A note on killing calculus on Riemannian manifolds

Abstract

In this article, it has been observed that a unit Killing vector field xi on an n-dimensional Riemannian manifold (M,g), influences its algebra of smooth functions C-infinity(M). For instance, if h is an eigenfunction of the Laplace operator Delta with eigenvalue lambda, then xi(h) is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian H-h(xi,xi) of a smooth function h is an element of C-infinity(M) defines a self adjoint operator (sic)xi and has properties similar to most of properties of the Laplace operator on a compact Riemannian manifold (M,g). We study several properties of functions associated to the unit Killing vector field xi. Finally, we find characterizations of the odd dimensional sphere using properties of the operator (sic)xi and the nontrivial solution of Fischer-Marsden differential equation, respectively.