On a class of even-dimensional manifolds structured by an affine connection

We deal with a 2m-dimensional Riemannian manifold (M,g) structured by an affine connection and a vector field , defining a -parallel connection. It is proved that is both a torse forming vector field and an exterior concurrent vector field. Properties of the curvature 2-forms are established. It is shown that M is endowed with a conformal symplectic structure Ω and defines a relative conformal transformation of Ω