Rigidity of transformation groups in differential geometry

TL;DR

This survey explores how symmetry and transformation groups help classify manifolds with various geometric structures, highlighting techniques from algebra, dynamics, and analysis.

Contribution

It provides a comprehensive overview of methods used to study the rigidity of transformation groups in differential geometry.

Findings

Various techniques have been successfully applied to study rigidity.

Symmetry plays a central role in classifying geometric structures.

The survey highlights the interplay of algebra, dynamics, and analysis.

Abstract

In this survey, symmetry provides a framework for classification of manifolds with differential-geometric structures. We highlight pseudo-Riemannian metrics, conformal structures, and projective structures. A range of techniques have been developed and successfully deployed in this subject, some of them based on algebra and dynamics and some based on analysis. We aim to illustrate this variety.