September 2015, № 9 (184)

Rustanov A.R., Gerasimenko S.A., Shchipkova N.N. THE GEOMETRY OF TENSOR CONHARMONIC CURVATURE AC-MANIFOLDS OF CLASS C₁₁The main objective is to study the geometry of the curvature tensor conharmonic almost contact metric manifolds class C₁₁. 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 class C₁₁; 3) get the identities satisfied by the curvature tensor conharmonic AC-class varieties C₁₁; 4) select and explore some subclasses of AC-class varieties C₁₁ of differential-geometric invariants of the second order. The paper addressed these problems. We prove the following theorem. Theorem 1: conharmonic flat AC-manifold of class C₁₁ is Ricci-flat manifold. Theorem 2: conharmonic flat AC-manifold of class C₁₁ is a manifold of Einstein. Theorem 3: conharmonic flat AC-manifold of class C₁₁ is a flat manifold. Theorem 4: AC-manifold of class C₁₁ is a manifold of class K₁ if and only if the AC-manifold of class C₁₁ is Ricci-flat manifold. Theorem 5: AC-manifold of class C₁₁ is a manifold of class K₂ if and only if the AC-manifold of class C₁₁ is Ricci-flat manifold. Theorem 6: AC-manifold of class C₁₁ is a manifold of class K₃ if and only if the AC-manifold of class C₁₁ is a flat manifold. Theorem 7: AC-manifold of class C₁₁, a manifold of class K₄ 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. Theorem 8. AC-manifold of class C₁₁ dimension greater than 5 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

Editor-in-chief

Sergey Aleksandrovich
MIROSHNIKOV

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