International Electronic Journal of Geometry

Recently, in paper [14], we have introduced the following deformed (α, β)-metric: F (α, β) = β 2 + α 2 (a + 1) α +β where α = p aijy iy j is a Riemannian metric; β = biy i is a 1-form, | | < 2 √ a + 1 is a real parameter and a ∈ 1 4 , +∞ is a real positive scalar. The aim of this paper is to find the nonholonomic frame for this important kind of (α, β)-metric and also to investigate the Frobenius norm for the Hessian generated by this kind of metri

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  • [1] Benling L.,: Projectively flat Matsumoto metric and its approximation, Acta Math. Scientia, (2007), 27B(4), 781-789.
  • [2] Bacso´, S., Matsumoto, M.: On Finsler space of Douglas type. A generalization of the notion of Berwald space, Publ. Math. Debrecen, 51(1997), 385-406.
  • [3] Buc ˘ataru, I., Miron, R.: Finsler-Lagrange geometry. Applications to dynamical systems, Ed. Academiei, (2007).
  • [4] Hashiguchi, M., Icijyo, Y.: 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] Matsumoto, M.: A slope of a mountain is a Finsler surface with respect ot time measure, J. Math. Kyoto Univ., 29 (1989), 17-25
  • [7] Matsumoto. M.: On C-reducible Finsler spaces. Tensor (N.S.), 24, (1972), 29–37.
  • [8] Matsumoto. M.: Projective changes of Finsler metrics and projectively flat Finsler spaces, Tensor, N.S., 34 (1980), 303-315.
  • [9] Matsumoto. M.: Foundations of Finsler Geometry and Special Finsler Spaces, Kaisheisha Press, Otsu, Japan, (1986).
  • [10] Senarath, P.: 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] Shen, Z.: On projectively flat (α, β)-metrics, Canadian Math. Bulletin, 52(1.1)(2009), 132-144.
  • [12] Song, W., Wang, X.: 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] Tayebi, A., Sadeghi, H.: On Generalized Douglas-Weyl (α, β)-metrics, Acta Mathematica Sinica, English Series, Vol. 31, No. 10, (2015), 1611-
  • 620, DOI: 10.1007/s10114-015-3418-2.
  • [14] Pi¸scoran, L.I., Najafi, B., Barbu, C., Tabatabaeifar, T.: 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] Hu, Y.Z.: Some Operator Inequalities; Seminaire de probablilites: Strasbourg, France, (1994); pp. 316–333.
  • [16] Crasmareanu, M.: New tools in Finsler geometry: stretch and Ricci solitons, Math. Rep. (Bucur.), 16(66)(2014), no. 1, 83-93. MR3304401
  • [17] Overath, P., von der Mosel, H.: On minimal immersions on Finsler space, Annals of Global Analysis and Geometry 48(4),
  • DOI:10.1007/s10455-015-9476-y, (2015).
  • [18] Ingarden, R. S.: On the geometrically absolute optical representation in the electron microscope, v. Sot. Sci. Lettres Wroclaw B45(1957), 1–60.
  • [19] Bucataru, I.: Nonholonomic frames in Finsler geometry, Balkan Journal of Geometry and Its Applications, Vol.7, No.1, (2002), pp. 13-27.