Involutive distributions and dynamical systems of second-order type

TL;DR

This paper explores conditions under which vector fields on manifolds with involutive distributions can be transformed into second-order differential equations, introducing coordinate-independent criteria and analyzing the global bundle structure involved.

Contribution

It provides a new coordinate-independent criterion for identifying quadratic type vector fields and examines the global bundle structure induced by these fields and distributions.

Findings

Established a criterion for quadratic type vector fields

Analyzed the global bundle structure of manifolds with involutive distributions

Connected coordinate transformations to second-order differential equations

Abstract

We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define associated connections and we give a coordinate-independent criterion for determining whether the vector field is of quadratic type. Further, we investigate the underlying global bundle structure of the manifold under consideration, induced by the vector field and the involutive distribution.