NDLI: Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

Content Provider	Springer Nature Link
Author	Rios, Pedro de M. Domitrz, Wojciech
Copyright Year	2013
Abstract	We study the global centre symmetry set (GCS) of a smooth closed submanifold $$M^m\subset \mathbb{R }^n, n \le 2m$$ . The GCS includes both the centre symmetry set defined by Janeczko (Geometria Dedicata 60:9–16, 1996) and the Wigner caustic defined by Berry (Philos Trans R Soc Lond A 287:237–271, 1977). The definition of GCS $$(M)$$ uses the concept of an affine $$\lambda $$ -equidistant of $$M, E_{\lambda }(M), \lambda \in \mathbb{R }$$ . When $$M=L$$ is a Lagrangian submanifold in the affine symplectic space $$(\mathbb{R }^{2m},\omega =\sum _{i=1}^m dp^i\wedge dq^i)$$ , we present generating families for singularities of $$E_{\lambda }(L)$$ and prove that the caustic of any simple stable Lagrangian singularity in a $$4m$$ -dimensional Lagrangian fibre bundle is realizable as the germ of an affine equidistant of some $$L\subset \mathbb{R }^{2m}$$ . We characterize the criminant part of GCS $$(L)$$ in terms of bitangent hyperplanes to $$L$$ . Then, after presenting the appropriate equivalence relation to be used in this Lagrangian case, we classify the affine-Lagrangian stable singularities of GCS $$(L)$$ . In particular we show that, already for a smooth closed convex curve $$L\subset \mathbb{R }^2$$ , many singularities of GCS $$(L)$$ which are affine stable are not affine-Lagrangian stable.
Ending Page	382
Page Count	22
Starting Page	361
File Format	PDF
ISSN	00465755
e-ISSN	15729168
Journal	Geometriae Dedicata
Issue Number	1
Volume Number	169
Language	English
Publisher	Springer Netherlands
Publisher Date	2013-05-03
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Singularities of differentiable mappings Lagrangian submanifolds; Maslov index Geometry Classification; finite determinacy of map germs Stability Normal forms Lagrangian singularities Symplectic geometry Centre symmetry set
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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