Support function, big front and caustic Duality an Singularities

Plática dada por Ricardo Uribe-Vargas (Université Borgogne, Franche-Comté, Francia) en el evento Singularities in generic geometry and its applications: Valencia V, para celebrar el 60 aniversario de Federico Sánchez Bringas el viernes 4 de agosto del 2017 en el Auditorio Carlos Graef del Amoxcalli en la Facultad de Ciencias de la UNAM

Abstract:
The so-called ”support function” of a given closed convex plane curve enables to describe the equidistant curves and their singularities. We show that the graph of the support function contains all local and global geometric information of the initial curve, of its equidistants and of its evolute (caustic). To any plane curve (without convexity restrictions) corresponds a curve on the unit cylinder (the graph of a ”multivalued support function”) and vice-versa. We define the ”support map”, which sends any plane curve to a curve on the unit cylinder and establish the correspondence between Euclidean differential geometry of plane curves and projective differential geometry of curves on the unit cylinder.
We geometrically construct the natural isomorphism between the front (in space-time) formed by the union of equidistants of a plane curve and the dual surface of its corresponding curve on the cylinder (the subvariety formed by the planes of R3 which are tangent to this space curve). Our results hold in Euclidean of higher dimensions for submanifolds of any dimension.
A corollary of our construction is the following:
Theorem. For any of singularities X (for example, A, D, E) the set of singularities of type X of the evolute of a smooth submanifold M of Rn is isomorphic to the set of singularities of type X in the front formed by the hyperplanes of Rn+1 which are tangent to the image of M by the support map (in the unit cylinder Cn ⊂ Rn+1) by the support map.
The results are explained by the natural contactomorphism between J1(Sn−1, R) and ST∗Rn and their relations with J1(Rn, R) and T∗Rn wherea Legendrian manifold of J1(Rn, R) is also a Lagrangian manifold of T∗Rn.

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Ricardo Uribe-Vargas en institut de mathematiques de bourgogne
https://math.u-bourgogne.fr/spip.php?article148

Ricardo Uribe-Vargas en Math-net
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Ricardo Uribe-Vargas en ResearchGate
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Ricardo Uribe-Vargas en Genealogy Project
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