On singular Finsler foliations of $(\alpha ,\beta )$-spaces

Abstract

We investigate singular Finsler foliations (SFFs) on a manifold equipped with an $(\alpha ,\beta )$-metric. To be precise, we verify that any SFF of an $(\alpha ,\beta )$-space is, under some hypotheses on the metric, a singular Riemannian foliation (SRF). This gives a partial answer to the general question "under which conditions a SFF is a SRF with respect to some Riemannian metric". Moreover, we extend the proof of Molino's conjecture to SFFs whenever they are also a SRFs. Finally, we prove equifocality of the regular leaves for a SFF under the same condition.

27 pages, 3 figures