Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

TL;DR

This paper introduces the Global Centre Symmetry set (GCS) for Lagrangian submanifolds, classifies its stable singularities, and explores their differences from affine-Lagrangian stable singularities, advancing geometric understanding of symmetries.

Contribution

It develops a new method for studying GCS singularities of Lagrangian submanifolds and classifies their stable types, extending previous affine equidistant analysis.

Findings

Classified all stable singularities of affine equidistants of Lagrangian submanifolds.

Identified differences between affine stable and affine-Lagrangian stable singularities.

Showed many GCS singularities in convex curves are not affine-Lagrangian stable.

Abstract

We define the Global Centre Symmetry set (GCS) of a smooth closed m-dimensional submanifold M of R^n, $n \leq 2m$, which is an affinely invariant generalization of the centre of a k-sphere in R^{k+1}. The GCS includes both the centre symmetry set defined by Janeczko and the Wigner caustic defined by Berry. We develop a new method for studying generic singularities of the GCS which is suited to the case when M is lagrangian in R^{2m} with canonical symplectic form. The definition of the GCS, which slightly generalizes one by Giblin and Zakalyukin, is based on the notion of affine equidistants, so, we first study singularities of affine equidistants of Lagrangian submanifolds, classifying all the stable ones. Then, we classify the affine-Lagrangian stable singularities of the GCS of Lagrangian submanifolds and show that, already for smooth closed convex curves in R^2, many singularities of the GCS which are affine stable are not affine-Lagrangian stable.