Germs of tubular neighborhood embeddings for submanifolds N of manifolds 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.

中文翻译：

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类欧拉矢量场、范式和各向同性嵌入

流形 M 的子流形 N 的管状邻域嵌入的胚芽​​与 N 附近的类欧拉向量场的胚芽一一对应。在许多情况下，这将几何结构的“范式结果”证明简化为构建一个与给定结构兼容的类欧拉矢量场。我们在各种例子中说明了这个原理，包括 Morse-Bott 引理、Weinstein 的拉格朗日嵌入定理和 Zung 的线性化定理，用于适当的 Lie groupoids。在本文的第二部分，我们将理论扩展到加权上下文，并应用于各向同性嵌入。