Geometry of conformal vector fields

Abstract

It is well known that the Euclidean space (Rn,〈,〉), the n-sphere Sn(c) of constant curvature c and Euclidean complex space form (Cn,J,〈,〉) are examples of spaces admitting conformal vector fields and therefore conformal vector fields are used in obtaining characterizations of these spaces. In this article, we study the conformal vector fields on a Riemannian manifold and present the existing results as well as some new results on the characterization of these spaces. Taking clue from the analytic vector fields on a complex manifold, we define φ-analytic conformal vector fields on a Riemannian manifold and study their properties as well as obtain characterizations of the Euclidean space (Rn,〈,〉) and the n-sphere Sn(c) of constant curvature c using these vector fields.

Keywords

• Conformal vector fields
• Ricci curvature
• Scalar curvature
• Obata’s theorem
• Laplacian
• φ-analytic vector fields