Projective Geometry

Abstract

Two sprays G and $\tilde G$ on a manifold are said to be pointwise projectively related if they have the same geodesics as point sets. For any geodesic c(t) of G, there is an orientation-preserving reparameterization t = t(s) such that c(s) := c(t(s)) is a geodesic of $\tilde G$, and vice versa. In this chapter, we will show that two sprays G and $\tilde G$ on a manifold are pointwise projectively related if and only if there is a scalar function P on T M \ {0} such that

$$ \tilde G = G - 2P\;Y.$$

(1)

Then we prove the Rapcsák theorem on projectively related Finsler metrics. This remarkable theorem plays an important role in the projective geometry of Finsler spaces. See [Th3] for a systematic survey on the early development in this field.