On Killing Vector Fields on Riemannian Manifolds - Sharief Deshmukh; Olga Belova (Mathematics, 2021)
We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field <semantics>w</semantics> on a connected Riemannian manifold <semantics>(M,g)</semantics> we show that for each non-constant smooth function <semantics>f∈C∞(M)</semantics> there exists a non-zero vector field <semantics>wf</semantics> associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold <semantics>(M,g)</semantics> with an appropriate lower bound on the integral of the Ricci curvature <semantics>S(wf,wf)</semantics> gives a characterization of the odd-dimensional unit sphere <semantics>S2m+1</semantics>. Also, we show on an n-dimensional compact Riemannian manifold <semantics>(M,g)</semantics> that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue <semantics>nc</semantics> and the unit Killing vector field <semantics>w</semantics> satisfying <semantics><mfenced separators="" open="∥" close="∥">∇w</mfenced>2≤(n−1)c</semantics> and Ricci curvature in the direction of the vector field <semantics>∇f−w</semantics> is bounded below by <semantics><mfenced separators="" open="(" close=")">n−1</mfenced>c</semantics> is necessary and sufficient for <semantics>(M,g)</semantics> to be isometric to the sphere <semantics>S2m+1(c)</semantics>. Finally, we show that the presence of a unit Killing vector field <semantics>w</semantics> on an n-dimensional Riemannian manifold <semantics>(M,g)</semantics> with sectional curvatures of plane sections containing <semantics>w</semantics> equal to 1 forces dimension n to be odd and that the Riemannian manifold <semantics>(M,g)</semantics> becomes a K-contact manifold. We also show that if in addition <semantics>(M,g)</semantics> is complete and the Ricci operator satisfies Codazzi-type equation, then <semantics>(M,g)</semantics> is an Einstein Sasakian manifold.