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From Fermat principle to the Lobačevskii-Fok space in particle physics

The Poincare model of the Lobacevskii geometry is derived from the Fermat principle. The Lobacevskii geometry is interpreted as the Lobacevskii-Fok velocity space geometry of moving particles. The relation of this geometry to the decay of the neutral pi-meson is considered. The generalization of the Lobacevskii geometry is performed and the new angle of parallelism is derived. The light confined circularly in the optical medium is defined as the optical black hole. The existence of the centrifugal force acting on the photon is discussed.