On concurrent vector fields on Riemannian manifolds

Amira Ishan; Amira Ishan

doi:10.3934/math.20231281

AIMS Mathematics

2023, Volume 8, Issue 10: 25097-25103. doi: 10.3934/math.20231281

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Research article

On concurrent vector fields on Riemannian manifolds

Amira Ishan ^,

Department of Mathematics, College of Science, Taif University, Taif 21944, Saudi Arabia

Received: 09 June 2023 Revised: 12 August 2023 Accepted: 22 August 2023 Published: 29 August 2023
MSC : 53C15, 53C40

It is shown that the presence of a non-zero concurrent vector field on a Riemannian manifold poses an obstruction to its topology as well as certain aspects of its geometry. It is shown that on a compact Riemannian manifold, there does not exist a non-zero concurrent vector field. Also, it is shown that a Riemannian manifold of non-zero constant scalar curvature does not admit a non-zero concurrent vector field. It is also shown that a non-zero concurrent vector field annihilates de-Rham Laplace operator. Finally, we find a characterization of a Euclidean space using a non-zero concurrent vector field on a complete and connected Riemannian manifold.
- concurrent vector fields,
- Euclidean spaces,
- de-Rham Laplace operator,
- Ricci curvature
Citation: Amira Ishan. On concurrent vector fields on Riemannian manifolds[J]. AIMS Mathematics, 2023, 8(10): 25097-25103. doi: 10.3934/math.20231281

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Abstract

It is shown that the presence of a non-zero concurrent vector field on a Riemannian manifold poses an obstruction to its topology as well as certain aspects of its geometry. It is shown that on a compact Riemannian manifold, there does not exist a non-zero concurrent vector field. Also, it is shown that a Riemannian manifold of non-zero constant scalar curvature does not admit a non-zero concurrent vector field. It is also shown that a non-zero concurrent vector field annihilates de-Rham Laplace operator. Finally, we find a characterization of a Euclidean space using a non-zero concurrent vector field on a complete and connected Riemannian manifold.

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