On Locally Symmetric Polynomial Metrics: Riemannian and Finslerian surfaces

Year 2024, Volume: 17 Issue: 2, 679 - 699, 27.10.2024

Csaba Vincze , Mark Olah , Abris Nagy

https://doi.org/10.36890/iejg.1454779

Abstract

In the paper we investigate locally symmetric polynomial metrics in special cases of Riemannian and Finslerian surfaces. The Riemannian case will be presented by a collection of basic results (regularity of second root metrics) and formulas up to Gauss curvature. In case of Finslerian surfaces we formulate necessary and sufficient conditions for a locally symmetric fourth root metric in 2D to be positive definite. They are given in terms of the coefficients of the polynomial metric to make checking the positive definiteness as simple and direct as possible. Explicit examples are also presented. The situation is more complicated in case of spaces of dimension more than two. Some necessary conditions and an explicit example are given for a positive definite locally symmetric polynomial metric in 3D. Computations are supported by the MAPLE mathematics software (LinearAlgebra).

Keywords

Finsler surfaces, locally symmetric polynomial metrics, positive definiteness

Supporting Institution

HUN-REN Hungarian Research Network

Thanks

Mark Olah has received funding from the HUN-REN Hungarian Research Network.

References

• [1] Vincze, Cs., Khoshdani, T., Oláh M.: On generalized Berwald surfaces with locally symmetric fourth root metrics. Balk. J. Geom. Appl., Vol. 24 (2), 63–78(2019). arXiv:1808.10855.
• [2] Balan, V., Brinzei, N.: Einstein equations for (h, v)-Bervald-Moór relativistic models. Balk. J. Geom. Appl., 11 , 20–27(2006).
• [3] Balan, V.: Spectra of symmetric tensors and m-root Finsler models. Linear Algebra and its Applications, 436 (1), 152–162(2012).
• [4] Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann–Finsler geometry. Springer-Verlag, (2000).
• [5] Brinzei, N.: Projective relations for m-th root metric spaces. arXiv:0711.4781v2 (2008).
• [6] Majidi, J., Tayebi A., Haji-Badali, A.: On Einstein-reversible m-th root Finsler metrics. Int. J. Geom. Methods Mod. Phys. 20(6), Paper No. 2350099, 14 pp(2023).
• [7] Matsumoto, M., Okubo, K.: Theory of Finsler spaces with m-th root metric. Tensor (N.S.), 56, 9–104(1995).
• [8] Shimada, H.: On Finsler spaces with the metric $L=\sqrt[m]{a_{i_1 \ldots i_m} y^{i_1}\cdot \ldots \cdot y^{i_m}}$. Tensor (N.S.), 33, 365–372(1979).
• [9] Tayebi, A.: On the theory of 4-th root Finsler metrics. Tbil. Math. J. 12 (1), 83–92(2019).
• [10] Tayebi, A., Najafi, B.: On m-th root metrics. J. Geom. Phys. 61, 1479–1484(2011).
• [11] Tamássy, L.: Finsler Spaces with Polynomial Metric. Hypercomplex Numbers in Geometry and Physics, 2 (6) Vol. 3, 85–92( 2006).
• [12] Tiwari, B., Kumar, M., Tayebi, A.: On generalized Kropina change of generalized m-th root Finsler metrics. Proc. Nat. Acad. Sci. India Sect. A91 (3), 443–450(2021).
• [13] Vincze, Cs.: On a special type of generalized Berwald manifolds: semi-symmetric linear connections preserving the Finslerian length of tangent vectors. European Journal of Mathematics, Finsler Geometry: New methods and Perspectives, Volume 3 Issue 4, 1098–1171, (December 2017).