nLab Sasakian manifold

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)).

Related concepts

• contact manifold

• conifold

• 5-manifold

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 (doi:10.1093/acprof:oso/9780198564959.001.0001)

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

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

See also

• Wikipedia, Sasakian manifold

Discussion related to spin structure and G₂-structure includes

• Ilka Agricola, Thomas Friedrich, 3-Sasakian manifolds in dimension seven, their spinors and $G_2$ structures, J. Geom. Phys. 60 (2010), 326-332 doi:10.1016/j.geomphys.2009.10.003, arXiv:0812.1651

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

• Olaf Lechtenfeld, Alexander Popov, Richard Szabo, Sasakian quiver gauge theories and instantons on Calabi-Yau cones (arXiv:1412.4409)

In M-theory:

• José Figueroa-O'Farrill, Andrea Santi, Sasakian manifolds and M-theory, Class. Quantum Grav. 33 (2016), 095004 (arXiv:1511.03460)