Nearly trans-Sasakian almost 𝐶(𝜆)-manifolds
https://doi.org/10.22405/2226-8383-2023-24-5-153-166
Abstract
The nearly trans-Sasakian manifolds, which are almost 𝐶(𝜆)-manifolds, are considered. On the space of the adjoint G-structure, the components of the Riemannian curvature tensor, the Ricci tensor of the nearly trans-Sasakian manifolds, and the almost 𝐶(𝜆)-manifolds are obtained.
Identities are obtained that are satisfied by the Ricci tensor of nearly trans-Sasakian manifolds.
It is proved that a Ricci-flat almost 𝐶(𝜆)-manifold is locally equivalent to the product of a Ricciflat K¨ahler manifold and a real line. Identities are obtained that are satisfied by the Ricci tensor of an almost 𝐶(𝜆)-manifold. It is proved that the Ricci curvature of an almost 𝐶(𝜆)-manifold in the direction of the structure vector is equal to zero if and only if it is cosymplectic, and hence locally equivalent to the product of a K¨ahler manifold and a real line. An identity is obtained that is satisfied by the Riemannian curvature tensor of a nearly trans-Sasakian manifold, which is an almost 𝐶(𝜆)-manifold. It is proved that for a nearly trans-Sasakian manifold M the following conditions are equivalent: 1) the manifold M is an almost 𝐶(𝜆)-manifold; 2) the manifold M is a closely cosymplectic manifold; 3) the manifold M is locally equivalent to the product of a nearly K¨ahler manifold and the real line. In the case when the manifold M is a trans-Sasakian almost
𝐶(𝜆)-manifold, the manifold M is cosymplectic, and hence locally equivalent to the product of a K¨ahler manifold and a real line. For an NTS-manifold of dimension greater than three, which is almost a 𝐶(𝜆)-manifold, the pointwise constancy of the Φ-holomorphic sectional curvature implies global constancy. A complete classification of such manifolds is obtained.
About the Authors
Aligadzhi Rabadanovich Rustanov
Institute of Digital Technologies and Modeling in Construction; Moscow State University of Civil Engineering
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Galina Vasilyevna Teplyakova
Institute of Mathematics and Digital Technologies; Orenburg State University
Russian Federation
Russian Federation
candidate of pedagogical sciences
Svetlana Vladimirovna Kharitonova
Institute of Mathematics and Digital Technologies; Orenburg State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate professor
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Review
For citations:
Rustanov A.R., Teplyakova G.V., Kharitonova S.V. Nearly trans-Sasakian almost 𝐶(𝜆)-manifolds. Chebyshevskii Sbornik. 2023;24(5):153-166. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-5-153-166
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