A natural question is whether there is any analog of the uniformization theorem for three-dimensional manifolds. In other words, is every compact three-manifold modeled on a well-known Riemannian manifold? The answer is, in general, no. Thurston showed exactly eight model geometries exist, each of which serves as a model for at least one compact three-manifold. Further, he conjectured that even though not all compact three-manifolds can be modeled in one of the eight model geometries, they can be divided into smaller pieces that do it. This was known as Thurston's geometrization conjecture, which Perelman proved in $2003$.

In today's talk, we will discuss these eight geometries; more precisely, we will consider various properties of their Riemannian metrics.