On Killing Vector Fields on Riemannian Manifolds

We study the influence of a unit Killing vector field on geometry Riemannian manifolds. For given w connected manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists non-zero wf associated with f. In particular, an eigenfunction f Laplace operator n-dimensional compact appropriate lower bound integral Ricci curvature S(wf,wf) gives characterization odd-dimensional sphere S2m+1. Also, if positive constant c and is eigenvalue nc satisfying ∇w2≤(n−1)c in direction ∇f−w bounded below by n−1c necessary sufficient to be isometric S2m+1(c). Finally, presence sectional curvatures plane sections containing equal 1 forces dimension n odd becomes K-contact manifold. also addition complete satisfies Codazzi-type equation, then Einstein Sasakian