It is shown that the problem of a possible violation of the Lorentz transformations at Lorentz factors γ > 5×10 10 , indicated by the situation which has developed in the physics of ultra-high energy cosmic rays (the absence of the GZK cutoff), has a nontrivial solution. Its essence consists in the discovery of the so-called generalized Lorentz transformations which seem to correctly link the i...

In this paper, we reviewtwo approaches that can describe, in a geometrical way, the kinematics of particles are affected by Planck-scale departures, named Finsler and Hamilton geometries. By relying on maps connect spaces velocities momenta, discuss properties configuration phase induced these two distinct particular, exemplify approach considering so-called q-de Sitter-inspired modified disper...

We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the ...

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems of Finsler manifolds under various curvature conditions. As applications, we derive Mckean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize a result on fundamental group due to Milnor to Finsler manifolds.