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On Riemannian and Ricci curvatures of homogeneous Finsler manifolds

Part of: Local differential geometry Global differential geometry Noncompact transformation groups

Published online by Cambridge University Press:  25 November 2024

A. Tayebi Open the ORCID record for A. Tayebi [Opens in a new window]
A. Tayebi*
Affiliation:
Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran
*
e-mail: akbar.tayebi@gmail.com
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Abstract

The famous Cheng-Shen’s conjecture in Riemann-Finsler geometry claims that every n-dimensional closed W-quadratic Randers manifold is a Berwald manifold. In this paper, first we study the Riemann and Ricci curvatures of homogeneous Finsler manifolds and obtain some rigidity theorems. Then, by using this investigation, we construct a family of W-quadratic Randers metrics which are not R-quadratic nor strongly Ricci-quadratic.

Keywords

Flag curvatureRiemann curvatureBerwald curvature

MSC classification

Primary: 53B40: Finsler spaces and generalizations (areal metrics)
Secondary: 22F30: Homogeneous spaces 53C30: Homogeneous manifolds

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 1 , March 2025 , pp. 73 - 90
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Akbar-Zadeh, H., Sur les espaces de Finsler á courbures sectionnelles constantes. Bull. Acad. Roy. Bel. Cl, Sci, 5e Série. LXXXIV(1988), 281–322.CrossRefGoogle Scholar
Atashafrouz, M. and Najafi, B., On Cheng-Shen conjecture in Finsler geometry . Int. J. Math. 31(2020), 2050030.CrossRefGoogle Scholar
Bácsó, S. and Matsumoto, M., Randers spaces with the h-curvature tensor $H$ dependent on position alone. Publ. Math. Debrecen. 57(2000), 185–192.CrossRefGoogle Scholar
Berwald, L., Untersuchung der Krümmung allgemeiner metrischer Räume auf Grund des in ihnen herrschenden Parallelismus . Math. Z. 25(1926), 40–73.CrossRefGoogle Scholar
Cheng, X. and Shen, Z., Finsler geometry- An approach via Randers spaces, Springer, Heidelberg and Science Press, Beijing, 2012.Google Scholar
Cheng, X., Shen, Z. and Tian, Y., A class of Einstein $(\alpha, \beta )$ -metrics. Israel. J. Math. 192(2012), 221–249.CrossRefGoogle Scholar
Deng, S. and Hou, Z., Homogeneous Einstein-Randers spaces of negative Ricci curvature. C. R. Math. Acad. Sci. Paris. 347(2009), 1169–1172.Google Scholar
Deng, S. and Hu, Z., On flag curvature of homogeneous Randers spaces . Canad. J. Math. 65(2013), 66–81.CrossRefGoogle Scholar
Hu, Z. and Deng, S., Ricci-quadratic homogeneous Randers spaces . Nonlinear Anal. 92(2013), 130–137.CrossRefGoogle Scholar
Huang, L., Flag curvatures of homogeneous Finsler spaces . Europ. J. Math. 3(2017), 1000–1029.CrossRefGoogle Scholar
Li, B. and Shen, Z., Randers metrics of quadratic Riemann curvature . Int. J. Math. 20(2009), 1–8.CrossRefGoogle Scholar
Mo, X., On the non-Riemannian quantity $H$ of a Finsler metric. Differ. Geom. Appl. 27(2009), 7–14.CrossRefGoogle Scholar
Robles, C., Einstein metrics of Randers type. PhD dissertation, University of British Columbia, 2003.Google Scholar
Shen, Z., Differential geometry of Spray and Finsler spaces, Kluwer Academic Publishers, 2001.CrossRefGoogle Scholar
Shen, Z., On a class of Landsberg metrics in Finsler geometry . Canad. J. Math. 61(2009), no. 6, 1357–1374.CrossRefGoogle Scholar
Tayebi, A. and Najafi, B., A class of homogeneous Finsler metrics . J. Geom. Phys. 140(2019), 265–270.CrossRefGoogle Scholar
Tayebi, A. and Najafi, B., On homogeneous isotropic Berwald metrics . European J. Math. 7(2021), 404–415.CrossRefGoogle Scholar
Tayebi, A. and Najafi, B., On homogeneous Landsberg surfaces . J. Geom. Phys. 168(2021), 104314.CrossRefGoogle Scholar
Wu, B. Y., Some rigidity theorems for Finsler manifolds of sectional flag curvature . Arch. Math (Brno) 46(2010), 99–104.Google Scholar
Yan, Z. and Deng, S., On homogeneous Einstein $\left(\alpha, \beta \right)$ -metrics. J. Geom. Phys. 103(2016), 20–36.CrossRefGoogle Scholar