# nLab Sasakian manifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

complex geometry

# Contents

## Idea

A Sasakian manifold is a contact manifold $(M,\theta)$ equipped with a metric such that its Riemannian cone is a Kähler manifold.

If the manifold is also an Einstein manifold one speaks of Sasaki-Einstein manifolds.

When the Riemannian cone is a hyper-Kähler manifold, then one talks of a 3-Sasakian manifold. These are Sasaki-Einstein.

## Examples

An example is an odd-dimensional sphere $S^{2n-1} \subset \mathbb{C}^n\setminus\{\mathbf{0}\}$, where the ambient complex manifold carries its standard Kähler structure.

## Properties

When the Riemannian cone is a hyper-Kähler manifold, then one talks of a 3-Sasakian manifold.

Such a manifold is then automatically Einstein and spin.

A 7-dimensional 3-Sasakian manifold carries a 1-parameter family of $G_2$-structures (see e.g. (Agricola-Friedrich 2010)).

## References

The concept goes back to

• Shigeo Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure I, Tohoku Math. J. (2) 12 (1960), 459-476 (euclid:1178244407)

• Shigeo Sasaki, Y. Hatakeyama, On differentiable manifolds with contact metric structures, J. Math. Soc. Japan 14 (1962), 249-271 (euclid:1261060580)

An early set of lecture notes is

• Shigeo Sasaki, Almost contact manifolds, Part 1, Lecture Notes, Mathematical Institute, Tohoku University (1965).

• Shigeo Sasaki, Almost contact manifolds, Part 2, Lecture Notes, Mathematical Institute, Tohoku University (1967).

• Shigeo Sasaki, Almost contact manifolds, Part 3, Lecture Notes, Mathematical Institute, Tohoku University (1968).

Even today, after 40 years, the breath, depth and the relative completeness of the Sasaki lectures is truly quite remarkable. $[...]$ As it is, the notes are not easily available and, consequently, not well-known. (Boyer-Galicki 07, p. 1)

Modern accounts include

• Charles Boyer, Krzysztof Galicki, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, 2007

• James Sparks, Sasaki-Einstein Manifolds, Surv. Diff. Geom. 16 (2011) 265-324 (arXiv:1004.2461)

• Charles Boyer, Sasakian geometry: The recent work of Krzysztof Galicki (arXiv:0806.0373)

Discussion related to spin structure and G2-structure includes

Relation to SU(2)-structure:

• Beniamino Cappelletti-Montano, Giulia Dileo, Nearly Sasakian geometry and $SU(2)$-structures (arXiv:1410.0942)

• Anna Fino, Hypo contact and Sasakian structures on Lie groups, talk at Workshop on CR and Sasakian Geometry, Luxembourg– 24 - 26 March 2008 (pdf)

Discussion relating to quiver gauge theories includes

In M-theory:

Last revised on April 4, 2019 at 09:43:35. See the history of this page for a list of all contributions to it.