A note on "Sur le noyau de l'op\'{e}rateur de courbure d'une vari\'{e}t\'{e} finsl\'{e}rienne" [C. R. Acad. Sci. Paris, t. 272 (1971), 807-810]

TL;DR

This paper examines the kernel of the h-curvature in Finsler geometry, providing a counterexample to previous claims and establishing conditions under which the kernel and nullity distribution coincide.

Contribution

It offers a counterexample to prior results and identifies conditions for the kernel of h-curvature and nullity distribution to be equivalent in Finsler geometry.

Findings

Counterexample showing kernel and nullity distribution do not always coincide

Conditions identified for their equivalence

Clarification of the relationship between curvature kernel and nullity in Finsler geometry

Abstract

In this note, adopting the pullback formalism of global Finsler geometry, we show by a counterexample that the kernel $\text{Ker}_R$ of the h-curvature $R$ of Cartan connection and the associated nullity distribution $\N_R$ do not coincide, contrary to Akbar-Zadeh's result \cite{akbar.nul3.}. We give sufficient conditions for $\text{Ker}_R$ and $\N_R$ to coincide.