Salah Chaib (Universidade do Minho (Portugal) and ENS de Lyon)

Title: Geodesic Completeness on Lie Groups

Abstract: In this talk, I will discuss geodesic completeness problems on Lie groups. For a fixed metric Lie group, geodesic completeness can be studied through the associated Euler–Arnold equation, a homogeneous quadratic vector field on the Lie algebra. I will present some results obtained on this problem, with particular emphasis on the three-dimensional Lorentzian case. 

A complementary problem is to determine those Lie groups for which every left-invariant semi-Riemannian metric is geodesically complete. This leads to the notion of absolute completeness and to several Lie-algebraic obstructions to this property. I will discuss some of these ideas, including the role of the adjoint representation and affine subalgebras.

The talk is based on joint work with Ana Cristina Ferreira and Ghani Zeghib, as well as ongoing work.