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The aim of this paper is to study from the point of view of linearconnections 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 Weylsubstructure: a map $W:$ $\mathcal{G}\to$ $\Omega^{1}(M)$ such that$W(\overline{g})=W(g)-du,$ $\overline{g}=e^{u}g;u\in C^{\infty}(M)$. Compatiblelinear connections are introduced as a natural extension of similar notionsfrom Riemannian geometry and such a connection is unique if a symmetrycondition is imposed. In the foliated case the local expression of this uniqueconnection is obtained. The notion of Vranceanu connection is introduced for apair (Weyl structure, distribution) and it is computed for the tangent bundleof Finsler spaces, particularly Riemannian, choosing as distribution thevertical bundle of tangent bundle projection and as 1-form the Cartan form.

Weyl substructures and compatible linear connections