Projective Differential Geometrical Structure of the Painlevé Equations

The necessary and sufficient conditions that an equation of the form y″=f(x, y, y′) can be reduced to one of the Painlevé equations under a general point transformation are obtained. A procedure to check these conditions is found. The theory of invariants plays a leading role in this investigation. The reduction of all six Painlevé equations to the form y″=f(x, y) is obtained. The structure of equivalence classes is investigated for all the Painlevé equations. Following Cartan the space of the normal projective connection which is uniquely associated with any class of equivalent equations is considered. The specific structure of the spaces under investigation allows us to immerse them into P3. Each immersion generates a triple of two-dimensional manifolds in P3. The surfaces corresponding to all the Painlevé equations are presented.