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SUB-WEYL GEOMETRY AND ITS LINEAR CONNECTIONS

The aim of this paper is to study from the point of view of linear connections the data with M a smooth (n+p)-dimensional real manifold, an n-dimensional manifold semi-Riemannian distribution on M, the conformal structure generated by g and W a Weyl substructure: a map such that W(ḡ) = W(g) - du, ḡ = eug;u ∈ C∞(M). Compatible linear connections are introduced as a natural extension of similar notions from Weyl 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 one-form the Cartan form.