Abstract
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 is an element of C-infinity(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 S2m+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 nc and the unit Killing vector field w satisfying parallel to del w parallel to(2) <= (n - 1) c and Ricci curvature in the direction of the vector field del f - w is bounded below by (n - 1)c is necessary and sufficient for (M, g) to be isometric to the sphere S2m+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.