Nonholonomic Frame for a Deformed $(\alpha,\beta )$-metric

Year 2021, Volume: 14 Issue: 2, 231 - 238, 29.10.2021

Laurian-loan Pıscoran

https://doi.org/10.36890/iejg.973879

Abstract

Recently, in paper [14], we have introduced the following deformed $(\alpha, \beta)$-metric:
$$
F_{\epsilon}(\alpha,\beta)=\frac{\beta^{2}+\alpha^{2}(a+1)}{\alpha}+\epsilon\beta
$$
where $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric; $\beta=b_{i}y^{i}$ is a 1-form, $\left|\epsilon\right|<2\sqrt{a+1}$ is a real parameter and $a\in \left(\frac{1}{4},+\infty\right)$ is a real positive scalar.
The aim of this paper is to find the nonholonomic frame for this important kind of $(\alpha, \beta)$-metric and also to investigate the Frobenius norm for the Hessian generated by this kind of metric.

Keywords

Finsler ( α ; β ) metric, nonholonomic frame, projectively flat

References

• [1] L. Benling, Projectively flat Matsumoto metric and its approximation, Acta Math. Scientia, (2007), 27B(4), 781-789.
• [2] S. Bacsό, M. Matsumoto, On Finsler space of Douglas type. A generalization of the notion of Berwald space, Publ. Math. Debrecen, 51(1997), 385-406.
• [3] I. Bucătaru, R. Miron, Finsler-Lagrange geometry. Applications to dynamical systems, Ed. Academiei, (2007).
• [4] M. Hashiguchi, Y. Icijyo, Randers spaces with rectilinear geodesics, Rep. Fac. Sci. Kagoshima Univ., 13(1980), 33-40.
• [5] Meyer D.C., Matrix analysis and applied linear algebra, SIAM, (2000).
• [6] M. Matsumoto, A slope of a mountain is a Finsler surface with respect ot time measure, J. Math. Kyoto Univ., 29 (1989), 17-25.
• [7] M. Matsumoto. On C-reducible Finsler spaces. Tensor (N.S.), 24, (1972), 29–37.
• [8] M. Matsumoto, Projective changes of Finsler metrics and projectively flat Finsler spaces, Tensor, N.S., 34 (1980), 303-315.
• [9] M. Matsumoto, Foundations of Finsler Geometry and Special Finsler Spaces, Kaisheisha Press, Otsu, Japan, (1986).
• [10] P. Senarath, Differential geometry of projectively related Finsler spaces, Ph.D. Thesis, Massey University, (2003), http://mro.massey.ac.nz/bitstream/handle/10179/1918/02_whole.pdf?sequence=1.
• [11] Z. Shen, On projectively flat (α,β)-metrics, Canadian Math. Bulletin, 52(1.1)(2009), 132-144.
• [12] W. Song, X. Wang, A New Class of Finsler Metrics with Scalar Flag Curvature, Journal of Mathematical Research with Applications, Vol. 32, No. 4, (2012), 485-492, DOI:10.3770/j.issn:2095-2651.2012.04.013.
• [13] A.Tayebi, H. Sadeghi, On Generalized Douglas-Weyl (α,β)-metrics, Acta Mathematica Sinica, English Series, Vol. 31, No. 10, (2015), 1611- 1620, DOI: 10.1007/s10114-015-3418-2.
• [14] L.I. Pi¸scoran, B. Najafi, C. Barbu, T. Tabatabaeifar, The deformation of an (α,β)-metric, International Electronic Journal of Geometry, Vol. 14 No. 1 Pag. 167–173 (2021), Doi: https://doi.org/10.36890/IEJG.777149
• [15] Y.-Z. Hu, Some Operator Inequalities; Seminaire de probablilites: Strasbourg, France, (1994); pp. 316–333.
• [16] M. Crasmareanu, New tools in Finsler geometry: stretch and Ricci solitons, Math. Rep. (Bucur.), 16(66)(2014), no. 1, 83-93. MR3304401
• [17] P. Overath, H. von der Mosel, On minimal immersion on Finsler space, Annals of Global Analysis and Geometry 48(4), DOI:10.1007/s10455- 015-9476-y, (2015).
• [18] R. S. Ingarden, On the geometrically absolute optical representation in the electron microscope, v. Sot. Sci. Lettres Wroclaw B45(1957), 1–60.
• [19] I. Bucataru, Nonholonomic frames in Finsler geometry, Balkan Journal of Geometry and Its Applications, Vol.7, No.1, (2002), pp. 13-27.