On Killing Vector Fields on Riemannian Manifolds

Sharief Deshmukh,Olga Belova

doi:10.3390/math9030259

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∈C∞(M) there exists a non-zero vector field wf 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(wf,wf) 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 ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c 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.

Highlights

Killing vector fields are known to play vital role in influencing the geometry as well as topology of Riemannian manifolds and being incompressible fields play important role in physics
If we restrict the length of a Killing vector fields such as constant length, it severely restricts the geometry of Riemannian manifolds on which they are set
Of w and the ellipsoid is invariant under the flow {φt }, the vector field w is tangent to the hypersurface M and with respect the induced metric g, we have g(w, w) = 1, i.e., w is a unit Killing vector field on the compact Riemannian manifold ( M, g)

Summary

Introduction

Killing vector fields are known to play vital role in influencing the geometry as well as topology of Riemannian manifolds (see [1,2,3,4,5,6,7,8,9,10]) and being incompressible fields play important role in physics (see [11]). There are unit Killing vector fields on odd-dimensional spheres. We consider the associated vector field w f corresponding to eigenfunction f of Laplace operator on a compact Riemannian manifold ( M, g) corresponding to eigenvalue dim M and show that if the integral of Ricci curvature in the direction of w f is bounded below by certain bound, forces dim M to be odd and gives a necessary and sufficient for ( M, g) to be isometric to the unit sphere S2m+1 (see Theorem 1). As a by-product of Theorem 3 with an additional condition that the Ricci operator is Codazzi type tensor on a Riemannian manifold ( M, g) that admits a unit Killing vector field w, we show that ( M, g) is an Einstein Sasakian manifold (see Corollary 2)

Preliminaries

Characterizations of Spheres

Killing Vector Fields and Sasakian Manifolds