Sasakian manifold

In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold ${\displaystyle (M,\theta )}$ equipped with a special kind of Riemannian metric ${\displaystyle g}$, called a Sasakian metric. They are studied as a natural odd-dimensional counterpart of Kähler manifolds (which are necessarily even-dimensional).