Weyl substructures and compatible linear connections

TL;DR

This paper investigates Weyl substructures and compatible linear connections on manifolds with semi-Riemannian distributions, introducing a unique connection under symmetry conditions and applying it to Finsler geometry.

Contribution

It extends the concept of compatible linear connections from Riemannian to Weyl substructures and introduces the Vranceanu connection for Weyl structures on distributions.

Findings

Unique compatible linear connection under symmetry conditions.

Explicit local expression of the connection in foliated cases.

Application to tangent bundles of Finsler spaces.

Abstract

The aim of this paper is to study from the point of view of linear connections the data $(M,\mathcal{D},g,W),$ with $M$ a smooth $(n+p)$ dimensional real manifold, $(\mathcal{D},g)$ a \textit{$n$}\textit{\emph{dimensional semi-Riemannian distribution}}\emph{}on $M,$ $\mathcal{G}$ the conformal structure generated by $g$ and $W$ a Weyl substructure: a map $W:$ $\mathcal{G}\to$ $\Omega^{1}(M)$ such that $W(\overline{g})=W(g)-du,$ $\overline{g}=e^{u}g;u\in C^{\infty}(M)$. Compatible linear connections are introduced as a natural extension of similar notions from Riemannian geometry and such a connection is unique if a symmetry condition is imposed. In the foliated case the local expression of this unique connection is obtained. The notion of Vranceanu connection is introduced for a pair (Weyl structure, distribution) and it is computed for the tangent bundle of Finsler spaces, particularly Riemannian, choosing as distribution the vertical bundle of tangent bundle projection and as 1-form the Cartan form.