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Euler-like vector fields, normal forms, and isotropic embeddings. (English) Zbl 1465.53037

According to [F. Bischoff et al., Compos. Math. 156, No. 4, 697–732 (2020; Zbl 1434.53029)], germs of tubular neighborhood embeddings for submanifolds \(N\subseteq M\) are in one-one correspondence with germs of Euler-like vector fields near \(N\) (any such field vanishes along \(N\), and its linear approximation is the Euler vector field on the normal bundle). This result allows to reduce the proof of ‘normal forms results’ for geometric structures to the construction of an Euler-like vector field compatible with the given structure. In the first expository part of this article, the author illustrates this principle in a variety of examples, including the Morse-Bott lemma, A. Weinstein’s Lagrangian embedding theorem [Adv. Math. 6, 329–346 (1971; Zbl 0213.48203); J. Differ. Geom. 16, 125–128 (1981; Zbl 0453.53030); Bull. Am. Math. Soc., New Ser. 16, 101–104 (1987; Zbl 0618.58020)], and Nguyen Tien Zung’s linearization theorem [Ann. Sci. Éc. Norm. Supér. (4) 39, No. 5, 841–869 (2006; Zbl 1163.22001)] for proper Lie groupoids. In the second part of this article, the author extends the theory to a weighted context (mainly, for degree-2 weightings), with an application to isotropic embeddings (see Theorems 5.10 and 5.12).

MSC:

53B25 Local submanifolds

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References:

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