A survey on unicorns in Finsler geometry

[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[7] R.L. Bryant, Finsler surfaces with prescribed curvature conditions, unpublished preprint (part of his Aisenstadt lectures) (1995).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.