Sasakian manifold

Builds on Riemannian manifold Hermitian manifold Contact manifold Reeb vector field

Definition

Let $(M, g)$ be a Riemannian manifold. The metric cone over $M$ is the manifold $C(M) = (0, \infty) \times M$ with the metric $\bar g = dr^2 + r^2 g$ , where $r$ is the coordinate on $(0, \infty)$ . The manifold $(M, g)$ is Sasakian if $C(M)$ has an integrable almost complex structure $J$ with two properties. The metric $\bar g$ is a Kähler metric for $J$ , and $J$ is invariant under the dilations $(r, x) \mapsto (\lambda r, x)$ for $\lambda > 0$ .

Remarks

Example: the unit sphere $S^{2n+1} \subset \C^{n+1}$ with the round metric. The map $(r, x) \mapsto r x$ identifies $C(S^{2n+1})$ with $\C^{n+1} \setminus \set{0}$ , and it carries $dr^2 + r^2 g$ to the flat metric, which is Kähler.

Non-example: the round sphere $S^2$ is not Sasakian. Its cone is $\R^3 \setminus \set{0}$ , of odd dimension $3$ , so it has no almost complex structure at all.

A Sasakian manifold has odd dimension $2n + 1$ . The vector field $\xi = J(r \partial_r)$ , restricted to $\set{1} \times M$ , is a unit Killing field, the Reeb vector field of the contact form $\eta = g(\xi, \cdot)$ , which makes $M$ a contact manifold.

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Tag	`FGRPYWW1V1`
URI	`m2://FGRPYWW1V1`
Kind	definition
Writing	W0 stub
MSC	53C25, 53D10
Defines	Sasakian manifold
Uses	Riemannian manifold, Hermitian manifold, Contact manifold, Reeb vector field
Changed	2 November 2026

Sasakian manifold

Definition

Remarks

Reading path

Further reading

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