On concurrent vector fields on Riemannian manifolds

It is shown that the presence of a non-zero concurrent vector field on a Riemannian manifold poses an obstruction to its topology as well as certain aspects of its geometry. It is shown that on a compact Riemannian manifold, there does not exist a non-zero concurrent vector field. Also, it is shown that a Riemannian manifold of non-zero constant scalar curvature does not admit a non-zero concurrent vector field. It is also shown that a non-zero concurrent vector field annihilates de-Rham Laplace operator. Finally, we find a characterization of a Euclidean space using a non-zero concurrent vector field on a complete and connected Riemannian manifold.