On a non-riemannian quantity of (α,β)-metrics

Year 2024, Volume: 42 Issue: 2, 566 - 571, 30.04.2024

Semail Ulgen

Abstract

In this paper, we study a non-Riemannian quantity χ-curvature of (α, β)-metrics, a special class of Finsler metrics with Riemannian metric α and a 1-form β . We prove that every (α, β)-metric has a vanishing χ-curvature under certain conditions.

Keywords

(α , β)- Finsler Metric , Vanishing χ-Curvature , Non-Riemannian Quantity

References

• REFERENCES
• [1] Matsumoto M. On C-reducible Finsler spaces. Tensor NS 1972;24:2937.
• [2] Antonelli PL, Miron R. Lagrange and Finsler Geometry: Applications to Physics and Biology. Dordrecht: Kluwer Academic Publishers; 1996. (Fundamental Theories of Physics; vol 76). [CrossRef]
• [3] Balan V, Stavrinos PC. Finslerian (α, β)-metrics in Weak Gravitational Models. In: Anastasiei M, Antonelli PL, editors. Finsler and Lagrange Geometries. Dordrecht: Kluwer Academic Publishers; 2003. [CrossRef]
• [4] Roman M, Shimada H, Sabau VS. On β-change of the Antonelli Shimada ecological metric. Tensor NS 2004;65:6573.
• [5] Bao D, Chern SS, Shen Z. An Introduction to Riemann-Finsler Geometry. New York: Springer; 2000. [CrossRef]
• [6] Kikuchi S. On the condition that a space with (α, β)-metric be locally Minkowskian. Tensor NS 1979;33:242246.
• [7] Matsumoto M. Finsler spaces with (α, β)-metric of Douglas type. Tensor NS 1998;60:123134.
• [8] Matsumoto M. The Berwald connection of a Finsler space with an (α, β)- metric. Tensor NS 1991;50:1821.
• [9] Sabau VS, Shimada H. Classes of Finsler spaces with (α, β)-metrics. Rep Math Phy 2001;47:3148. [CrossRef]
• [10] Shen Z. On a class of landsberg metrics in finsler geometry. Can J Math 2009;61:13571374. [CrossRef]
• [11] Gabrani M, Rezaei B, Sevim ES. A Class of Finsler Metrics with Almost Vanishing H- and Ξ-curvatures. Results Math 2021;76:117. [CrossRef]
• [12] Gabrani M, Rezai B, Sengelen Sevim E. General spherically symmetric Finsler metrics with constant Ricci and flag curvature. Differ Geom Appl 2021;76:119. [CrossRef]
• [13] Najafi B, Shen Z, Tayebi A. Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties. Geom Dedicata 2008;131:8797. [CrossRef]
• [14] Shen Z. On Some Non-Riemannian Quantities in Finsler Geometry. Can Math Bull 2013;56:184193. [CrossRef]
• [15] Tang D. On the non-Riemannian quantity H in Finsler geometry. Differ Geom Appl 2011;29:207213. [CrossRef]
• [16] Pişcoran LI, Mishra VN. S-curvature for a new class of (α, β)-metrics. Rev R Acad Cienc Exactas Fís Nat Ser A Mat 2017;111:11871200. [CrossRef]
• [17] Najafi B, Tayebi A. A new quantity in Finsler geometry. C R Math 2011;349:8183. [CrossRef]
• [18] Bao D, Chern SS. On a notable connection in Finsler Geometry. Houston J Math 1993;19:135180.
• [19] Tayebi A, Azizpour E, Esrafilian E. On a Family of connections in Finsler geometry. Publ Math Debrecen 2008;72:115. [CrossRef]
• [20] Ingarden RS. On the geometrically absolute optical representation in the electron microscope. Trav Soc Sci Lettr Wroclaw. 1957;B45:360.
• [21] Bao D, Chern SS, Shen Z. An Introduction to Riemann-Finsler Geometry. New York: Springer; 2000. [CrossRef]
• [22] Cheng X, Mo X, Shen Z. On the flag curvature of Finsler metrics of scalar curvature. J London Math Soc 2003;68:762780. [CrossRef]
• [23] Cheng X, Shen Z. A class of Finsler metrics with isotropic S-curvature. Isr J Math 2009;169:317340. [CrossRef]
• [24] Cheng X, Shen Z. Randers metrics of scalar flag curvature. J Australian Math Soc 2009;87:359370. [CrossRef]
• [25] Chern SS, Shen Z. Riemann-Finsler Geometry. Singapore: World Scientific; 2005. [CrossRef]
• [26] Tamassy L. Metrical almost linear connections in TM for Randers spaces. Bull Soc Sci Lett Lodz Ser Rech Deform 2006;51:147152.
• [27] Shen Z. Lectures on Finsler Geometry. Singapore: World Scientific Publishers;2001. [CrossRef]