5 - Projective invariants of submanifolds

Summary

This chapter concerns projective geometry and projective topology of submanifolds of dimension greater than 1 in projective space. We start with a panorama of classical results concerning surfaces in ℝℙ³. This is a thoroughly studied subject, and we discuss only selected topics connected with the main themes of this book. Section 5.2 concerns relative, affine and projective differential geometry of non-degenerate hypersurfaces. In particular, we construct a projective differential invariant of such a hypersurface, a (1, 2)-tensor field on it. Section 5.3 is devoted to various geometrical and topological properties of a class of transverse fields of directions along non-degenerate hypersurfaces in affine and projective space, the exact transverse line fields. In Section 5.4 we use these results to give a new proof of a classical theorem: the complete integrability of the geodesic flow on the ellipsoid and of the billiard inside the ellipsoid. Section 5.5 concerns Hilbert's fourth problem: to describe Finsler metrics in a convex domain in projective space whose geodesics are straight lines. We describe integral-geometric and analytic solutions to this celebrated problem in dimension 2 and discuss the multi-dimensional case as well. The last section is devoted to Carathéodory's conjecture on two umbilic points on an ovaloid and recent conjectures of Arnold on global geometry and topology of non-degenerate closed hypersurfaces in projective space.