September 2015, № 9 (184)

Gerasimenko S.A., Rustanov A.R., Shchipkova N.N. ANALOGS IDENTITIES GRAY FOR TENSOR CONHARMONIC CURVATURE AC-MANIFOLDS OF CLASS C₁₁The main objective is to study the geometry of the curvature tensor conharmonic AC-class varieties C₁₁. For this purpose, the following two problems are solved: 1) get contact analogues identities Gray conharmonic curvature tensor introduced in consideration of Ishi; 2) on the basis of these identities to isolate and study the subclasses of AC-class varieties C₁₁. The paper identified three classes of almost contact metric manifolds class C₁₁, named as GK₁-, GK₂- and GK₃-manifolds. In Theorem 1 we obtain conditions on the curvature tensor components conharmonic on the space of the associated G-structure in which almost contact metric structure belongs to a class C₁₁ selected classes. Theorem 2 is proved that the variety of AC-class C₁₁ is GK₃-manifold and the GK₂-manifold. In Theorem 3 proved that AC-manifold of class C₁₁, which is GK₁-manifold is a manifold with Einstein's cosmological constant. In particular, in the case of completeness and continuity is compact and has a finite fundamental group. Finally, in Theorem 4 is proved that the variety of AC-class C₁₁ dimension greater than 5 is GK₁-manifold if and only if it is Ricci flat manifold.Key words: identities Gray, the Riemann-Christoffel tensor, the Ricci tensor, the tensor conharmonic curvature.

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About this article

Authors: Gerasimenko S.A., Rustanov A.R., Shchipkova N.N.

Year: 2015

Editor-in-chief

Sergey Aleksandrovich
MIROSHNIKOV

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