The Study of S - Curvature on a Homogeneous Finsler Space with Randers-Matsumoto Metric
DOI:
https://doi.org/10.12723/mjs.64.2Keywords:
Randers-Matsumoto metric, Homogeneous Finsler space, invariant vector field, S-curvature, E-curvatureAbstract
In this article, we have focused on the study of S-curvature of Randers-Matsumoto metric on a homogeneous Finsler space. We have deduced the condition for an isometry of Finsler homogeneous space with Randers-Matsumoto metric to be an isometry of Riemannian homogeneous space and proved that the group of isometries of Finsler space are closed subgroups of that of Riemannian space. We have examined the existence of invariant vector field. Further, we have derived the formula for S-curvature on the reductive homogeneous space, discussed the condition for isotropic S-curvature and derived the E-curvature of the Randers-Matsumoto metric for the homogenous space by using S-curvature formula.
References
P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic, Dordrecht, The Netherlands, (1993).
K. Chandru and S. K. Narasimhamurthy, On curvatures of Homogeneous Finsler-Kropina space, Gulf Journal of Mathematics, 5, pp. 73-83, (2017).
X. Cheng and Z. Shen, A class of Finsler metrics with isotropic -curvature, Israel Journal of Mathematics, 169, pp. 317–340, (2009).
S. S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, Vol. 6, World Scientific, Singapore, (2005).
S. Deng and Z. Hu, Curvatures of homogeneous Randers spaces, Advances in Mathematics, 240, pp. 194–226, (2013).
S. Deng and X. Wang, The -curvature of Homogeneous -metrics, Balkan Journal of Geometry and Its Applications, 15, No. 2, pp. 47-56, (2010).
S. Deng and M. Xu, Recent progress on homogeneous Finsler spaces with positive curvature, European Journal of Mathematics, 3, pp. 974–999, (2017).
M. K. Gupta and P. N. Pandey, On hypersurface of a Finsler space with a special metric, Acta Mathematica Hungarica, 120, no. 1-2, pp. 165–177, (2008).
M. K. Gupta, A. Singh and P. N. Pandey, On a Hypersurface of a Finsler Space with Randers Change of Matsumoto Metric, Geometry, Hindawi Publishing Corporation, 2013, Article ID 842573, 6 pages, (2013).
M. Hashiguchi and Y. Ichijyo, Randers spaces with rectilinear geodesics, Reports of the Faculty of Science of Kagoshima University, no. 13, pp. 33–40, (1980).
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, San Francisco London, ISBN 0123384605, (1978).
M. Matsumoto, On C-reducible Finsler spaces, Tensor N. S., 24, pp. 29–37, (1972).
M. Matsumoto, The induced and intrinsic Finsler connections of a hypersurface and Finslerian projective geometry, Journal of Mathematics of Kyoto University, 25, no. 1, pp. 107–144, (1985).
M. Matsumoto, A slope of a mountain is a Finsler surface with respect to a time measure, Journal of Mathematics of Kyoto University, 29, no. 1, pp. 17–25, (1989).
H. G. Nagaraja and P. Kumar, On Randers change of Matsumoto metric, Bulletin of Mathematical Analysis and Applications, 4, no. 1, pp. 148–155, (2012).
S. Ohta, Vanishing -curvature of Randers spaces, Differential Geometry and its Applications, 29, pp. 174–178, (2011).
L. I. Piscoran and V. N. Mishra, -curvature for a new class of -metrics, RACSAM, 111, pp. 1187–1200, (2017).
M. Rafie-Rad and B. Rezaei, On Einstein Matsumoto metrics, Nonlinear Analysis RWA, 13, no. 2, pp. 882–886, (2012).
G. Shanker and S. Rani, On -curvature of a homogeneous Finsler space with square metric, International Journal of Geometric Methods in Modern Physics, World Scientific Publishing Company, 17, No. 2, 2050019, (2020).
Y. B. Shen and Z. Shen, Introduction to Modern Finsler Geometry, Higher Education Press Limited Company: World Scientific Publishing Co., ISBN 9789814704908, (2016).
Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Advances in Mathematics, 128, pp. 306-328, (1997).
C. Shibata, On invariant tensors of ��-changes of Finsler metrics, Journal of Mathematics of Kyoto University, 24, no. 1, pp. 163–188, (1984).
H. Shimada and S. V. Sabau, Introduction to Matsumoto metric, Nonlinear Analysis, 63, pp. 165–168, (2005).
A. Tayebi, E. Peyghan and H. Sadeghi, On Matsumoto-type Finsler metrics, Nonlinear Analysis, 13, no. 6, pp. 2556– 2561, (2012).
M. Xu and S. Deng, Killing frames and S-curvature of Homogeneous Finsler spaces, Glasgow Math. J., 57, pp. 457–464, (2015).
M. Xu and S. Deng, Homogeneous -spaces with positive flag curvature and vanishing S-curvature, Non-linear Analysis: Theory, Methods and Applications, 127, pp. 45-54, (2015).
Z. Yan and S. Deng, On homogeneous Einstein -metrics, Journal of Geometry and Physics, 103, pp. 20–36, (2016).
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