Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

Abstract

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere Sm(c) of constant curvature c. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation.

Keywords

• Riemannian manifold
• sphere
• conformal vector field
• de-Rham Laplace operator
• Fischer–Marsden differential equation
• Obata’s differential equation