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).

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

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

where ${\displaystyle t}$ is the parameter in ${\displaystyle {\mathbb {R} }^{>0}}$.

A manifold ${\displaystyle M}$ equipped with a 1-form ${\displaystyle \theta }$ is contact if and only if the 2-form

on its cone is symplectic (this is one of the possible definitions of a contact structure). A contact Riemannian manifold is Sasakian, if its Riemannian cone with the cone metric is a Kähler manifold with Kähler form

As an example consider

where the right hand side is a natural Kähler manifold and read as the cone over the sphere (endowed with embedded metric). The contact 1-form on ${\displaystyle S^{2n-1}}$ is the form associated to the tangent vector ${\displaystyle i{\vec {N}}}$, constructed from the unit-normal vector ${\displaystyle {\vec {N}}}$ to the sphere (${\displaystyle i}$ being the complex structure on ${\displaystyle {\mathbb {C} }^{n}}$).