Disaffinity Vectors on a Riemannian Manifold and Their Applications

Sharief Deshmukh, Amira Ishan and Bang-Yen Chen ()
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Sharief Deshmukh: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Amira Ishan: Department of Mathematics, College of Science, Taif University, Taif 21944, Saudi Arabia
Bang-Yen Chen: Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA

Mathematics, 2024, vol. 12, issue 23, 1-10

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; Killing vector field (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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