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Geodesic vector fields and Eikonal equation on a Riemannian manifold
- Source :
- Indagationes Mathematicae. 30:542-552
- Publication Year :
- 2019
- Publisher :
- Elsevier BV, 2019.
-
Abstract
- In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.
- Subjects :
- Geodesic
Eikonal equation
Euclidean space
General Mathematics
010102 general mathematics
Mathematical analysis
010103 numerical & computational mathematics
Einstein manifold
Riemannian manifold
01 natural sciences
Killing vector field
Mathematics::Metric Geometry
Vector field
Mathematics::Differential Geometry
0101 mathematics
Ricci curvature
Mathematics
Subjects
Details
- ISSN :
- 00193577
- Volume :
- 30
- Database :
- OpenAIRE
- Journal :
- Indagationes Mathematicae
- Accession number :
- edsair.doi...........2aa4a67c086344b3d4b61ba90c66cbc7