Notes on Projective Differential Geometry

Projective differential geometry was initiated in the 1920s, especially by Élie Cartan and Tracey Thomas. Nowadays, the subject is not so well-known. These notes aim to remedy this deficit and present several reasons why this should be done at this time. The deeper underlying reason is that projective differential geometry provides the most basic application of what has come to be known as the ‘Bernstein-Gelfand-Gelfand machinery’. As such, it is completely parallel to conformal differential geometry. On the other hand, there are direct applications within Riemannian differential geometry. We shall soon see, for example, a good geometric reason why the symmetries of the Riemann curvature tensor constitute an irreducible representation of SL(n,R) (rather than SO(n) as one might näıvely expect). Projective differential geometry also provides the simplest setting in which overdetermined systems of partial differential equations naturally arise. Let M be a smooth real manifold of dimension n. There are two ways to define a projective differential geometry on M . One is geometric and intuitive. The other is more operational and useful in practice. Their equivalence is the subject of the following proposition.