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In this paper we characterize
-contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat
-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature
$\varepsilon \,=\,\pm 1$
denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a
-contact Lorentzian manifold.
In this paper we show that a contact metric three-manifold is a generalised (k, μ)-space on an everywhere dense open subset if and only if its characteristic vector field ξ determines a harmonic map from the manifold into its unit tangent sphere bundle equipped with the Sasaki metric. Moreover, we classify the contact metric three-manifolds whose characteristic vector field ξ is strongly normal (or equivalently, is harmonic and minimal).
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