Geodesic transformations of distributions of sub-Riemannian manifolds

Let $M$ be a sub-Riemannian contact-type manifold endowed with a distribution $D$. Using an endomorphism $N: D\to D$ of the distribution $D$, one can prolong the inner connection, which transfers admissible vectors along admissible curves on the manifold $M$, up to a connection in the vector bundle $(D,\pi,M)$, where $\pi:D\to M$ 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.