synthetic differential geometry
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On a finite-dimensional real vector space , canonically regarded also as a smooth manifold, the Euler vector field is that vector field whose flow is the linear multiplication action of the vector space
For any linear basis of the dual vector space to , which may be regarded as coordinate functions on the manifold , then the Euler vector field is equivalenty given by
More generally, for a real vector bundle over a smooth manifold , its Euler vector field is the vector field on the total space which over each point restricts to the above Euler vector field on the fiber space .
For a coordinate chart over the base manifold with coordinate functions , so that has coordinate functions , the Euler vector field on is again given by the form (1).
For a submanifold embedding of smooth manifolds, a vector field on is called Euler-like (with respect to ) if it vanishes on and its linear expansion around is an Euler vector field, in the above sense, on the normal bundle .
A smooth function is homogeneous of degree iff its derivative along the Euler vector field (above) satisfies
This basic fact is known as Euler’s homogeneous function theorem (e.g. Courant 1936) and this is the origin of the notion of the Euler vector field.
The Euler vector field originates in the proof of Leonhard Euler‘s theorem on homogeneous functions:
Richard Courant: Homogeneous Functions, pp. 108 in vol 1 of: Differential & Integral Calculus, 2 volumes (1936) [vol1: ISBN:978-1-118-03149-0, ark:/13960/t4fn6m190, vol2: ISBN:978-0-471-60840-0, ark:/13960/t2g82wq0c, pdf]
Wolfram MathWorld: Euler’s Homogeneous Function Theorem
Discussion in the generality of dg-manifolds (-algebroids):
On Euler-like vector fields:
Henrique Bursztyn, Hudson Lima, Eckhard Meinrenken, section 2 of: Splitting theorems for Poisson and related structures, J. Reine Angew. Math. 754 (2019) 281–312 [arXiv:1605.05386, doi:10.1515/crelle-2017-0014]
Eckhard Meinrenken: Euler-like vector fields, normal forms, and isotropic embeddings, Indagationes Mathematicae 32 1 (2021) 224-245, Indagationes Mathematicae [arXiv:2001.10518, doi:10.1016/j.indag.2020.08.006]
Arthur Lei Qiu: Euler-like vector fields (2021) [notes:pdf, pdf, slides:pdf, pdf]
L. A. Visscher: Euler-like vector fields and normal forms, Master thesis (2021) [pdf, pdf]
Created on August 28, 2024 at 16:46:35. See the history of this page for a list of all contributions to it.