
Generalized Wagner's curvature tensor of almost contact metric spaces
S. V. Galaev^{} ^{} Saratov State University
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 onetoone 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 $N:D\rightarrow D$.
Bibliography: 15 titles.
Keywords:
almost contact metric structure, $N$prolonged connection, generalized Wagner curvature tensor, Tanaka–Webster connection, Schouten–vanKampen connection.
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UDC:
514.76 Received: 08.02.2016 Accepted:13.09.2016
Citation:
S. V. Galaev, “Generalized Wagner's curvature tensor of almost contact metric spaces”, Chebyshevskii Sb., 17:3 (2016), 53–63
Citation in format AMSBIB
\Bibitem{Gal16}
\by S.~V.~Galaev
\paper Generalized Wagner's curvature tensor of almost contact metric spaces
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 3
\pages 5363
\mathnet{http://mi.mathnet.ru/cheb497}
\elib{http://elibrary.ru/item.asp?id=27452082}
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