CMUP, Univ. Porto & CMAT, Univ. Minho
In this talk we will discuss geodesic completeness of left-invariant metrics for real and complex Lie groups. We will start by establishing the Euler-Arnold formalism in the holomorphic setting. We will present a new method for reobtaining the well-known classification for the real Lie group SL(2, R) and, as a new addition, how it can be used to investigate the maximum domain of definition of every single geodesic for every possible metric. We will also discuss the notion of geodesic completeness for holomorphic metrics and establish a full classification for the Lie group SL(2, C) for which it can be seen that holomorphic complete metrics are rare.