### V.23 Mostow’s Strong Rigidity Theorem

*David Fisher*

### 1 What Are Rigidity Theorems?

A typical *rigidity theorem* is a statement that some class of objects is much smaller than one might expect. To make this notion clear, let us look at some examples of MODULI SPACES [IV.8] that might lead us to expect that spaces of a certain type would in general be large.

### 2 Some Moduli Spaces

A *flat metric* on an *n*-dimensional MANIFOLD [I.3 §6.9] is a METRIC [III.56] that is locally isometric to the usual metric on the Euclidean space ^{n}. In other words, every point *x* in the manifold is contained in a neighborhood *N*_{x} such that there is a distance-preserving bijection ...