# Manchester Geometry Seminar 2012/2013

**Thursday 6 December 2012. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm*
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New Holonomy Groups in Pseudo-Riemannian Geometry and Integrable Systems on Lie Algebras

Alexey Bolsinov (Loughborough University)

`A.Bolsinov@lboro.ac.uk`

The purpose of the talk is to discuss a rather unexpected relationship between three geometric objects: holonomy groups, projectively equivalent metrics and integrable systems on Lie algebras. This relationship is used to construct new examples of holonomy groups in pseudo-Riemannian geometry. Namely, we prove the following result: Let $g$ be a non-degenerate bilinear form on a vector space $V$, and let $L\,:V \to V$ be a $g$-symmetric operator. Then the centraliser of $L$ in $SO(V,g)$ is a holonomy group for a suitable Levi-Civita connection.