Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

Abstract

We study the global centre symmetry set (GCS) of a smooth closed submanifold 'M POT.M'⊂'R POT.N', n≤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 λ-equidistant of M, 'E IND.λ'(M), λ∈R . When M=L is a Lagrangian submanifold in the affine symplectic space ('R POT.2M', [...]) , we present generating families for singularities of 'E IND.λ'(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⊂'R POT.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⊂'R POT.2' , many singularities of GCS(L) which are affine stable are not affine-Lagrangian stable.