Sasakian structures are defined for arbitrary contact manifolds using Kähler structures on symplectic R^x-bundles, recovering the classical case and giving a canonical Sasakian product.
Grabowski, Graded contact manifolds and contact Courant algebroids,J
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Sasaki structures on general contact manifolds
Sasakian structures are defined for arbitrary contact manifolds using Kähler structures on symplectic R^x-bundles, recovering the classical case and giving a canonical Sasakian product.