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mmr 2020, 6(4): 621-630 |
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Tayebi A, Bahadori M, Sadeghi H. λ-Projectively Related Finsler Metrics and Finslerian Projective Invariants. mmr 2020; 6 (4) :621-630
URL: http://mmr.khu.ac.ir/article-1-2868-en.html
URL: http://mmr.khu.ac.ir/article-1-2868-en.html
1- University of Qom , akbar.tayebi@gmail.com
2- University of Qom
2- University of Qom
Abstract: (2690 Views)
Introduction
In this paper, by using the concept of spherically symmetric Finsler metric, we define the notion of
-projectively related metrics as an extension of projectively related metrics. We construct some non-trivial examples of -projectively related metrics. Let F and 
be two -projectively related metrics on a manifold M. We find the relation between the geodesics of F and and prove that any geodesic of F is a multiple of a geodesic of and the other way around. There are several projective invariants of Finsler metrics, namely, Douglas metrics, Weyl metrics and generalized Douglas-Weyl curvature. We prove that the Douglas metrics, Weyl metrics and generalized Douglas-Weyl metrics are -projective invariants.
Material and methods
First we obtain the spray coefficients of a spherically symmetric Finsler metric. By considering it, we define-projectively related metrics which is a generalization of projectively related Finsler metrics. Then we find the geodesics of two -projectively related metrics. We obtain the relation between Douglas, Weyl and generalized Douglas-Weyl curvatures of two -projectively related metrics.
Results and discussion
We find the Douglas curvature, Weyl curvature and generalized Douglas-Weyl curvature of two-projectively related Finsler metrics. These calculations tell us that these class of Finsler metrics are -projective invariants.
Conclusion
The following conclusions were drawn from this research.
In this paper, by using the concept of spherically symmetric Finsler metric, we define the notion of
-projectively related metrics as an extension of projectively related metrics. We construct some non-trivial examples of -projectively related metrics. Let F and be two -projectively related metrics on a manifold M. We find the relation between the geodesics of F and and prove that any geodesic of F is a multiple of a geodesic of and the other way around. There are several projective invariants of Finsler metrics, namely, Douglas metrics, Weyl metrics and generalized Douglas-Weyl curvature. We prove that the Douglas metrics, Weyl metrics and generalized Douglas-Weyl metrics are -projective invariants.Material and methods
First we obtain the spray coefficients of a spherically symmetric Finsler metric. By considering it, we define
-projectively related metrics which is a generalization of projectively related Finsler metrics. Then we find the geodesics of two -projectively related metrics. We obtain the relation between Douglas, Weyl and generalized Douglas-Weyl curvatures of two -projectively related metrics.Results and discussion
We find the Douglas curvature, Weyl curvature and generalized Douglas-Weyl curvature of two
-projectively related Finsler metrics. These calculations tell us that these class of Finsler metrics are -projective invariants.Conclusion
The following conclusions were drawn from this research.
- We prove that the Douglas curvature, Weyl curvature and generalized Douglas-Weyl curvature are -projective invariants.
- Let F and be two -projectively related metrics on a manifold M. We show that F is a Berwald metric if and only if is a Berwald metric. ./files/site1/files/64/12.pdf
Keywords: Projective invariant, Projectively flat metric, Projectively related metrics, Douglas metric, Weyl metric, Generalized Douglas-Weyl metric.
Type of Study: Original Manuscript |
Subject:
alg
Received: 2018/10/31 | Revised: 2021/02/16 | Accepted: 2019/07/23 | Published: 2021/01/29 | ePublished: 2021/01/29
Received: 2018/10/31 | Revised: 2021/02/16 | Accepted: 2019/07/23 | Published: 2021/01/29 | ePublished: 2021/01/29
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