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Gerasimenko S.A., Rustanov A.R., Shchipkova N.N.
ANALOGS IDENTITIES GRAY FOR TENSOR CONHARMONIC CURVATURE AC-MANIFOLDS OF CLASS C11 [№ 9 ' 2015]
The 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 Theoremš1 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. Theoremš2 is proved that the variety of AC-class C11 is GK3-manifold and the GK2-manifold. In Theoremš3 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 Theoremš4 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.

Rustanov A.R., Gerasimenko S.A., Shchipkova N.N.
THE GEOMETRY OF TENSOR CONHARMONIC CURVATURE AC-MANIFOLDS OF CLASSšC11 [№ 9 ' 2015]
The 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.

Rustanov A.R., Kazakova O.N., Kharitonova S.V.
THE AXIOM F-HOLOMORPHIC PLANES FOR NORMAL LCACS-MANIFOLDS [№ 4 ' 2015]
Almost contact metric structures have a rich geometry, as well as numerous applications in various areas of mathematics and theoretical physics. Sufficiently important aspect of almost contact geometry is the study of almost contact manifolds satisfying the axiom æ-holomorphic planes.

Rustanov A.R., Kharitonova S.V., ëÁzÁËÏvÁ ï.N.
TWO CLASSES OF ALMOST C (λ)-MANIFOLDS [№ 3 ' 2015]
Almost contact metric manifolds have a rich differential geometric structures. The paper deals with almost contact metric manifolds, which are almost c (λ)-manifolds. D. Janssen and L. Vanhecke began investigated of almost c (λ)-manifolds. The curvature tensor is crucial for almost c (λ)-manifolds and curvature identities satisfied by this tensor are very important for understanding the differential geometric properties of almost c (λ)-manifolds. The results obtained in this paper identities expressing additional symmetry properties of the Riemannian curvature tensor of almost c (λ)-manifolds, allow to solve an actual problem of classifying almost c (λ)-manifolds, namely, to distinguish the class CR1 and CR2-class almost c (λ)-manifolds. We obtain the following identities curvature almost manifold.

Rustanov A.R., Shchipkova N.N.
GEOMETRY OF CLASS C11AC-MANIFOLDS CONCIRCULAR CURVATURE TENSOR [№ 9 ' 2014]
This paper describes the geometry of concircular curvature tensor in AC-manifolds of class C11. We have calculated its components for the space of the associated G-structure, derivated identities satisfied by the concircular curvature tensor. On the basis of these identities allocated some subclasses of AC-manifolds of class C11 and derivated local characterization of selected classes.

Rustanov A.R., Shchipkova N.N.
DIFFERENTIAL GEOMETRY OF CONTACT METRIC MANIFOLDS OF Nó11 CLASS [№ 1 ' 2013]
This paper considers a new class of contact metric manifolds, which generalizes the class of áó-manifolds of the ó11 class by the classification of Chinya and Gonzalez. The complete group of structural equations for NC11-manifolds derived, and components of Riemann-Christoffel tensor, Ricci tensor and the scalar curvature are computed basing on these equations. Properties of NC11-manifolds are derived. Some identities of the Riemann curvature tensor are derived, too.

Rustanov A.R., Shchipkova N.N.
CURVATURE IDENTITIES OF CLASS C11 MANIFOLD [№ 6 ' 2011]
The article deals with the AU-manifold of class C11, which generalize the class of skew-symplectic manifolds. The identities of the curvature of this class manifold are obtained. As a consequence, the expression of the tensor F-sectional curvature tensor by the Riemannš— Christoffel tensor was got.

Rustanov A.R., Shchipkova N.N.
THE DIFFERENTIAL GEOMETRY OF THE ALMOST CONTACT METRIC VARIETIES OF THE CLASS S11. [№ 9 ' 2010]
The work examined the new class of almost contact metric varieties, the generalizing class of cosimplect varieties. The authors got the complete classification of the AS-varieties of the class S11 of a constant F-holomorphic sectional curvature.


Editor-in-chief
Sergey Aleksandrovich
MIROSHNIKOV

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