Sasaki 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.

A Sasakian metric is defined using the construction of the Riemannian cone. Given a Riemannian manifold ${\displaystyle (M,g)}$, its Riemannian cone is a product

of ${\displaystyle M}$ with a half-line ${\displaystyle {\mathbb {R}}^{>0}}$, equipped with the cone metric

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