On the k-nullity foliations in Finsler geometry

Abstract

Here, a Finsler manifold $(M,F)$ is considered with corresponding <br />curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. Certain <br />subspaces of the tangent spaces of $M$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. <br /> 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 <br />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 Chern and Kuiper and enlarged to Finsler setting by Akbar-Zadeh and contains it as a special case.