GENERALIZED WAGNER’S CURVATURE TENSOR OF ALMOST CONTACT METRIC SPACES

Abstract

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 \(N:D\rightarrow D\).