会場

埼玉大学 大学院理工学研究科棟5階 数学研究室1 （ このページ の15番の建物）

講演者

Ricardo Uribe-Vargas, Université de Bourgogne

タイトル

Support function, big front singularities and duality.

アブストラクト

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 $R^3$ which are tangent to this space curve). Our results hold in Euclidean spaces of higher dimensions for submanifolds of any dimension.
A corollary of our construction is the following:
Theorem. For any class of singularities $X$ (for example, $A, D, E$) the set of singularities of type $X$ of the evolute of a smooth submanifold $M$ of $R^n$ is isomorphic to the set of singularities of type $X$ in the front formed by the hyperplanes of $R^{n+1}$ which are tangent to the image of $M$ by the support map (in the unit cylinder $C_n\subset R^{n+1}$) by the support map.

Seminar dinner