Premium
On Polar Duality, Lagrange and Legendre Singularities and Stereographic Projection to Quadrics
Author(s) -
UribeVargas Ricardo
Publication year - 2003
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611503014230
Subject(s) - stereographic projection , mathematics , submanifold , subvariety , hypersurface , quadric , hyperplane , projection (relational algebra) , mathematical analysis , gauss map , geometry , duality (order theory) , euclidean space , pure mathematics , legendre polynomials , surface of revolution , surface (topology) , algorithm , statistics , variety (cybernetics)
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 quadrics of revolution (and to a more general class of quadrics called quasi‐revolution quadrics ). V. D. Sedykh found a relation between Lagrangian and Legendrian singularities by stereographic projection to a sphere in Euclidean space. We generalise Sedykh' results 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 here R n denotes Euclidean space; (2) when the quadric is a paraboloid the relation between Lagrangian and Legendrian singularities is the most natural one, and the calculations, formulas and proofs are simpler; (3) using the classical theory of poles, polars and polar duality, we construct the natural isomorphism between the front of the Lagrange submanifold of the normal map (considered as a subvariety in J 0 R n = R n × R and the front of the Legendre submanifold of the tangential map (considered as a subvariety in the space( R n + 1 ) ∨ of affine n ‐dimensional subspaces in R n +1). As a consequence of our results we obtain a formula to calculate the vertices of smooth curves in R n . Our results may be applied to calculate and study umbilic points of surfaces in R 3 , or more generally to study the contact of submanifolds in Euclidean space R n with k ‐spheres, for k = 1, …, n − 1, in terms of the contact of submanifolds in R n +1 with ( k + 1)‐planes. We remark that it is possible to obtain additional geometric information when stereographic projection is replaced by an inversion. 2000 Mathematics Subject Classification 32S05, 51L15, 53A20, 53D05, 53D10, 53D12, 57R17, 58K05, 58K30, 58K35