6 - Projective structures on smooth manifolds

Summary

In this chapter we consider M, a smooth manifold of dimension n. How does one develop projective differential geometry on M? If M is a PGL(n + 1,ℝ)- homogeneous space, locally diffeomorphic to ℝℙⁿ, then the situation is clear, but the supply of such manifolds is very limited. Informally speaking, a projective structure on M is a local identification of M with ℝℙⁿ (without the requirement that the group PGL(n + 1, ℝ) acts on M). Projective structure is an example of the classic notion of a G-structure widely discussed in the literature. Our aim is to study specific properties of projective structures; see [121, 220] for a more general theory of G-structures.

There are many interesting examples of manifolds that carry projective structures; however, the general problem of existence and classification of projective structures on an n-dimensional manifold is wide open for n ≥ 3. There is a conjecture that every three-dimensional manifold can be equipped with a projective structure; see [196]. This is a very hard problem and its positive solution would imply, in particular, the Poincaré conjecture.

In this chapter we give a number of equivalent definitions of projective structures and discuss some of their main properties. We introduce two invariant differential operators acting on tensor densities on a manifold and give a description of projective structures in terms of these operators.