$\nabla^{N}$-Einstein almost contact metric manifolds

On an almost contact metric manifold $M$, an $N$-connection $\nabla^{N}$ defined by the pair $(\nabla,N)$, where $\nabla$ is the interior metric connection and $N: TM \to TM$ is an endomorphism of the tangent bundle of the manifold $M$ such that $N\vec\xi=\vec0$, $N(D)\subset D$, is considered. Special attention is paid to the case of a skew-symmetric $N$-connection $\nabla^{N}$, which means that the torsion of an $N$-connection considered as a trivalent covariant tensor is skew-symmetric. Such a connection is uniquely defined and corresponds to the endomorphism $N = 2\psi$, where the endomorphism $\psi$ is defined by the equality $\omega(X,Y)=g(\psi X,Y)$ and is called in this work the second structure endomorphism of an almost contact metric manifold. The notion of a $\nabla^{N}$-Einstein almost contact metric manifold is introduced. For the case $N = 2\psi$, conditions under which almost contact manifolds are $\nabla^{N}$-Einstein manifolds are found.