On polar duality, Lagrange and Legendre singularities and stereographic projection to quadrics

摘要

We establish the correspondence between Euclidean differential geometry of submanifolds in R~n and projective differential geometry of submanifolds in R~(n+1) under stereographic projection to revolution quadrics (defined below). There is a relation between Lagrangian and Legendrian singularities by stereographic projection to a sphere in Euclidean space [14] that we generalise in several directions, for instance: (1) such a relation holds for any stereographic projection of a hyperplane to a quasi-revolution quadric in R~n * R, where R~n denotes Euclidean space;