Contact metric manifolds with η-parallel torsion tensor

Abstract

We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, µ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CRintegrable. Next we show that if the metric of a non-Sasakian (k, µ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to E n+1 × S n (4) in higher dimensions.

Ricci Solitons and Contact Metric Manifolds

“…In this direction, Sharma [15] proved that if the metric g of K-contact manifold is a gradient soliton, then it is shrinking and the metric g is Einstein-Sasakian. This result has been generalised by Ghosh et al [12] for a (κ, μ)-space (see [3]). Moreover, Sharma-Ghosh [16] studied Sasakian 3-metric as a Ricci soliton and proved that it is expanding and homothetic to the standard Sasakian metric on the Heisenberg group nil 3 .…”

“…Subtracting (35) from (33), using (12) and noting that Trl = 2nκ, it is immediate that (ϕX)Trl = 0, as n > 1. Taking ϕX instead of X and remembering that ξ Trl = 0…”

A Generalization of the Goldberg Conjecture for CoKähler Manifolds

“…Recently, Sharma [25], Ghosh et al [13] and Ghosh [12] obtained many interesting results regarding (gradient) Ricci solitons on (κ, μ)-contact metric manifolds. Moreover, gradient Ricci solitons on (κ, μ) -almost Kenmotsu manifolds were first studied by the present author jointly with De and Liu [27].…”

“…Later, Sharma in [25] generalized Boyer and Galicki's result by proving that any complete K-contact manifold admitting a gradient Ricci soliton is a compact SasakiEinstein manifold. Moreover, non-K-contact (κ, μ)-contact metric manifolds admitting a gradient Ricci soliton were studied by Ghosh et al in [13]. For more related results regarding (gradient) Ricci solitons in the framework of contact metric manifolds, see Cho [8], Cho and Sharma [9] and Ghosh [12].…”

SOME CLASSES OF (Κ,μ)-Contact METRIC MANIFOLD SATISFYING SEMISYMMETRIC CONDITIONS.

“…-contact metric manifolds have studied by several authors in the following papers [1,10,11,19] [13,20].…”