Sarita Rani, Gauree Shanker

DOI Number
First page
Last page


The study of curvature properties of homogeneous Finsler spaces with $(\alpha, \beta)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on a homogeneous Finsler space with Randers changed square metric has been proved. Further, an explicit formula for $S$-curvature of Randers changed square metric has been established. Finally, using the formula of $S$-curvature, the mean Berwald curvature of afore said $(\alpha, \beta)$-metric has been calculated. 


Homogeneous Finsler space, square metric, Randers change, invariant vector field, S-curvature, mean Berwald curvature.

Full Text:



S. I. Amari and H. Nagaoka: Methods of Information Geometry. Translations of Mathematical

Monographs, AMS, 191, Oxford Univ. Press, 2000.

P. L. Antonelli, R. S. Ingarden and M. Matsumoto: The Theory of Sprays and Finsler spaces

with Applications in Physics and Biology. Vol. 58, Springer Science & Business Media, 2013.

D. Bao, S. S. Chern and Z. Shen: An Introduction to Riemann-Finsler Geometry. Springer-Verlag,

New York, 2000.

L. Berwald: Uber dien-dimensionalen Geometrien konstanter Krummung, in denen die Geraden die

kurzesten sind. Mathematische Zeitschrift. 30 (1) (1929), 449-469.

X. Cheng and Z. Shen: A class of Finsler metrics with isotropic S-curvature. Israel Journal of

Mathematics. 169 (2009), 317-340.

S. S. Chern: Finsler geometry is just Riemannian geometry without quadratic restriction. Notices of

AMS. 43 (9) (1996), 959-963.

S. S. Chern and Z. Shen: Riemann-Finsler Geometry. Nankai Tracts in Mathematics, Vol. 6, World

Scientic, Singapore, 2005.

S. Deng: The S-curvature of homogeneous Randers spaces. Dierential Geometry and its Applications. 27 (2009), 75-84.

S. Deng: Homogeneous Finsler Spaces. Springer Monographs in Mathematics, New York, 2012.

S. Deng and Z. Hou: The group of isometries of a Finsler space. Pacic J. Math. 207 (2002), 149-155.

S. Deng and Z. Hou: Invariant Randers metrics on Homogeneous Riemannian manifolds. Journal of

Physics A: Mathematical and General. 37 (2004), 4353-4360; Corrigendum, ibid, 39 (2006), 5249-5250.

S. Deng and X. Wang: The S-curvature of homogeneous (; )-metrics. Balkan Journal of Geometry

and Its Applications. 15 (2) (2010), 39-48.

R. Gardner and G. Wilkens: A pseudo-group isomorphism between control systems and certain

generalized Finsler structures. Contemporary Mathematics. 196 (1996), 231-244.

M. Hashiguchi and Y. Ichijyo: Randers spaces with rectilinear geodesics. Rep. Fac. Sci., Kagoshima

Univ., (Math., Phys. & Chem.). 13 (1980), 33-40.

Z. Hu and S. Deng: Homogeneous Randers spaces with isotropic S-curvature and positive

ag curvature. Mathematische Zeitschrift. 270 (2012), 989-1009.

G. Kron: Non-Riemannian dynamics of rotating electrical machinery. Studies in Applied Mathematics.

(1934), 103-194.

M. Matsumoto: On C-reducible Finsler-spaces. Tensor, N. S. 24 (1972), 29-37.

M. Matsumoto: On Finsler spaces with Randers metric and special forms of important tensors. Journal of mathematics of Kyoto University. 14 (3) (1974), 477-498.

K. Nomizu: Invariant ane connections on homogeneous spaces. Amer. J. Math. 76 (1954), 33-65.

G. Randers: On an asymmetric metric in the four-space of general relativity. Phys. Rev. 59 (1941),


G. Shanker and K. Kaur: Naturally reductive homogeneous space with an invariant (; )-metric.

Lobachevskii Journal of Mathematics. 40 (2) (2019), 210-218.

G. Shanker and K. Kaur: Homogeneous Finsler space with innite series (; )-metric. To appear

in Applied Sciences.

G. Shanker and S. Rani: On the rigidity of spherically symmetric Finsler metrics with isotropic

E-curvature. Preprint, arXiv:1808.10667 [math.DG].

G. Shanker, S. Rani and K. Kaur: Dually flat Finsler spaces associated with Randers- change. To

appear in Journal of Rajasthan Academy of Physical Sciences. 18 (1-2), 2019.

Z. Shen: Volume comparison and its applications in Riemann-Finsler geometry. Advances in Mathematics. 128 (1997), 306-328.

Z. Shen: Finsler metrics with K = 0 and S = 0. Canadian Journal of Mathematics. 55 (2003), 112-132.

Z. Shen: Non-positively curved Finsler manifolds with constant S-curvature. Mathematische Zeitschrift. 249 (2005), 625-639.

Z. Shen and C. Yu: On Einstein square metrics. Publicationes Mathematicae Debrecen. 85 (3-4)

(2014), 413-424.

Q. Xia: On a class of projectively flat Finsler metrics. Dierential Geometry and its Applications, 44

(2016), 1-16.

M. Xu and S. Deng: Homogeneous (; )-spaces with positive flag curvature and vanishing S- curvature,. Nonlinear Analysis. 127 (2015), 45-54.

C. Yu and H. Zhu: Projectively flat general (; )-metrics with constant flag curvature. Journal of

Mathematical Analysis and Applications. 429 (2) (2015), 1222-1239.

DOI: https://doi.org/10.22190/FUMI2003673R


  • There are currently no refbacks.

© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)