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Year 2021, Volume: 14 Issue: 1, 167 - 173, 15.04.2021
https://doi.org/10.36890/iejg.777149

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References

  • [1] Amari S.I. and Nagaoka H.: Method of Information Geometry. AMS translation of Math., Monograph., Oxford Univ. Press, (2000).
  • [2] Bucătaru I.: Nonholonomic frames for Finsler spaces with $(\alpha,\beta)$-metrics. Proceedings of the conference on Finsler and Lagrange geometries, Iaşi, August 2001, Kluwer Acad. Publ. pp.69-78, (2003).
  • [3] Cheng X., Shen Z. and Zhou Y.: On a class of locally dually flat Finsler metrics. International J. Math. 21(11), pp. 1-13 (2010).
  • [4] Cheng X. and Tian Y.: Locally dually flat Finsler metrics with special curvature properties. Differential Geometry and its Applications. 29(1), Pages S98-S106 (2011).
  • [5] Crasmareanu M.: Lagrange spaces with indicatrices as constant mean curvature surfaces or minimal surfaces. An. Şt. Univ. Ovidius Constan¸ta. 10(1), 63-72 (2002).
  • [6] Constantinescu O. and Crasmareanu M.: Examples of connics arrising in Finsler and Lagrangian geometries. Anal. Stiint. ale Univ. Ovidius Constan¸ta. 17(2), pp. 45-60 (2009).
  • [7] Matsumoto M.: Foundations of Finsler Geometry and Special Finsler Spaces. Kaiseisha Press, Otsu, (1986).
  • [8] Miron R.: Finsler- Lagrange spaces with $(\alpha,\beta)$-metrics and Ingarden spaces. Reports on Mathematical Physics. 58 (1), pp. 417-431 (2006).
  • [9] Miron R.: Variational problem in Finsler spaces with $(\alpha,\beta)$-metrics. Algebras, Groups and Geometries, Hadronic Press. Vol. 20, pp. 285-300 (2003).
  • [10] Miron R. and Tavakol R.: Geometry of Space-Time and Generalized-Lagrange Spaces. Publicationes Mathematicae. 44 (1-2), pp.167-174 (1994).
  • [11] Nicolaescu B.: The variational problem in Lagrange spaces endowed with $(\alpha,\beta)$-metrics. Proceedings of the 3-rd international colloquium "Mathematics and numerical physics", pp.202-207, (2004).
  • [12] Pişcoran L.I. and Mishra V.N.: Projectivelly flatness of a new class of $(\alpha,\beta)$-metrics. Georgian Mathematical Journal. 26 (1), 133-139 (2019).
  • [13] Pişcoran L.I. and Mishra V.N.: S-curvature for a new class of $(\alpha,\beta)$-metrics. RACSAM. doi:10.1007/s13398-016-0358-3, (2017).
  • [14] Shen Z.: Finsler geometry with applications to information geometry. Chin. Ann. Math. 27(81), pp.73-94 (2006).
  • [15] Chen S.S. and Shen Z.: Riemann-Finsler geometry. Singapore: World Scientific, (2005).
  • [16] Shen Z.: On projectively flat $(\alpha,\beta)$-metrics. Can. Math. Bull. 52(1), pp. 132-144 (2009).
  • [17] Shimada H. and Sabău S.: Remarkable classes of $(\alpha,\beta)$-metric space. Rep. Math. Phys. 47, pp. 31-48 (2001).
  • [18] Vacaru S.I.: Finsler and Lagrange geometries in Einstein and string gravity. Int. J. Geom. Methods Mod. Phys. 5,No. 4 , 473-511 (2008).
  • [19] Xia Q.: On locally dually flat $(\alpha,\beta)$-metrics. Diff. Geom. Appl. 29(2), pp. 233-243 (2011).
  • [20] Tayebi A., Sadeghi H. and Peyghan H.: On Finsler metrics with vanishing S-curvature. Turkish J. Math. 38, pp.154-165 (2014).

The Deformation of an $(\alpha, \beta)$-Metric

Year 2021, Volume: 14 Issue: 1, 167 - 173, 15.04.2021
https://doi.org/10.36890/iejg.777149

Abstract

In this paper, we will continue our investigation on the new recently introduced $(\alpha, \beta)$-metric $F=\beta+\frac{a\alpha^{2}+\beta^{2}}{\alpha}$ in \cite{Pis}; where $\alpha$ is a Riemannian metric; $\beta$ is a 1-form, and $a\in \left(\frac{1}{4},+\infty\right)$ is a real positive scalar. We will investigate the deformation of this metric, and we will investigate its properties.

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Thanks

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References

  • [1] Amari S.I. and Nagaoka H.: Method of Information Geometry. AMS translation of Math., Monograph., Oxford Univ. Press, (2000).
  • [2] Bucătaru I.: Nonholonomic frames for Finsler spaces with $(\alpha,\beta)$-metrics. Proceedings of the conference on Finsler and Lagrange geometries, Iaşi, August 2001, Kluwer Acad. Publ. pp.69-78, (2003).
  • [3] Cheng X., Shen Z. and Zhou Y.: On a class of locally dually flat Finsler metrics. International J. Math. 21(11), pp. 1-13 (2010).
  • [4] Cheng X. and Tian Y.: Locally dually flat Finsler metrics with special curvature properties. Differential Geometry and its Applications. 29(1), Pages S98-S106 (2011).
  • [5] Crasmareanu M.: Lagrange spaces with indicatrices as constant mean curvature surfaces or minimal surfaces. An. Şt. Univ. Ovidius Constan¸ta. 10(1), 63-72 (2002).
  • [6] Constantinescu O. and Crasmareanu M.: Examples of connics arrising in Finsler and Lagrangian geometries. Anal. Stiint. ale Univ. Ovidius Constan¸ta. 17(2), pp. 45-60 (2009).
  • [7] Matsumoto M.: Foundations of Finsler Geometry and Special Finsler Spaces. Kaiseisha Press, Otsu, (1986).
  • [8] Miron R.: Finsler- Lagrange spaces with $(\alpha,\beta)$-metrics and Ingarden spaces. Reports on Mathematical Physics. 58 (1), pp. 417-431 (2006).
  • [9] Miron R.: Variational problem in Finsler spaces with $(\alpha,\beta)$-metrics. Algebras, Groups and Geometries, Hadronic Press. Vol. 20, pp. 285-300 (2003).
  • [10] Miron R. and Tavakol R.: Geometry of Space-Time and Generalized-Lagrange Spaces. Publicationes Mathematicae. 44 (1-2), pp.167-174 (1994).
  • [11] Nicolaescu B.: The variational problem in Lagrange spaces endowed with $(\alpha,\beta)$-metrics. Proceedings of the 3-rd international colloquium "Mathematics and numerical physics", pp.202-207, (2004).
  • [12] Pişcoran L.I. and Mishra V.N.: Projectivelly flatness of a new class of $(\alpha,\beta)$-metrics. Georgian Mathematical Journal. 26 (1), 133-139 (2019).
  • [13] Pişcoran L.I. and Mishra V.N.: S-curvature for a new class of $(\alpha,\beta)$-metrics. RACSAM. doi:10.1007/s13398-016-0358-3, (2017).
  • [14] Shen Z.: Finsler geometry with applications to information geometry. Chin. Ann. Math. 27(81), pp.73-94 (2006).
  • [15] Chen S.S. and Shen Z.: Riemann-Finsler geometry. Singapore: World Scientific, (2005).
  • [16] Shen Z.: On projectively flat $(\alpha,\beta)$-metrics. Can. Math. Bull. 52(1), pp. 132-144 (2009).
  • [17] Shimada H. and Sabău S.: Remarkable classes of $(\alpha,\beta)$-metric space. Rep. Math. Phys. 47, pp. 31-48 (2001).
  • [18] Vacaru S.I.: Finsler and Lagrange geometries in Einstein and string gravity. Int. J. Geom. Methods Mod. Phys. 5,No. 4 , 473-511 (2008).
  • [19] Xia Q.: On locally dually flat $(\alpha,\beta)$-metrics. Diff. Geom. Appl. 29(2), pp. 233-243 (2011).
  • [20] Tayebi A., Sadeghi H. and Peyghan H.: On Finsler metrics with vanishing S-curvature. Turkish J. Math. 38, pp.154-165 (2014).
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Laurian-loan Pıscoran

Najafi Behzad This is me 0000-0003-2788-3360

Cătălin Barbu 0000-0002-2094-1938

Tabatabaeifar Tayebeh This is me 0000-0002-5334-0135

Project Number ----
Publication Date April 15, 2021
Acceptance Date December 26, 2020
Published in Issue Year 2021 Volume: 14 Issue: 1

Cite

APA Pıscoran, L.-l., Behzad, N., Barbu, C., Tayebeh, T. (2021). The Deformation of an $(\alpha, \beta)$-Metric. International Electronic Journal of Geometry, 14(1), 167-173. https://doi.org/10.36890/iejg.777149
AMA Pıscoran Ll, Behzad N, Barbu C, Tayebeh T. The Deformation of an $(\alpha, \beta)$-Metric. Int. Electron. J. Geom. April 2021;14(1):167-173. doi:10.36890/iejg.777149
Chicago Pıscoran, Laurian-loan, Najafi Behzad, Cătălin Barbu, and Tabatabaeifar Tayebeh. “The Deformation of an $(\alpha, \beta)$-Metric”. International Electronic Journal of Geometry 14, no. 1 (April 2021): 167-73. https://doi.org/10.36890/iejg.777149.
EndNote Pıscoran L-l, Behzad N, Barbu C, Tayebeh T (April 1, 2021) The Deformation of an $(\alpha, \beta)$-Metric. International Electronic Journal of Geometry 14 1 167–173.
IEEE L.-l. Pıscoran, N. Behzad, C. Barbu, and T. Tayebeh, “The Deformation of an $(\alpha, \beta)$-Metric”, Int. Electron. J. Geom., vol. 14, no. 1, pp. 167–173, 2021, doi: 10.36890/iejg.777149.
ISNAD Pıscoran, Laurian-loan et al. “The Deformation of an $(\alpha, \beta)$-Metric”. International Electronic Journal of Geometry 14/1 (April 2021), 167-173. https://doi.org/10.36890/iejg.777149.
JAMA Pıscoran L-l, Behzad N, Barbu C, Tayebeh T. The Deformation of an $(\alpha, \beta)$-Metric. Int. Electron. J. Geom. 2021;14:167–173.
MLA Pıscoran, Laurian-loan et al. “The Deformation of an $(\alpha, \beta)$-Metric”. International Electronic Journal of Geometry, vol. 14, no. 1, 2021, pp. 167-73, doi:10.36890/iejg.777149.
Vancouver Pıscoran L-l, Behzad N, Barbu C, Tayebeh T. The Deformation of an $(\alpha, \beta)$-Metric. Int. Electron. J. Geom. 2021;14(1):167-73.