Nonlinear splittings on fibre bundles

TL;DR

This paper introduces nonlinear splittings on fibre bundles, generalizing Ehresmann connections, and explores their properties, applications in geometry and mechanics, and their relation to curvature and reduction techniques.

Contribution

It defines nonlinear splittings, analyzes their properties, and connects them to various geometric and mechanical systems, extending the framework of connections.

Findings

Nonlinear splittings generalize Ehresmann connections.

They relate to Lagrangian systems and Finsler geometry.

A curvature map for nonlinear splittings is introduced.

Abstract

We introduce the notion of a nonlinear splitting on a fibre bundle as a generalization of an Ehresmann connection. We present its basic properties and we pay attention to the special cases of affine, homogeneous and principal nonlinear splittings. We explain where nonlinear splittings appear in the context of Lagrangian systems and Finsler geometry and we show their relation to Routh symmetry reduction, submersive second-order differential equations and unreduction. We define a curvature map for a nonlinear splitting, and we indicate where this concept appears in the context of nonholonomic systems with affine constraints and Lagrangian systems of magnetic type.