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

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