Almost Conformal Vector Fields on a Riemannian Manifold

We introduce the notion of an almost conformal vector field, which generalizes conformal vector fields and recently introduced m-modified conformal vector fields on a Riemannian manifold. The definition of an almost conformal vector field ζ on an n-dimensional Riemannian manifold N,g requires two smooth functions σ and f called the potential and copotential and a skew symmetric tensor φ called the associated tensor of ζ. Many examples of almost conformal vector fields which are not conformal vector fields are provided. We find conditions using σ, f and φ under which an almost conformal vector field ζ on an n-dimensional compact Riemannian manifold N,g is either conformal or a Killing vector field. We also find conditions under which a compact Riemannian manifold N,g admitting an almost conformal vector field is isometric to the sphere Sn(c). Finally, we find conditions under which an almost conformal vector field ζ on a noncompact Riemannian manifold N,g is a Killing vector field.