Methods of Riemannian geometry in the Finsler geometry

• Applicant: Professor Dr. Vladimir Matveev
• Subject Area: Mathematics
• Term: from 2016 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 318916629

• Final Report Year: 2022

Final Report Abstract

We view the following four results as main scientific outcome of the project. • We have shown that a generic Finsler metric has infinite-dimensional holonomy group. • We have proved the Landsberg Unicorn conjecture under the additional assumption that for every point the restriction of the Finsler metric to the point is symmetric with respect to the standard action of SO(n - 1). • We have proved that for generic Finlser metric the holonomy group is infinite-dimensional. • We have described all 2-dimensional Finsler metrics admitting 3 projective vector fields. • We have shown that on a closed surface of negative Euler characteristic two projectively equivalent Finsler metrics are trivially projectively equivalent.

Publications

• Differentiability of projective transformations in dimension 2. Adv. Geom. 20 (2020), no. 4, 553-557
  Lang, Julius
  (See online at https://doi.org/10.1515/advgeom-2019-0023)
• Finsler metrics on surfaces admitting three projective vector fields. Differential Geom. Appl. 69 (2020), 101590, 13 pp
  Lang, Julius
  (See online at https://doi.org/10.1016/j.difgeo.2019.101590)
• On Finsler surfaces that are both Douglas and generalized Berwald. Publ. Math. Debrecen 97 (2020), no. 3-4, 381-391
  Bartelmess, Nina; Lang, Julius
  (See online at https://doi.org/10.5486/PMD.2020.8784)
• Almost all Finsler metrics have infinite dimensional holonomy group. J. Geom. Anal. 31 (2021), no. 6, 6067-6079
  Hubicska, B.; Matveev, V. S.; Muzsnay, Z.
  (See online at https://doi.org/10.1007/s12220-020-00517-9)
• Conformally related Douglas metrics in dimension two are Randers. Arch. Math. (Basel) 116 (2021), no. 2, 221-231
  Matveev, Vladimir S., Saberali, Samaneh
  (See online at https://doi.org/10.1007/s00013-020-01533-5)
• Proof of Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski norms with SO(k) x SO(n-k)-symmetry. Canadian Journal of Mathematics
  Xu, Ming, Matveev, Vladimir S.
  (See online at https://doi.org/10.4153/S0008414X21000304)
• Projectively equivalent Finsler metrics on surfaces of negative Euler characteristic. J. Topol. Anal. 14 (2022), no. 1, 287-296
  Lang, Julius
  (See online at https://doi.org/10.1142/s1793525320500491)