Construction of the holonomy invariant foliated cocycles on the tangent bundle via formal integrability

Abstract

This paper is dedicated to exhaustive structural analysis of the holonomy invariant foliated cocycles on thetangent bundle of an arbitrary (m + n) -dimensional manifold. For this purpose, by applying Spencer theory of formalintegrability, sufficient conditions for the metric associated with the semispray S are determined to extend to a transversemetric for the lifted foliated cocycle on TM. Accordingly, this geometric structure converts to a holonomy invariantfoliated cocycle on the tangent space, which is totally adapted to the Helmholtz conditions.