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

As shown by Bursztyn et al. (2019), 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-l...

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Bibliographic Details
Published in:Indagationes mathematicae 2021-02, Vol.32 (1), p.224-245
Main Author: Meinrenken, Eckhard
Format: Article
Language:English
Subjects:
Canonical forms
Darboux theorem
Fields (mathematics)
Isotropic embeddings
Lie groupoids
Linearizations
Manifolds (mathematics)
Morse lemma
Theorems
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Summary:As shown by Bursztyn et al. (2019), 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.
ISSN:0019-3577
1872-6100
DOI:10.1016/j.indag.2020.08.006