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 eigenfunction the Laplace operator Δ with eigenvalue λ, then ξ(h) also same eigenvalue. Additionally, Hessian Hh(ξ,ξ) function h∈C∞(M) defines self adjoint ⊡ξ and properties similar to most compact (M,g). We study several associated ξ. Finally, we find characterizations odd dimensional sphere using nontrivial solution Fischer–Marsden differential equation, respectively.