Semi - Riemannian Geometry with Nonholonomic Constraints

TL;DR

This paper explores the geometry of semi-Riemannian manifolds with nonholonomic constraints, introducing new differential operators and explicit formulas for specific cases, expanding understanding of their geometric and Hamiltonian properties.

Contribution

It introduces a framework for analyzing semi-Riemannian manifolds with nonholonomic constraints, including new formulas and methods for studying their exponential maps and Hamiltonian systems.

Findings

Derived explicit formulas for specific semi-Riemannian nonholonomic manifolds.

Analyzed solutions of Hamiltonian systems and their projections.

Developed differential operators for studying exponential maps.

Abstract

In the present article the geometry of semi-Riemannian manifolds with nonholonomic constraints is studied. These manifolds can be considered as analogues to the sub-Riemannian manifolds, where the positively definite metric is substituted by a nondegenerate metric. To study properties of the exponential map the Christoffel symbols and other differential operators were introduced. We study solutions of the Hamiltonian system and their projections into the underlying manifold. The explicit formulae were found for a specific example of a semi-Riemannian manifold with nonholonomic constraints.