On some aspects of the geometry of non integrable distributions and applications

TL;DR

This paper explores the geometry of non-integrable distributions on Riemannian manifolds, introducing intrinsic connections and second fundamental forms to characterize involutive and geodesic distributions, with applications to constrained mechanical systems.

Contribution

It introduces a novel comparison of two covariant derivatives and curvatures related to distributions, providing new tools for geometric and mechanical analysis.

Findings

Second fundamental form characterizes involutive distributions.

Comparison of intrinsic and ambient curvatures offers new insights.

Framework applies to mechanical systems with constraints.

Abstract

We consider a regular distribution $\mathcal{D}$ in a Riemannian manifold $(M,g)$. The Levi-Civita connection on $(M,g)$ together with the orthogonal projection allow to endow the space of sections of $\mathcal{D}$ with a natural covariant derivative, the intrinsic connection. Hence we have two different covariant derivatives for sections of $\mathcal{D}$, one directly with the connection in $(M,g)$ and the other one with this intrinsic connection. Their difference is the second fundamental form of $\mathcal{D}$ and we prove it is a significant tool to characterize the involutive and the totally geodesic distributions and to give a natural formulation of the equation of motion for mechanical systems with constraints. The two connections also give two different notions of curvature, curvature tensors and sectional curvatures, which are compared in this paper with the use of the second fundamental form.