Geodesic vector fields and Eikonal equation on a Riemannian manifold |
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Affiliation: | 1. Aalto University, Department of Mathematics and Systems Analysis, P.O. Box 11100, FI 00076, Aalto, Finland;2. Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India;3. Guangdong Technion — Israel Institute of Technology, 241 Daxue Road, 515063 Shantou, Guangdong, People’s Republic of China |
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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. |
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Keywords: | Geodesic vector field Eikonal equation Killing vector field Einstein manifold |
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