$ is proved. Two counterexamples are given: the first shows that $\N_{R^\ast}$ does not coincide with the kernel distribution of \, $\overast{R}$; the second illustrates that $\N_{P^\ast}$ is not completely integrable. We give a simple class of a non-Berwaldian Landsberg spaces with singularities.