Speaker
Dr
Wolfgang Globke
(University of Adelaide)
Description
Let $M$ be pseudo-Riemannian homogeneous Einstein manifold of
finite volume, and suppose a connected Lie group $G$ acts
transitively and isometrically on $M$.
We study such spaces $M$ in three important cases.
First, we assume $\langle\cdot,\cdot\rangle $ is invariant, in which
case the Einstein property requires that $G$ is either solvable
or semisimple.
Next, we investigate the case where $G$ is solvable. Here,
$M$ is compact and $M=G/\Gamma$ for a lattice $\Gamma$ in $G$.
We show that in dimensions less or equal to $7$,
compact quotients $M=G/\Gamma$
exist only for nilpotent groups $G$.
We conjecture that this is true for any dimension.
In fact, this holds if Schanuel's Conjecture on transcendental
numbers is true.
Finally, we consider semisimple Lie groups $G$,
and find that $M$ splits as a pseudo-Riemannian product of Einstein
quotients for the compact and the non-compact factors of $G$.
This is joint work with Yuri Nikolayevsky (La Trobe University).
Primary author
Dr
Wolfgang Globke
(University of Adelaide)
Co-author
Prof.
Yuri Nikolayevsky
(La Trobe University)
