Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups

Abstract

This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler–Arnold formalism in the holomorphic setting. We study the real Lie group SL(2, R) and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group SL(2, ℂ).