Klein geometries, parabolic geometries and differential equations of finite type

We define the infinitesimal and geometric orders of an effective Klein geometry G/H. Using these concepts, we prove (i) For any integer m ≥ 2, there exists an effective Klein geometry G/H of infinitesimal order m such that G/H is a projective variety. (ii) An effective Klein geometry G/H of geometric order M defines a differential equation of order M + 1 on G/H whose global solution space is G. © 2008 Heldermann Verlag.