Almost contact metric spaces with $N$-connection

On a manifold with an almost contact metric structure $(\varphi,\vec\xi,\eta,g,X,D)$ and an endomorphism $N:D\to D$, a notion of the $N$-connection is introduced. The conditions under which an $N$-connection is compatible with an almost contact metric structure $\nabla^N\eta=\nabla^Ng=\nabla^N\vec\xi=0$ are found. The relations between the Levi–Civita connection, the Schouten–van-Kampen connection and the $N$-connection are investigated. Using the $N$-connection the conditions under which an almost contact metric structure is an almost contact Kahlerian structure are investigated.