Smooth and PL-Rigidity Problems on Locally Symmetric Spaces

TL;DR

This survey reviews known results and open problems related to the rigidity of negatively curved locally symmetric spaces, focusing on smooth and PL-rigidity, and explores the topology of associated metric spaces.

Contribution

It compiles and discusses existing results and open questions on rigidity phenomena and the topology of metric spaces in negatively curved locally symmetric spaces.

Findings

Summary of known rigidity results

Identification of key open problems

Insights into the topology of metric and Teichmuller spaces

Abstract

This is a survey on known results and open problems about Smooth and PL-Rigidity Problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved metrics on a manifold.