A survey on unicorns in Finsler geometry

Document Type : Review Article

Author

Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran

Abstract

This survey is an inspiration of my joint paper with Behzad Najafi published in Science in China. I explain some of interesting results about the unicorn problem.

Keywords

Main Subjects

References

[1] G. S. Asanov, Finsleroid-Finsler spaces of positive definite and relativistic types, Rep. Math. Phys. 58 (2006), 275-300.
[2] G. S. Asanov, Finsleroid-Finsler space with Berwald and Landsberg conditions, preprint, http://arxiv.org/abs/math.DG/0603472.
[3] D. Bao, On two curvature-driven problems in Riemann-Finsler geometry, Adv. Stud. Pure. Math. 48 (2007), 19-71.
[4] D. Bao, S. S. Chern and Z. Shen, Rigidity issues on Finsler surfaces, Rev. Roumaine Math. Pures Appl. 42 (1997), 707-735.
[5] L. Berwald, Uber die ¨ n-dimensionalen Geometrien konstanter Kr¨ummung, in denen die Geraden die k¨urzesten sind, Math. Z. 30 (1929), 449-469.
[6] L. Berwald, Uber Parallel¨ubertragung in R¨aumen mit allgemeiner Massbestimmung, Jber. Deutsch. Math.- ¨ Verein. 34 (1925), 213-220.
[7] R.L. Bryant, Finsler surfaces with prescribed curvature conditions, unpublished preprint (part of his Aisenstadt lectures) (1995).
[8] G. Chen and L. Liu, The generalized unicorn problem in the almost regular Douglas (α, β)-spaces, Differ. Geom. Appl. 69 (2020), 101589.
[9] G. Chen and L. Liu, On conformal transformations between two almost regular (α, β)-metrics, Bull. Korean Math. Soc. 55 (2018), 1231-1240.
[10] M. Crampin, On Landsberg spaces and the Landsberg-Berwald problem, Houston J. Math. 37(4) (2011), 1103-1124.
[11] S. G. Elgendi, Solutions for the Landsberg unicorn problem in Finsler geometry, arXiv:1908.10910.
[12] S. G. Elgendi, On the problem of non Berwaldian Landsberg spaces, Bull. Australian. Math. Soc, 102 (2020), 331-341.
[13] S. G. Elgendi and N. L. Youssef, A note on L. Zhou’s result on Finsler surfaces with K = 0 and J = 0, Differ. Geom. Appl. 77 (2021), 101779.
[14] M. Hashiguchi and Y. Ichijy¯o, Randers spaces with rectilinear geodesics, Rep. Fac. Sci. Kagoshima Univ. 13 (1980), 33-40.
[15] Y. Ichijy¯o, Finsler spaces modeled on a Minkowski space, J. Math. Kyoto. Univ. 16 (1976), 639-652.
[16] Y. Ichijy¯o, On special Finsler connections with vanishing hv-curvature tensor, Tensor, N. S. 32 (1978), 146-155.
[17] B. Li and Z. Shen, On a class of weakly Landsberg metrics, Sci. China Ser. A. 50(4) (2007), 573-589.
[18] M. Matsumoto, On transformations of locally Minkowskian space, Tensor (NS). 22 (1971), 103-111.
[19] M. Matsumoto, Remarks on Berwald and Landsberg spaces, Contemp. Math. 196 (1996), 79-82.
[20] M. Matsumoto, An improvment proof of Numata and Shibata’s theorem on Finsler spaces of scalar curvature, Publ. Math. Debrecen. 64 (2004), 489-500.
[21] M. Matsumoto, Conformally Berwald and conformally flat Finsler spaces, Publ. Math. Debrecen. 58 (2001), 275-285.
[22] B. Najafi and A. Tayebi, Weakly stretch Finsler metrics, Publ. Math. Debrecen. 91 (2017), 441-454.
[23] S. V. Sabau, K. Shibuya and H. Shimada, On the existence of generalized unicorns on surfaces, Differ. Geom. Appl. 28 (2010), 406-435.
[24] Z. Shen, On a class of Landsberg metrics in Finsler geometry, Canadian. J. Math. 61 (2009), 1357-1374.
[25] C. Shibata, On Finsler spaces with Kropina metrics, Report. Math. 13 (1978), 117-128.
[26] C. Shibata, On invariant tensors of β-changes of Finsler metrics, J. Math. Kyoto Univ. 24 (1984), 163-188.
[27] Z. I. Szab´o, All regular Landsberg metrics are Berwald, Ann. Glob. Anal. Geom. 34 (2008), 381-386; correction, ibid, 35 (2009), 227-230.
[28] Z. I. Szab´o, Positive definite Berwald spaces. Structure theorems on Berwald spaces, Tensor (N.S.), 35 (1981), 25-39.
[29] J. Szilasi and Cs. Vincze, On conformal equivalence of Riemann-Finsler metrics, Publ. Math. Debrecen. 52 (1998), 167-185.
[30] A. Tayebi and B. Najafi, Classification of 3-dimensional Landsbergian (α, β)-metrics, Publ. Math. Debrecen. 96 (2020), 45-62.
[31] A. Tayebi and Najafi, On a class of homogeneous Finsler metrics, J. Geom. Phys. 140 (2019), 265-270.
[32] A. Tayebi and B. Najafi, The weakly generalized unicorns in Finsler geometry. Sci. China Math. (2021). https://doi.org/10.1007/s11425-020-1853-5.
[33] A. Tayebi and B. Najafi, On homogeneous Landsberg surfaces, J. Geom. Phys. 168 (2021), 104314.
[34] A. Tayebi and H. Sadeghi, On generalized Douglas-Weyl (α, β)-metrics, Acta. Math. Sinica. English. Series. 31 (2015), 1611-1620.
[35] A. Tayebi and H. Sadeghi, On a class of stretch metrics in Finsler geometry, Arabian. J. Math. 8 (2019), 153-160.
[36] A. Tayebi and T. Tabatabaeifar, Unicorn metrics with almost vanishing H- and Ξ-curvatures, Turkish. J. Math. 41 (2017), 998-1008.
[37] C. Vincze, A new proof of Szab´o’s theorem on the Riemann-metrizability of Berwald manifolds, Acta. Math. Acad. Paedagog. Nyh´azi. 21 (2005), 199-204.
[38] C. Vincze, An observation on Asanov’s unicorn metrics, Publ. Math. Debrecen. 90 (2017), 251-268.
[39] C. Vincze, On Asanov’s Finsleroid-Finsler metrics as the solutions of a conformal rigidity problem, Differ. Geom. Appl. 53 (2017), 148-168.
[40] M. Xu and S. Deng, The Landsberg equation of a Finsler space, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XXII (2021), 31-51.
[41] D. Zheng and Q. He, Projective change between arbitrary (α, β)-metric and Randers metric, Publ. Math. Debrecen. 83 (2013), 179-196.
[42] L. Zhou, The Finsler surface with K = 0 and J = 0, Differ. Geom. Appl. 35 (2014), 370-380.
[43] S. Zhou, J. Wang and B. Li, On a class of almost regular Landsberg metrics, Sci. China Ser. A. 62 (2019), 935-960.
[44] Y. Zou and X. Cheng, The generalized unicorn problem on (α, β)-metrics, J. Math. Anal. Appl. 414 (2014), 574-589.
AUT Journal of Mathematics and Computing
Volume 2, Issue 2
September 2021
Pages 239-250

Files

Share

How to cite

Statistics