Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4

One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved. © 2017, Pleiades Publishing, Ltd

Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4

One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved. © 2017, Pleiades Publishing, Ltd

Topological and geometrical properties of spaces with symmetric and nonsymmetric f-quasimetrics

The properties of spaces equipped with a topology defined by a distance function are studied. The considered distance function is not necessarily symmetric but satisfies the so-called f-triangle inequality, which is a weakened version of the usual triangle inequality. Sufficient conditions for metrizability of such spaces are proposed. A construction of a quasimetric topologically equivalent to a given f-quasimetric is proposed. © 2017 Elsevier B.V

Topological and geometrical properties of spaces with symmetric and nonsymmetric f-quasimetrics

The properties of spaces equipped with a topology defined by a distance function are studied. The considered distance function is not necessarily symmetric but satisfies the so-called f-triangle inequality, which is a weakened version of the usual triangle inequality. Sufficient conditions for metrizability of such spaces are proposed. A construction of a quasimetric topologically equivalent to a given f-quasimetric is proposed. © 2017 Elsevier B.V