Nearly trans-Sasakian almost $C(\lambda)$-manifolds

A. R. Rustanov ; G. V. Teplyakova ; S. V. Kharitonova

Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 153-166.

Résumé

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.

Détail

Keywords: nearly trans-Sasakian manifold, almost $C(\lambda)$-manifold, Kenmotsu manifold, cosymplectic manifold, Sasakian manifold.

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