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 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.

Rustanov A.R., Gerasimenko S.A., Shchipkova N.N.
THE GEOMETRY OF TENSOR CONHARMONIC CURVATURE AC-MANIFOLDS OF CLASS C₁₁ ^{[№ 9 ' 2015]}
The 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.

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 Ф-holomorphic planes.

Rustanov A.R., Kharitonova S.V., Каzакоvа О.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 NС₁₁ CLASS ^{[№ 1 ' 2013]}
This paper considers a new class of contact metric manifolds, which generalizes the class of АС-manifolds of the С₁₁ 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 Riemann — 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.