sasaki-einstein manifold

[yyoon@fas.harvard.edu]

Why is Sasaki-Einstein manifold named so?
Did Sasaki and Einstein come out with this idea?
I am just curious.

[Aaron Bergman]

In article <1146745977.855257.31540@j73g2000cwa.googlegroups.c om>,
yyoon@fas.harvard.edu wrote:

> Why is Sasaki-Einstein manifold named so?
> Did Sasaki and Einstein come out with this idea?
> I am just curious.

There's such a thing as a Sasakian manifold, and there's such a thing as
an Einstein manifold. A Sasaki-Einstein manifold is one that is both.

Sasaki is, in a sense, an odd dimensional version of a Kaehler manifold.
An Einstein manifold is a solution to the Einstein field equations with
a cosmological constant. In other words, the Ricci tensor is
proportional to the metric.

The string theory relevance here is that the base of a Ricci flat cone
is Einstein and the base of a Kaehler cone is Sasaki. Thus, the base of
a Calabi-Yau cone is Sasaki-Einstein.

Aaron

[Igor]

yyoon@fas.harvard.edu wrote:

> Why is Sasaki-Einstein manifold named so?
> Did Sasaki and Einstein come out with this idea?
> I am just curious.

I'm not sure that it applies to this particular case, but the term
Einstein manifold usually relates to vacuum solutions to the GR
equations with a non-zero cosmological constant. The name seems to be
used more often in pure math literature rather than physics.