Contact metric manifolds with η-parallel torsion tensor
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 CR-integrable. Next
we show that if the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci …
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 CR-integrable. Next
we show that if the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci …
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 CR- integrable. 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.
Springer
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