Metric geometry and Finsler structures of low regularity

• Applicant: Professor Dr. Vladimir Matveev
• Subject Area: Mathematics
• Term: from 2017 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 390960259

• Final Report Year: 2022

Final Report Abstract

We view the following three results as main scientific outcome of the project. • We have shown that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric. In particular, we theoretically explained a known phenomenon that it is almost impossible to experimentally distinguish the Brownian motion coming from a Finsler metric from the one coming from an appropriate Riemannian metric. • We have also shown that a C 1 smooth Riemanian metric whose Riemannian curvature is zero in the weak sense has constant components in a coordinate system, and established existence, uniqueness and regularity of solutions of the corresponding Dirichlet-Neumann and Dirichlet problems inside a bounded convex domain. • We found local necessary and sufficient conditions for a bilinear form to be flat. More precisely, we gave explicit necessary and sufficient conditions for a tensor field of type (0,2) which is not necessary symmetric or skewsymmetric, and is possibly degenerate, to have constant entries in a local coordinate system.

Publications

• Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds. The Journal of Geometric Analysis, 31(12), 12446-12484.
  Ma, Tianyu; Matveev, Vladimir S. & Pavlyukevich, Ilya
  
• Almost every path structure is not variational. General Relativity and Gravitation, 54(10).
  Kruglikov, Boris S. & Matveev, Vladimir S.
  
• On the equation DutHDu=G. Nonlinear Analysis, 214, 112554.
  Bandyopadhyay, S.; Dacorogna, B.; Matveev, V. & Troyanov, M.
  
• Bernhard Riemann 1861 revisited: existence of flat coordinates for an arbitrary bilinear form. Mathematische Zeitschrift, 305(1).
  Bandyopadhyay, S.; Dacorogna, B.; Matveev, V. S. & Troyanov, M.
  
• Canonical curves and Kropina metrics in Lagrangian contact geometry. Nonlinearity, 37(1), 015007.
  Ma, Tianyu; Flood, Keegan J.; Matveev, Vladimir S. & Žádník, Vojtěch