Abstract:
The paper deals with a sub-Riemannian manifold $M$ of contact type with a given associated connection $\nabla^A$. Additionally it is assumed that the structure endomorphism $\Psi$ defined by the equality $\omega$(\vec{x},\vec{y})=g(\Psi\vec{x},\vec{y})$ is covariantly constsnt with respect to the connection $\nabla^A$. The obtained sub-Riemannian manifold is an analog of a Sasaki manifold. It is proved that the manifold $M$ is $\eta$Einstein if and only if it is $\eta-$Einstein with respect to the connection $\nabla^N$ , where $N: D \to D$ is an endomorphism of the distribution $D$ of the manifold $M$ such that $\nabla^A N=0$. Let $\eta$ be a 1-form defining the distribution $D$ of the manifold $M$. If $rk(d\eta) = 2p, 2p < n 1, 2p \neq 0$, where n is the dimension of the manifold $M$, on the sub-Riemannian manifold may be defined in a natural way an endomorphism $N$ satisfying $N^2=N+1$ that is called a golden affinor structure. It is shown that the endomorphism N is covariantly constant∇with respect to the connection $\nabla^A$. As the central example, is considered the distribution $D$ of a sub-Riemannian manifold $M$ with zero Schouten tensor. The distribution $D$ of the manifold $M$ is itself a sub-Riemannian manifold with a golden affinor structure.
