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 $\mathbf{w}$ on a connected Riemannian manifold $(M,g)$ we show that for each non-constant smooth function $f\in C^{\infty }\left(M\right)$ there exists a non-zero vector field ${\mathbf{w}}^f$ 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 $S({\mathbf{w}}^f,{\mathbf{w}}^f)$ gives a characterization of the odd-dimensional unit sphere ${\mathbf{S}}^{2m+1}$. 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 $\mathbf{w}$ satisfying ${∥\nabla \mathbf{w}∥}^2\le (n-1)c$ and Ricci curvature in the direction of the vector field $\nabla f-\mathbf{w}$ is bounded below by $\left(n-1\right)c$ is necessary and sufficient for $(M,g)$ to be isometric to the sphere ${\mathbf{S}}^{2m+1}\left(c\right)$. Finally, we show that the presence of a unit Killing vector field $\mathbf{w}$ on an n-dimensional Riemannian manifold $(M,g)$ with sectional curvatures of plane sections containing $\mathbf{w}$ 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.