Projective connections

Abstract

We have seen, in Chapter III, that the paths of an A _n are not changed by a projective transformation

$$ '\Gamma _{\mu \lambda }^\chi = \Gamma _{\mu \lambda }^\chi + 2p\left( {_\mu A_\lambda ^\chi } \right). $$

((1.1))

$$ {p_\lambda }arbitrary vector $$

The problem arose of associating with this group of transformations a single “projective” connection, which will take the place of the infinite number of L _n connections. The introduction of the parameters $II_{\mu \lambda}^{\chi}$ was one step, but it was not yet sufficient, because they depend on a special choice of coordinates. We must try to continue in the same direction of research, changing the group of transformations. This has been done and has lead to important results. It seems however more useful to attack the problem from another side and use as the starting point the fundamental principle of differential geometry, as formulated in Ch. I. We take an X _n and associate with every point a local projective space D _n . We ask for the linear connections associable with this configuration. It can then be shown that an infinite number of L _n related by (1.1) can be obtained from this projective connection.