幾何学コロキウム(2017/6/16): Support function, big front singularities and duality

開催日時
2017年   6月 16日 16時 30分 ~ 2017年   6月 16日 18時 00分
場所
理3-204号室
講演者
Ricardo Uribe-Vargas 氏 ( Bourgogne 大学 )
 
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 construction the natural isomorphism
between the front (in space-time) formed by the union of equidistants
of a plane curve with 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.

Theorem. For any class of singularities $X$ 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.

The talk will be elementary and with many pictures.


関連項目
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研究集会・セミナー・集中講義の一覧へ