Geodesic completeness of pseudo-Riemannian Lie groups

"Geodesic completeness of pseudo-Riemannian Lie groups", organizado polo CITMAga. Será impartido por Ana Cristina Ferreira (Universidade do Minho)

Data: Venres 1 de decembro

Hora: 13:00 h.

Lugar: Aula 9, Facultade de Matemáticas, USC.

Abstract:

A striking difference between Riemannian and pseudo-Riemannian metrics is that pseudo-Riemannian ones often fail to be geodesically complete even in the compact case.  We will present some developments in the classification of Lie groups with all their left-invariant pseudo-Riemannian metrics complete. More concretely, we will discuss the specifics of geodesic completeness when the manifold in question is a Lie group and recall the Euler-Arnold theorem as well as the seminal work of Marsden for the compact (homogeneous) case. We will see how an interpretation in Riemannian terms of his techniques provided us with tools for characterising completeness even for general manifolds. As for Lie groups, we will show how a certain notion of "linear growth'' allowed us to establish large classes of Lie groups whose left-invariant metrics are all complete. Time permitting, we will also discuss the generalisation of the Euler-Arnold formalism to the holomorphic-Riemann setting and discuss the classification of geodesic completeness for 3-dimensional (non-unimodular) Lie groups.
This is a series of joint works with S. Chaib, A. Elshafei, H. Reis, M. Sánchez and A. Zeghib.