Homogeneous Kähler structures on principal R^×-bundles reduce to Sasakian structures exactly when the Euler vector field is pre-geodesic and the line bundle is oriented, and the same dictionary covers co-Kähler structures as the 'invariant' case.
A survey on cosymplectic geometry
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We give an up-to-date overview of geometric and topological properties of cosymplectic and coKaehler manifolds. We also mention some of their applications to time-dependent mechanics.
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On Homogeneous K\"ahler Manifolds
Homogeneous Kähler structures on principal R^×-bundles reduce to Sasakian structures exactly when the Euler vector field is pre-geodesic and the line bundle is oriented, and the same dictionary covers co-Kähler structures as the 'invariant' case.