Geodesic Transformations of Distributions of Sub-Riemannian Manifolds

• Published in: Journal of mathematical sciences (New York, N.Y.) N.Y.), 2023-12, Vol.277 (5), p.722-726
• Main Author: Galaev, S. V.
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics; Prolongation; Riemann manifold
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-023-06878-0

Let M be a sub-Riemannian contact-type manifold endowed with a distribution D . Using an endomorphism N : D → D of the distribution D , one can prolong the intrinsic connection, which transfers admissible vectors along admissible curves on the manifold M , up to a connection in the vector bundle ( D, π,M ), where π : D → M is the natural projection. The connection obtained is called the N -prolonged connection. The setting of an N -prolonged connection is equivalent to the setting of an N -prolonged sub-Riemannian on the distribution D . Using the structure equations of the N -prolonged structure, we calculate the coefficients of the Levi-Civita connection obtained by the prolongation of the Riemannian manifold. We prove that if a distribution D of a sub-Riemannian manifold is not integrable, then two N -prolonged, contact-type, sub-Riemannian structures, one of which is determined by the zero endomorphism and the other by an arbitrary nonzero endomorphism, belong to distinct geodesic classes.