Applications of Disaffinity Vectors to Certain Riemannian Manifolds

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

Mathematics, 2024, vol. 12, issue 24, 1-13

Abstract: A disaffinity vector on a Riemannian manifold is a vector field whose affinity tensor vanishes. In this paper, we prove that every disaffinity vector on a compact Riemannian manifold is an incompressible vector field. Then, we discover a sufficient condition for an incompressible vector field to be disaffinity. Next, we study trans-Sasakian 3-manifolds whose Reeb vector field is disaffinity and obtain two sufficient conditions for a trans-Sasakian 3-manifold to be homothetic to a Sasakian 3-manifold. Finally, we prove that a complete Riemannian manifold admitting a non-harmonic disaffinity function satisfying the Eikonal equation and a Ricci curvature inequality is isometric to a Euclidean space.

Keywords: disaffinity function; disaffinity vector; incompressible vector field; Eikonal equation; trans-Sasakian manifolds; Sasakian manifold; euclidean space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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