Ph.D Thesis | |

Ph.D Student | Bader Uri |
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Subject | Simple Lie-Group Actions on a Pseudo-Riemannian Manifolds |

Department | Department of Mathematics |

Supervisor | PROF. Amos Nevo |

This work is devoted to the study of conformal actions of semisimple Lie-groups on Pseudo-Riemannian manifolds. It is known that (unlike the Riemannian case) a non-compact group might have an isometric action on a compact pseudo-Riemannian manifold. We construct a family of conformal actions of non-compact simple Lie-groups which are not (conformally equivalent to) isometric actions. This family serves as the "standard model actions". The main goal of our work is to show that, in a certain sense, all actions (conformal actions of simple Lie-groups on compact pseudo-Riemannian manifolds) are built out of these building blocks - isometric actions, and the standard models. This goal is achieved, assuming the manifold is Lorentzian. For the general Pseudo-Riemannian manifold, we achieve that goal under the extra assumption that the split rank of the group is maximal (to be defined by means of the signature of the Pseudo-Riemannian structure).

The main tools in this work are Lie-Theory (in particular finite representation of Lie-groups, weight spaces decomposition and the dynamics of projective actions) from the one hand, and the construction and comparison between various cohomological invariant (such as conformal, volume and projective 1-cocycles) on the other hand.

Our work splits naturally into three parts. The first part consists of the study of conformal actions on compact Pseudo-Riemannian manifolds of any signature. The second and third parts consist of the study of actions on Lorentzian manifolds in the homogeneous and genral cases correspondingly.