Generalized Wagner's curvature tensor of almost contact metric spaces

On a manifold with an almost contact metric structure $(M, \vec{\xi}, \eta, \varphi,g)$ and an endomorphism $N:D\rightarrow D$ the notion of an $N$-prolonged connection $\nabla^N=(\nabla,N)$, where $\nabla$ is an interior connection, is introduced. An endomorphism $N:D\rightarrow D$ found such that the curvature tensor of the $N$-prolonged connection coincides with the Wagner curvature tensor. It is proven that the curvature tensor of the interior connection equals zero if and only if on the manifold $M$ exists an atlas of adapted charts for that the coefficients of the interior connection are zero. A one-to-one correspondence between the set of $N$-prolonged and the set of $N$-connections is constructed. It is shown that the class of $N$-connections includes the Tanaka–Webster Schouten–van Kampen connections. An equality expressing the $N$-connection in the terms of the Levi–Civita connection is obtained. The properties of the curvature tensor of the $N$-connection are investigated; this curvature tensor is called in the paper the generalized Wagner curvature tensor. It is shown in particular that if the generalized Wagner curvature tensor in the case of a contact metric space is zero, then there exists a constant admissible vector field oriented in any direction. It is shown that the generalized Wagner curvature tensor may be zero only in the case of the zero endomorphism $N:D\rightarrow D$.
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