Projective connections

We compare the approaches of E. Cartan and of T.Y. Thomas and J.H.C. Whitehead to the study of ‘projective connections’. Although the quoted phrase has quite different meanings in the two contexts considered, we show that a class of projectively equivalent symmetric affine connections–or, more generally, sprays–on a manifold (the latter meaning) gives rise, in a global way, to a unique Cartan connection on a principal bundle over the manifold (the former meaning). The principal bundle on which the Cartan connection is defined is itself a geometric object, and exists independently of any particular connection. In the course of the discussion we derive a Cartan normal projective connection for a system of second-order ordinary differential equations (extending the results of Cartan from a single equation to many) and we generalize the concept of a normal Thomas–Whitehead connection from affine to general sprays.