Disaffinity Vectors on a Riemannian Manifold and Their Applications

Abstract

A disaffinity vector on a Riemannian manifold (M,g) is a vector field whose affinity tensor vanishes. In this paper, we observe that nontrivial disaffinity functions offer obstruction to the topology of M and show that the existence of a nontrivial disaffinity function on M does not allow M to be compact. A characterization of the Euclidean space is also obtained by using nontrivial disaffinity functions. Further, we study properties of disaffinity vectors on M and prove that every Killing vector field is a disaffinity vector. Then, we prove that if the potential field ζ of a Ricci soliton M,g,ζ,λ is a disaffinity vector, then the scalar curvature is constant. As an application, we obtain conditions under which a Ricci soliton M,g,ζ,λ is trivial. Finally, we prove that a Yamabe soliton M,g,ξ,λ with a disaffinity potential field ξ is trivial.

Keywords

• affinity tensor
• disaffinity function
• disaffinity vector
• Ricci soliton
• Yamabe soliton
• isometric to Euclidean space