Let M be a 2m+1-dimensional Riemannian manifold and let ∇ be the Levi-Civita connection and ξ be the Reeb vector field, η the Reeb covector field and X be the structure vector field satisfying a certain property on M. In this paper the following properties are proved: (i) ξ and X define a 3-covariant vanishing structure; (ii) the Jacobi bracket corresponding to ξ vanishes; (iii) the harmonic operator acting on Xb givesΔXb=f ||X||2 Xb,which proves that Xb is an eigenfunction of Δ, having f||X||2 as eigenvalue;(iv) the 2-form Ω and the Reeb covector η define a Pfaffian transformation, i.e. LX Ω=0,LX η=0; (v) ∇2 X defines the Ricci tensor;(vi) one has ∇X X = f X, f = scalar, which show that X is an affine geodesic vector field;(vii) the triple (X,ξ,φX) is an involutive 3-distribution on M, in the sense of Cartan.