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Rustanov A.R., Gerasimenko S.A., Shchipkova N.N. THE GEOMETRY OF TENSOR CONHARMONIC CURVATURE AC-MANIFOLDS OF CLASSšC11The main objective is to study the geometry of the curvature tensor conharmonic almost contact metric manifolds classšC11. To this end, following tasks: 1) calculate the basic essential components of curvature tensor conharmonic space of the associated G-structure; 2) explore conharmonic flat manifolds of classšC11; 3) get the identities satisfied by the curvature tensor conharmonic AC-class varietiesšC11; 4) select and explore some subclasses of AC-class varietiesšC11 of differential-geometric invariants of the second order. The paper addressed these problems. We prove the following theorem. Theoremš1: conharmonic flat AC-manifold of classšC11 is Ricci-flat manifold. Theoremš2: conharmonic flat AC-manifold of classšC11 is a manifold of Einstein. Theoremš3: conharmonic flat AC-manifold of classšC11 is a flat manifold. Theoremš4: AC-manifold of classšC11 is a manifold of classšK1 if and only if the AC-manifold of classšC11 is Ricci-flat manifold. Theoremš5: AC-manifold of classšC11 is a manifold of classšK2 if and only if the AC-manifold of classšC11 is Ricci-flat manifold. Theoremš6: AC-manifold of classšC11 is a manifold of classšK3 if and only if the AC-manifold of classšC11 is a flat manifold. Theoremš7: AC-manifold of classšC11, a manifold of classšK4 is a manifold of class Einstein's cosmological constant. In particular, in the case of completeness and continuity is compact and has a finite fundamental group. Theoremš8. AC-manifold of classšC11 dimension greater thanš5 is K4-manifold if and only if it is Ricci flat manifold.Key words: tensor Riemannian curvature, the Ricci tensor, the tensor conharmonical curvature, the conharmonically flat manifold, flat manifold.
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About this article
Authors: Rustanov A.R., Gerasimenko S.A., Shchipkova N.N.
Year: 2015
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Editor-in-chief |
Sergey Aleksandrovich MIROSHNIKOV |
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