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A pre-Lie algebroid is an anchored bundle provided with an almost Lie bracketsuch that the anchor is compatible with the Lie bracket of vector fields. Wefirstly show how most geometrical structures intensively studied in theframework of Lie algebroid can easily be extended in the pre-Lie algebroidcontext. The principal purpose of this paper is to show that how all theseresults only depend of the foliated structure and do not depend of the pre-Liealgebroid structure that we can put on a foliated anchored bundle. Asapplication, we obtain a Finsler connection on a foliated anchored bundle whichinduces the classical Finsler connection on each leaf and a similar result forthe Chern connection.

Geometrical structures on the prolongation of a pre-Lie algebroid on  fibered manifolds and application to Partial Finsler geometry on foliated  anchored bundle