In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, a Sasakian manifold (named after
Shigeo Sasaki) is a
contact manifold equipped with a special kind of
Riemannian metric , called a ''Sasakian'' metric.
Definition
A Sasakian metric is defined using the construction of the ''Riemannian cone''. Given a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, its Riemannian cone is the product
:
of
with a half-line
,
equipped with the ''cone metric''
:
where
is the parameter in
.
A manifold
equipped with a 1-form
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
:
Examples
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
is the form associated to the tangent vector
, constructed from the unit-normal vector
to the sphere (
being the complex structure on
).
Another non-compact example is
with coordinates
endowed with contact-form
and the Riemannian metric
As a third example consider:
where the right hand side has a natural Kähler structure, and the group
acts by reflection at the origin.
History
Sasakian manifolds were introduced in 1960 by the Japanese geometer
Shigeo Sasaki.
There was not much activity in this field after the mid-1970s, until the advent of
String theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
. Since then Sasakian manifolds have gained prominence in physics and algebraic geometry, mostly due to a string of papers by
Charles P. Boyer
Charles Place Boyer (born April 1942) is an American mathematician, specializing in differential geometry and moduli spaces. He is known as one of the four mathematicians who jointly proved in 1992 the Atiyah–Jones conjecture.
Boyer graduated f ...
and Krzysztof Galicki and their co-authors.
The Reeb vector field
The
homothetic vector field on the cone over a Sasakian manifold is defined to be
:
As the cone is by definition Kähler, there exists a complex structure ''J''. The ''Reeb vector field'' on the Sasaskian manifold is defined to be
:
It is nowhere vanishing. It commutes with all holomorphic
Killing vectors on the cone and in particular with all
isometries of the Sasakian manifold. If the orbits of the vector field close then the space of orbits is a Kähler orbifold. The Reeb vector field at the Sasakian manifold at unit radius is a unit vector field and tangential to the embedding.
Sasaki–Einstein manifolds
A Sasakian manifold
is a manifold whose Riemannian cone is Kähler. If, in addition, this cone is
Ricci-flat In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are ...
,
is called ''Sasaki–Einstein''; if it is
hyperkähler,
is called 3-Sasakian. Any 3-Sasakian manifold is both an Einstein manifold and a spin manifold.
If ''M'' is positive-scalar-curvature Kahler–Einstein manifold, then, by an observation of
Shoshichi Kobayashi, the circle bundle ''S'' in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes the projection from ''S'' to ''M'' into a Riemannian submersion. (For example, it follows that
there exist Sasaki–Einstein metrics on suitable
circle bundles over the 3rd through 8th
del Pezzo surfaces.) While this Riemannian submersion construction provides a correct local picture of any Sasaki–Einstein manifold,
the global structure of such manifolds can be more complicated. For example, one can more generally
construct Sasaki–Einstein manifolds by starting from a Kahler–Einstein orbifold ''M.'' Using this observation, Boyer, Galicki, and
János Kollár constructed infinitely many homeotypes of Sasaki-Einstein 5-manifolds.
The same construction shows that the moduli space of Einstein metrics on the 5-sphere has at least several hundred connected components.
Notes
References
*
Shigeo Sasaki, "On differentiable manifolds with certain structures which are closely related to almost contact structure", ''Tohoku Math. J.'' 2 (1960), 459-476.
*
Charles P. Boyer
Charles Place Boyer (born April 1942) is an American mathematician, specializing in differential geometry and moduli spaces. He is known as one of the four mathematicians who jointly proved in 1992 the Atiyah–Jones conjecture.
Boyer graduated f ...
, Krzysztof Galicki, ''Sasakian geometry''
* Charles P. Boyer, Krzysztof Galicki,
3-Sasakian Manifolds, ''Surveys Diff. Geom.'' 7 (1999) 123-184
* Dario Martelli, James Sparks and
Shing-Tung Yau,
Sasaki-Einstein Manifolds and Volume Minimization, ''ArXiv hep-th/0603021''
External links
EoM page, ''Sasakian manifold''
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Riemannian geometry
Symplectic geometry
Structures on manifolds