Euler-like vector fields, deformation spaces and manifolds with filtered structure

TL;DR

This paper explores the relationship between Euler-like vector fields, deformation spaces, and manifolds with filtered structures, extending known results to manifolds with Lie filtrations using deformation to the normal cone.

Contribution

It introduces a deformation to the normal cone suited for manifolds with Lie filtrations and connects it to Euler-like vector fields and tubular neighborhood embeddings.

Findings

Established a bijection between Euler-like vector fields and tubular neighborhoods.

Generalized deformation to the normal cone for manifolds with Lie filtrations.

Linked deformation spaces to filtered manifold structures.

Abstract

The first purpose of this note is to comment on a recent article of Bursztyn, Lima and Meinrenken, in which it is proved that if M is a smooth submanifold of a manifold V, then there is a bijection between germs of tubular neighborhoods of M and germs of "Euler-like" vector fields on V. We shall explain how to approach this bijection through the deformation to the normal cone that is associated to the embedding of M into V. The second purpose is to study generalizations to smooth manifolds equipped with Lie filtrations. Following in the footsteps of several others, we shall define a deformation to the normal cone that is appropriate to this context, and relate it to Euler-like vector fields and tubular neighborhood embeddings.