We deal with a 2m‐dimensional Riemannian manifold (M, g) structured by an affine connection and a vector field 𝒯, defining a 𝒯‐parallel connection. It is proved that 𝒯 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 𝒯 defines a relative conformal transformation of Ω.