On Locally Projectively Flat Finsler Space of a Special Exponential Metric with Constant Flag Curvature

Brijesh Kumar Tripathi; Sadika Khan

doi:10.52737/18291163-2024.16.15-1-11

Authors

Brijesh Kumar Tripathi L. D. College of Engineering
Sadika Khan Gujarat Technological University

DOI:

https://doi.org/10.52737/18291163-2024.16.15-1-11

Keywords:

Finsler Space, $(\alpha,\beta)$-Metric, Special Exponential Metric, Locally Projectively Flat, Flag Curvature, Minkowskian Space

Abstract

From the point of view of Hilbert's fourth problem, Finsler metrics on an open subset of $\mathbb{R}^n$ with positive geodesics that are straight lines are known as locally projectively flat Finsler metrics. In this article, we have studied such projectively flat $(\alpha,\beta)$-metrics in the form of the special exponential Finsler metric, where $\alpha$ is a Riemannian metric and $\beta$ is a differential 1-form. We found that the special exponential metric is locally projectively flat if and only if $\alpha$ is locally projectively flat and $\beta$ is parallel with respect to $\alpha$. Furthermore, we obtained the flag curvature and proved that the special exponential metric is locally Minkowskian.

References

A. Ajaykumar and P. Kumar, Geometry of locally projectively flat Finsler space with certain (α,β)-metric. J. Appl. Math. and Informatics, 41 (2023), no. 1, pp. 193-203.

P.L. Antonelli, R.S. Ingarden and M. Matsumoto, The theory of sprays and Finsler spaces with applications in physics and biology, Kluwer Acad. Publishers, Netherlands, 1993. DOI: https://doi.org/10.1007/978-94-015-8194-3

S. Báscó, X. Cheng and Z. Shen, Curvature properties of (α,β)-metrics. Advanced Studies in Pure Mathematics, Math. Soc. Japan, 48 (2007), pp. 73-110. DOI: https://doi.org/10.2969/aspm/04810073

L. Benling, Projectively flat Matsumoto metric and its approximation. Acta Math. Sci., 27 (2007), no. 4, pp. 781-789. DOI: https://doi.org/10.1016/S0252-9602(07)60075-7

T.Q. Binh and X. Cheng, On a class of projectively flat (α,β)-Finsler metrics. Publ. Math. Debrecen., 73 (2008), no. 3 and 4, pp. 391-400. DOI: https://doi.org/10.5486/PMD.2008.4295

S.S. Chern and Z. Shen, Riemann Finsler geometry, World Scientific Publishing Company, 6, 2005. DOI: https://doi.org/10.1142/5263

B. C. Chethana and S. K. Narsimhamurthy, Locally projectively flat special (α,β)-metric. Palest. J. Math., 10 (2021), pp. 69-74.

Y. Feng and Q. Xia, On a class of locally projectively flat Finsler metrics (ii). Differ. Geom. Appl., 62 (2019), pp. 39-59. DOI: https://doi.org/10.1016/j.difgeo.2018.09.004

M. Kitayama, M. Azuma and M. Matsumoto, On Finsler spaces with (α,β)-metric, Regularity, geodesics and main scalars. J. Hokkaido Univ. Edu., 46 (1995), no. 1, pp. 1-10.

B. Li and Z. Shen, On a class of locally projectively flat Finsler metrics. Int. J. Math., 27 (2016), pp. 165-178. DOI: https://doi.org/10.1142/S0129167X1650052X

M. Matsumoto, Finsler spaces with (α,β)-metric of Douglas type, Tensor (N. S.), 60 (1998), no. 2, pp. 123-134.

S. Numata, On Landsberg spaces of scalar flag curvature. J. Korean Math. Soc., 12 (1975), pp. 97-100.

Z. Shen, Projectively flat Randers metrics with constant flag curvature. Math. Annal., 325 (2003), no.1, pp. 19-30. DOI: https://doi.org/10.1007/s00208-002-0361-1

Z. Shen, On projectively flat $(alpha,beta)$-metrics. Canad. Math. Bull., 52 (2009), no. 1, pp. 132-144. DOI: https://doi.org/10.4153/CMB-2009-016-2

Z. Shen and G.C. Yildirim, On a class of projectively flat metrics with constant flag curvature. Canad. J. Math., 60 (2008), no. 2, pp. 443-456. DOI: https://doi.org/10.4153/CJM-2008-021-1

B.K. Tripathi, Hypersurface of a Finsler space with exponential form of (α,β)-metric. Ann. Univ. Craivo Math. Comput. Sci. Ser., 47 (2020), no. 1, pp. 132-140.

B.K. Tripathi, S. Khan and V.K. Chaubey, On projectively flat Finsler space with a cubic (α,β)-metric. Filomat, 37 (2023), no. 26, pp. 8975-8982. DOI: https://doi.org/10.2298/FIL2326975T

B.K. Tripathi and P. Kumar, Douglas spaces of some (α,β)-metric of a Finsler spaces. J. Adv. Math. Stud., 15 (2022), no. 4, pp. 444-455.

B.K. Tripathi and K.B. Pandey, Characterization of locally dually flat special Finsler (α,β)-metrics. International J. Math. Combin., 4 (2016), pp. 44-52.

B.K. Tripathi, D. Patel and T.N. Pandey, On weakly Berwald space with special exponential (α,β)-metric. J. Rajasthan Acad. Phys. Sci., 22 (2023), no. 1 and 2, pp. 97-113.

Q. Xia, On a class of projectively flat Finsler metrics. Differ. Geom. Appl., 44 (2016), pp. 1-16. DOI: https://doi.org/10.1016/j.difgeo.2015.10.002

C. Yu, Deformations and Hilbert's fourth problem, Math. Annal., 365 (2016), pp. 1379-1408. DOI: https://doi.org/10.1007/s00208-015-1324-7