Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

We study the global centre symmetry set ( GCS ) of a smooth closed submanifold . 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 uses th...

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Bibliographic Details
Published in:Geometriae dedicata 2014-04, Vol.169 (1), p.361-382
Main Authors: Domitrz, Wojciech, Rios, Pedro de M.
Format: Article
Language:English
Subjects:
Algebraic Geometry
Convex and Discrete Geometry
Differential Geometry
Hyperbolic Geometry
Mathematics
Mathematics and Statistics
Original Paper
Projective Geometry
Topology
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Summary:We study the global centre symmetry set ( GCS ) of a smooth closed submanifold . 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 uses the concept of an affine -equidistant of . When is a Lagrangian submanifold in the affine symplectic space , we present generating families for singularities of and prove that the caustic of any simple stable Lagrangian singularity in a -dimensional Lagrangian fibre bundle is realizable as the germ of an affine equidistant of some . We characterize the criminant part of GCS in terms of bitangent hyperplanes to . Then, after presenting the appropriate equivalence relation to be used in this Lagrangian case, we classify the affine-Lagrangian stable singularities of GCS . In particular we show that, already for a smooth closed convex curve , many singularities of GCS which are affine stable are not affine-Lagrangian stable.
ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-013-9861-2