Weakly conformal Finsler geometry |
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Authors: | Mehdi Rafie‐Rad |
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Institution: | 1. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran;2. Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran |
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Abstract: | An extension of conformal equivalence for Finsler metrics is introduced and called weakly conformal equivalence and is used to define the weakly conformal transformations. The conformal Lichnerowicz‐Obata conjecture is refined to weakly conformal Finsler geometry. It is proved that: If X is a weakly conformal complete vector field on a connected Finsler space (M, F) of dimension , then, at least one of the following statements holds: (a) There exists a Finsler metric F1 weakly conformally equivalent to F such that X is a Killing vector field of the Finsler metric, (b) M is diffeomorphic to the sphere and the Finsler metric is weakly conformally equivalent to the standard Riemannian metric on , and (c) M is diffeomorphic to the Euclidean space and the Finsler metric F is weakly conformally equivalent to a Minkowski metric on . The considerations invite further dynamics on Finsler manifolds. |
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Keywords: | Finsler metric Randers metric conformal geometry MSC (2010) 53C60 |
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