A Note on Killing Calculus on Riemannian Manifolds

In this article, it has been observed that a unit Killing vector field ξ on an n -dimensional Riemannian manifold ( M , g ) , influences its algebra of smooth functions C ∞ ( M ) . For instance, if h is an eigenfunction of the Laplace operator Δ with eigenvalue λ , then ξ ( h ) is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian H h ( ξ , ξ ) of a smooth function h ∈ C ∞ ( M ) defines a self adjoint operator ⊡ ξ 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 ξ . Finally, we find characterizations of the odd dimensional sphere using properties of the operator ⊡ ξ and the nontrivial solution of Fischer–Marsden differential equation, respectively.