Fiber-wise linear F-manifolds, compatible flat connections and Euler fields

Abstract

We define a fiber-wise linear (shortly, FWL) F-manifold as an F-manifold on the total space of a vector bundle π: E \rightarrow M for which the multiplication and unit field are FWL tensor fields. We develop a duality between FWL F-manifolds on E and the total space E^{*} of the dual vector bundle and we enrich it with FWL Euler fields and FWL compatible connections. We present examples in dimension two and three. We construct prolongations of F-manifolds and we prove that if the initial F-manifold admits a suitable Euler field or compatible flat connection then all prolongations inherit FWL Euler fields and FWL compatible flat connections.

The following modifications were made with respect to the previous version: the title is changed, the material on generalized geometry is removed (is left for another work) and instead compatible connections and iterations on prolongations are included (Sections 2.2, 3.3, 5.2, 7.3, 8). The introduction and abstract are modified accordingly