Euler-like vector fields, normal forms, and isotropic embeddings

Abstract

As shown in [4], germs of tubular neighborhood embeddings for submanifolds N ⊆ M are in one-one correspondence with germs of Euler-like vector fields near N . In many contexts, this reduces the proof of 'normal forms results' for geometric structures to the construction of an Euler-like vector field compatible with the given structure. We illustrate this principle in a variety of examples, including the Morse-Bott lemma, Weinstein's Lagrangian embedding theorem, and Zung's linearization theorem for proper Lie groupoids. In the second part of this article, we extend the theory to a weighted context, with an application to isotropic embeddings.