Sasakian manifold

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This article describes an additional structure on a differential manifold
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Definition

A Sasakian manifold is a differential manifold $M$ equipped with a contact form $\theta$ (making it a contact manifold) as well as a Riemannian metric (making it a Riemannian manifold) such that the following compatibility condition is satisfied (a Riemannian metric satisfying this special condition is termed a Sasakian metric):

The Riemannian cone naturally gets the structure of a Kahler manifold with Kahler form:

$t^{2}d\theta +2tdt\wedge \theta$