nLab Euler vector field

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

On vector spaces

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

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

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

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

On vector bundles

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

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

For submanifold embeddings

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

(BLM19, Def. 2.6)

Properties

Relation to homogeneous functions

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

ϵf=nf. \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.

References

The Euler vector field originates in the proof of Leonhard Euler‘s theorem on homogeneous functions:

Discussion in the generality of dg-manifolds ( L L_\infty -algebroids):

On Euler-like vector fields:

Created on August 28, 2024 at 16:46:35. See the history of this page for a list of all contributions to it.