KaydedilenlerDerginizi ekleyin

Dergiler / International Electronic Journal of Geometry / 2021 / Volume: 14 Issue: 2

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

Laurian-loan Pıscoran(Technical University of Cluj Napoca, North Univ. Center of Baia Mare)0000-0003-2269-718X
Sayfa
—
Yayın Tarihi
2021-10-29
🏛️Bu makalenin bibliyografik verileri ve tam metin bağlantısı DergiPark üzerinden sağlanmıştır.
DergiPark Kaydı ↗
✨

Yapay Zeka Hızlı Bakış & Çıkarımlar

AI Insight
⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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.

📌 3 Maddede Bu Makale

🎯
Amaç & Hipotez: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.
📊
Temel Bulgular & Veri: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.
💡
Sonuç & Katkı: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.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Finsler ( α · β ) metric · nonholonomic frame

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.

🔖

Bu Makaleye Atıf Yap

Pıscoran, L. (2021). Nonholonomic Frame for a Deformed $(\alpha,\beta )$-metric. International Electronic Journal of Geometry, Volume: 14 Issue: 2. https://doi.org/10.36890/iejg.973879

📄 Tam Metin PDF Okuyucu

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (19)

  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.