Nearly trans-Sasakian almost $C(\lambda)$-manifolds
A. R. Rustanova,
G. V. Teplyakovab,
S. V. Kharitonovab a Institute of Digital Technologies and Modeling in Construction, Moscow State University of Civil Engineering (Moscow)
b Institute of Mathematics and Digital Technologies, Orenburg State University (Orenburg)
Abstract:
The nearly trans-Sasakian manifolds, which are almost
$C(\lambda)$-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
$C(\lambda)$-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
$C(\lambda)$-manifold is locally equivalent to the product of a Ricci-flat Kähler manifold and a real line. Identities are obtained that are satisfied by the Ricci tensor of an almost
$C(\lambda)$-manifold. It is proved that the Ricci curvature of an almost
$C(\lambda)$-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 Kähler 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
$C(\lambda)$-manifold. It is proved that for a nearly trans-Sasakian manifold M the following conditions are equivalent: 1) the manifold M is an almost
$C(\lambda)$-manifold; 2) the manifold M is a closely cosymplectic manifold; 3) the manifold M is locally equivalent to the product of a nearly Kähler manifold and the real line. In the case when the manifold M is a trans-Sasakian almost
$C(\lambda)$-manifold, the manifold M is cosymplectic, and hence locally equivalent to the product of a Kähler manifold and a real line. For an NTS-manifold of dimension greater than three, which is almost a
$C(\lambda)$-manifold, the pointwise constancy of the
$\Phi$-holomorphic sectional curvature implies global constancy. A complete classification of such manifolds is obtained.
Keywords:
nearly trans-Sasakian manifold, almost $C(\lambda)$-manifold, Kenmotsu manifold, cosymplectic manifold, Sasakian manifold.
UDC:
514.76 Received: 03.09.2023
Accepted: 21.12.2023
DOI:
10.22405/2226-8383-2023-24-5-153-166