Differential Geometry, Lie Theory and Low-Dimensional Topology, Melbourne, Australia, 2016

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Details of talk

TitleCompact pseudo-Riemannian Einstein manifolds in low dimensions
PresenterWolfgang Globke (The University of Adelaide)
Author(s)Dr Wolfgang Globke

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 the two cases where $G$ is solvable or
semisimple. In the solvable case, $M$ is compact and $M=G/\Gamma$ for a lattice
$\Gamma$ in $G$. Solvable Lie algebras $\mathfrak{g}$ with invariant Einstein
scalar product exist only for $\dim\mathfrak{g}\leq 7$ and Witt index $\leq 2$.
The existence of a compact $M$ then requires the existence of a lattice in the
solvable group $G$, and in dimensions $\leq 7$, this requires $G$ to be
nilpotent. We conjecture that this is true for any dimension. In fact, this
holds if Schanuel's Conjecture on transcendental numbers is true.
In the case where $G$ is semisimple, $M$ 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.