Almost Fedosov and metriplectic structures in the geometry of semisprays

Given a pair (semispray S, almost symplectic form ω on a tangent bundle, the family of nonlinear connections N such that ω is recurrent with respect to (S,N) with a fixed recurrent factor is determined by using the Obata tensors. In particular, we obtain a characterization for a pair (N, ω) to be recurrent as well as for the triple (S, cN, ω) where cN is the canonical nonlinear connection of the semispray S. In the particular case of vanishing recurrence factor we get the family of almost Fedosov structures associated to a fixed semispray and almost symplectic structure. For a triple (semispray S, almost symplectic form ω, metric g), a characterization for existence of a corresponding almost metriplectic structure is obtained.