---
tag: B21YEPS76G
kind: concept
name: Sasakian manifold
msc: [53C25, 53D10]
specializes: [7CBT4SM83Y]
sources:
  - url: https://en.wikipedia.org/wiki/Sasakian_manifold
same_as:
  wikidata: Q7424986
  wikipedia: Sasakian manifold
---

## Summary

A Sasakian manifold is an odd-dimensional Riemannian manifold whose metric cone is a Kähler manifold.
It is the odd-dimensional partner of Kähler geometry, and it carries a natural [[contact structure|HBQ8BZ7XB5]] and a [[Reeb vector field|GGM63KXCF2]].
The round odd-dimensional spheres are the basic examples.
Sasaki–Einstein manifolds are of interest in the study of Einstein metrics and in physics.

## Explanations

### L1 human

Take a manifold $M$ and build a cone over it, like a cone over a circle in the plane.
The cone has one dimension more than $M$, and its metric grows with the distance $r$ from the tip.
$M$ is Sasakian when this cone is a Kähler manifold, a complex manifold with a compatible metric whose fundamental $2$-form is closed.
The cone over the unit sphere $S^{2n+1}$ is $\C^{n+1}$ with the origin removed, so the sphere is Sasakian.