Geodesic Vector Fields on a Riemannian Manifold

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to veloci...

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Bibliographic Details
Published in:Mathematics (Basel) 2020-01, Vol.8 (1), p.137
Main Authors: Deshmukh, Sharief, Peska, Patrik, Bin Turki, Nasser
Format: Article
Language:English
Subjects:
Acceleration
Differential geometry
eikonal equation
Euclidean geometry
Euclidean space
Fields (mathematics)
geodesic vector field
Geodesy
Geometry
Inequality
Integrals
isometric to euclidean space
isometric to sphere
Manifolds (mathematics)
Mathematics
Partial differential equations
Riemann manifold
Theory of relativity
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Summary:A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields.
ISSN:2227-7390
2227-7390
DOI:10.3390/math8010137