A characteristic property of Sasakian manifolds

We study the case when a unit vector field ξ on a Riemannian manifold (M,g) defines an isometric embedding ξ:(M,g)→(T1M, G), where G is the Riemannian g-natural metric. The main goal is to find conditions under which the submanifold ξ(M)⊂(T1M, G) can be totally geodesic. It is proved that the Reeb vector field of a K-contact metric structure on M gives rise to totally geodesic ξ(M) if and only if the structure is Sasakian. As a by-product, we find the expression for the second fundamental form of ξ(M)⊂(T1M, G).