Geodesic completeness of pseudo and holomorphic Riemannian metrics on Lie groups

TL;DR

This paper investigates the conditions for geodesic completeness of left-invariant metrics on real and complex Lie groups, providing new classifications and detailed analyses for specific groups like SL(2,R) and SL(2,C).

Contribution

It extends the Euler-Arnold formalism to the holomorphic setting and offers a comprehensive classification of geodesic completeness for SL(2,C).

Findings

Reobtained known characterization of geodesic completeness for SL(2,R).

Provided detailed analysis of geodesic domains for various metrics.

Established a full classification of geodesic completeness for SL(2,C).

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 $\mathrm{SL}(2, \mathbb{R})$ 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 $\mathrm{SL}(2, \mathbb{C})$.