The Geometries of 3‐Manifolds

The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [66-70]. He has shown that geometry has an important role to play in the theory in addition to the use of purely topological methods. The basic aim of this article is to discuss the various geometries which arise and explain their significance for the theory of 3-manifolds. The idea is that many 3-manifolds admit 'nice' metrics which give one new insight into properties of the manifolds. For the purposes of this article, the nicest metrics are those of constant curvature. An observer in a manifold with a constant curvature metric will see the same picture wherever he stands and in whichever direction he looks. Such manifolds have special topological properties. However, we will also need to consider nice metrics which are not of constant curvature. In this article, I will explain what is meant by a 'nice' metric and describe their classification in dimension three which is due to Thurston. Then I will discuss some of the 3-manifolds which admit these nice metrics and the relationship between their geometric and topological properties. In this introduction all manifolds and metrics will be assumed to be smooth so that the objects of interest are all Riemannian manifolds. It has been known since the nineteenth century that in dimension two there is a very close relationship between geometry and topology. I will start by describing some basic facts about closed surfaces. I will discuss these in more detail in §1. Each