2 - The geometry of the projective line

Summary

What are geometric objects? On the one hand, curves, surfaces, various geometric structures; on the other, tensor fields, differential operators, Lie group actions. The former objects originated in classical geometry while the latter ones are associated with algebra. Both points of view are legitimate, yet often separated.

This chapter illustrates unity of geometric and algebraic approaches. We study geometry of a simple object, the projective line. Such notions as non-degenerate immersions of a line in projective space and linear differential operators on the line are intrinsically related, and this gives two complementary viewpoints on the same thing.

Following F. Klein, we understand geometry in terms of group actions. In the case of the projective line, two groups play prominent roles: the group PGL(2,ℝ) of projective symmetries and the infinite-dimensional full group of diffeomorphisms Diff(ℝℙ¹). We shall see how these two types of symmetry interact.

Invariant differential operators on ℝℙ¹

The language of invariant differential operators is an adequate language for differential geometry. The best-known invariant differential operators are the de Rham differential of differential forms and the commutator of vector fields. These operators are invariant with respect to the action of the group of diffeomorphisms of the manifold. The expressions that describe these operations are independent of the choice of local coordinates.