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 m ⊂ R 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 affinedp i ∧ dq i ), we present generating families for singularities of E 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 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 2 , many singularities of GCS(L) which are affine stable are not affine-Lagrangian stable.

The Wigner caustic on shell and singularities of odd functions

“…We recall basic definitions of the theory of Lagrangian singularities (see [1], [7]). First, (T R 2m ,ω) with canonical projection π :…”

Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space

“…The choice of linear equation forẋ in (2.1) is not unique, but this is the simplest one. Among other possibilities, the choicė x = λx + − (1 − λ)x − is particularly well suited for the study of affine equidistants of Lagrangian submanifolds in symplectic space [3]. Now, let M be a smooth closed n-dimensional submanifold of the affine space R q (2n ≥ q) and consider the product…”

“…Affine equidistants of smooth submanifolds, in particular the Wigner caustic, have a way in mathematical physics and in the definition of affine-invariant global centre symmetry sets of these submanifolds and, in every case, precise knowledge of their singularities is an important issue [10,7,8,3,2]. Thus, stable singularities of affine equidistants of M n ⊂ R q have been extensively studied [1,6,7,8,9,3], culminating in its complete classification for all pairs (2n, q) of nice dimensions [4].…”

“…Thus, stable singularities of affine equidistants of M n ⊂ R q have been extensively studied [1,6,7,8,9,3], culminating in its complete classification for all pairs (2n, q) of nice dimensions [4].…”

Centre Symmetry Sets of Families of Plane Curves

“…T he CSS, which is invariant under affine transformations of JRk 1 1 , has been studied in detail for many cases in, for example, [10,7,8,9,6]. In this article, we are principally concerned with k = 1, that is a plane curve M , but allowing the curve to vary in a generic 1-parameter family.…”