nLab Euler vector field

Definition

On vector spaces

On a finite-dimensional real vector space $V$, canonically regarded also as a smooth manifold, the Euler vector field $\epsilon \colon V \longrightarrow T V$ is that vector field whose flow $\phi_\epsilon$ is the linear multiplication action of the vector space

$$\array{ V \times \mathbb{R} &\xrightarrow{\; \phi_\epsilon \;}& V \\ (v,t) &\mapsto& t\cdot v \mathrlap{\,.} }$$

For $\big\{v^i \in V^\ast\big\}_{i =1}^{dim(V)}$ any linear basis of the dual vector space to $V$, which may be regarded as coordinate functions on the manifold $V$, then the Euler vector field is equivalenty given by

(1)$$\epsilon \;=\; \textstyle{\sum_{i=1}^{dim(V)}} v^i \partial_{v^i} \,.$$

On vector bundles

More generally, for $V$ a real vector bundle over a smooth manifold $X$, its Euler vector field is the vector field on the total space $V$ which over each point $x \in X$ restricts to the above Euler vector field on the fiber space $V_x$.

For $U \hookrightarrow X$ a coordinate chart over the base manifold with coordinate functions $\big\{x^j\big\}_{j=1}^{dim(X)}$, so that $V_{\vert U}$ has coordinate functions $(x^j, v^i)$, the Euler vector field on $V_{\vert U}$ is again given by the form (1).

For submanifold embeddings

For $Y \xrightarrow{\iota} X$ a submanifold embedding of smooth manifolds, a vector field on $X$ is called Euler-like (with respect to $\iota$) if it vanishes on $Y$ and its linear expansion around $Y$ is an Euler vector field, in the above sense, on the normal bundle $N_\iota(Y)$.

(BLM19, Def. 2.6)

Properties

Relation to homogeneous functions

A smooth function $f \colon V \xrightarrow{} \mathbb{R}$ is homogeneous of degree $n \in \mathbb{N}$ iff its derivative along the Euler vector field $\epsilon$ (above) satisfies

$$\epsilon f \;=\; n f \,.$$

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.

Related concepts

• Liouville-Poincaré 1-form

References

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 ($L_\infty$-algebroids):

• Dmitry Roytenberg: §2 in: On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson Brackets and Beyond, Contemp. Math. 315, Amer. Math. Soc. (2002) [arXiv:math/0203110, ams:conm-315]

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]