On Killing Vector Fields on Riemannian Manifolds

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold ( M , g ) we show that for each non-constant smooth function f ∈ C ∞ ( M ) there exists a non-zero vector field w f associated with f . In particular, we show that for an eigenfunction f of the Laplace operator on an n -dimensional compact Riemannian manifold ( M , g ) with an appropriate lower bound on the integral of the Ricci curvature S ( w f , w f ) gives a characterization of the odd-dimensional unit sphere S 2 m + 1 . Also, we show on an n -dimensional compact Riemannian manifold ( M , g ) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue n c and the unit Killing vector field w satisfying ∇ w 2 ≤ ( n − 1 ) c and Ricci curvature in the direction of the vector field ∇ f − w is bounded below by n − 1 c is necessary and sufficient for ( M , g ) to be isometric to the sphere S 2 m + 1 ( c ) . Finally, we show that the presence of a unit Killing vector field w on an n -dimensional Riemannian manifold ( M , g ) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold ( M , g ) becomes a K-contact manifold. We also show that if in addition ( M , g ) is complete and the Ricci operator satisfies Codazzi-type equation, then ( M , g ) is an Einstein Sasakian manifold.