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.