About Foliations on Distributions of Sub-Finsler Manifolds of Contact Type

Abstract

A sub-Finsler structure of contact type is defined on a smooth manifold with a distribution of codimension 1 given on it and is reduced to specifying a smooth function on this distribution that repeats the standard properties of the fundamental function of a Finsler manifold. For the convenience of studying the sub-Finsler structure, the definition of a (structural) vector field that does not vanish anywhere and is transversal to the distribution is postulated. On a manifold with a sub-Finsler structure – on a sub-Finsler manifold – the parallel transport of vectors belonging to the distribution along curves tangent to the distribution is determined. The connection that ensures this parallel transfer is called internal connectivity in this work. Using the internal connection and the structure vector field, a contact-type sub-Riemannian structure with a Sasaki-type metric is defined on the distribution of a sub-Finsler manifold as on the total space of a vector bundle. The connections between the geometry of foliations that naturally arise on the distributions of sub-Finsler manifolds and the geometry of sub-Finsler manifolds of contact type are studied. In particular, we prove that a vertical foliation on the distribution of a sub-Finsler manifold is completely geodesic if and only if the specified manifold is a Landsberg manifold with a projectable sub-Finsler structure.