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Disaffinity Vectors on a Riemannian Manifold and Their Applications

Author

Listed:
  • Sharief Deshmukh

    (Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
    These authors contributed equally to this work.)

  • Amira Ishan

    (Department of Mathematics, College of Science, Taif University, Taif 21944, Saudi Arabia
    These authors contributed equally to this work.)

  • Bang-Yen Chen

    (Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Sharief Deshmukh & Amira Ishan & Bang-Yen Chen, 2024. "Disaffinity Vectors on a Riemannian Manifold and Their Applications," Mathematics, MDPI, vol. 12(23), pages 1-10, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3659-:d:1527179
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    References listed on IDEAS

    as
    1. Hanan Alohali & Sharief Deshmukh & Bang-Yen Chen & Hemangi Madhusudan Shah, 2024. "Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications," Mathematics, MDPI, vol. 12(17), pages 1-14, August.
    2. Adara M. Blaga & Sharief Deshmukh, 2024. "Some Properties of the Potential Field of an Almost Ricci Soliton," Mathematics, MDPI, vol. 12(19), pages 1-15, September.
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