The Global Solutions to Cartan's Realization Problem

TL;DR

This paper presents a systematic method leveraging Lie algebroids and groupoids to solve Cartan's realization problem, enabling classification and symmetry analysis of solutions, exemplified by extremal Kähler metrics.

Contribution

It introduces a new approach based on Lie algebroids and groupoids for solving Cartan's realization problem, including classification and symmetry determination.

Findings

Method finds local and complete solutions.

Determines symmetries and moduli spaces.

Applied to classify extremal Kähler metrics.

Abstract

We introduce a systematic method to solve a type of Cartan's realization problem. Our method builds upon a new theory of Lie algebroids and Lie groupoids with structure group and connection. This approach allows to find local as well as complete solutions, their symmetries, and to determine the moduli spaces of local and complete solutions. We apply our method to the problem of classification of extremal K\"ahler metrics on surfaces.