Fundamental vector field

In the study of mathematics, and especially of differential geometry, fundamental vector fields are instruments that describe the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions.

Important to applications in mathematics and physics¹ is the notion of a flow on a manifold. In particular, if ${\displaystyle M}$ is a smooth manifold and ${\displaystyle X}$ is a smooth vector field, one is interested in finding integral curves to ${\displaystyle X}$. More precisely, given ${\displaystyle p\in M}$ one is interested in curves ${\displaystyle \gamma _{p}:\mathbb {R} \to M}$ such that:

${\displaystyle \gamma _{p}'(t)=X_{\gamma _{p}(t)},\qquad \gamma _{p}(0)=p,}$

for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If ${\displaystyle X}$ is furthermore a complete vector field, then the flow of ${\displaystyle X}$, defined as the collection of all integral curves for ${\displaystyle X}$, is a diffeomorphism of ${\displaystyle M}$. The flow ${\displaystyle \phi _{X}:\mathbb {R} \times M\to M}$ given by ${\displaystyle \phi _{X}(t,p)=\gamma _{p}(t)}$ is in fact an action of the additive Lie group ${\displaystyle (\mathbb {R} ,+)}$ on ${\displaystyle M}$.

Conversely, every smooth action ${\displaystyle A:\mathbb {R} \times M\to M}$ defines a complete vector field ${\displaystyle X}$ via the equation:

${\displaystyle X_{p}=\left.{\frac {d}{dt}}\right|_{t=0}A(t,p).}$

It is then a simple result² that there is a bijective correspondence between ${\displaystyle \mathbb {R} }$-actions on ${\displaystyle M}$ and complete vector fields on ${\displaystyle M}$.

In the language of flow theory, the vector field ${\displaystyle X}$ is called the infinitesimal generator.³ Intuitively, the behaviour of the flow at each point corresponds to the "direction" indicated by the vector field. It is a natural question to ask whether one may establish a similar correspondence between vector fields and more arbitrary Lie group actions on ${\displaystyle M}$.

Let ${\displaystyle G}$ be a Lie group with corresponding Lie algebra ${\displaystyle {\mathfrak {g}}}$. Furthermore, let ${\displaystyle M}$ be a smooth manifold endowed with a smooth action ${\displaystyle A:G\times M\to M}$. Denote the map ${\displaystyle A_{p}:G\to M}$ such that ${\displaystyle A_{p}(g)=A(g,p)}$, called the orbit map of ${\displaystyle A}$ corresponding to ${\displaystyle p}$.⁴ For ${\displaystyle X\in {\mathfrak {g}}}$, the fundamental vector field ${\displaystyle X^{\#}}$ corresponding to ${\displaystyle X}$ is given by any of the following equivalent definitions:²⁴⁵

• ${\displaystyle X_{p}^{\#}=d_{e}A_{p}(X)}$
• ${\displaystyle X_{p}^{\#}=d_{(e,p)}A\left(X,0_{T_{p}M}\right)}$
• ${\displaystyle X_{p}^{\#}=\left.{\frac {d}{dt}}\right|_{t=0}A\left(\exp(tX),p\right)}$

where ${\displaystyle d}$ is the differential of a smooth map and ${\displaystyle 0_{T_{p}M}}$ is the zero vector in the vector space ${\displaystyle T_{p}M}$.

The map ${\displaystyle {\mathfrak {g}}\to \Gamma (TM),X\mapsto -X^{\#}}$ can then be shown to be a Lie algebra homomorphism.⁵

The Lie algebra of a Lie group ${\displaystyle G}$ may be identified with either the left- or right-invariant vector fields on ${\displaystyle G}$. It is a well-known result³ that such vector fields are isomorphic to ${\displaystyle T_{e}G}$, the tangent space at identity. In fact, if we let ${\displaystyle G}$ act on itself via right-multiplication, the corresponding fundamental vector fields are precisely the left-invariant vector fields.

In the motivation, it was shown that there is a bijective correspondence between smooth ${\displaystyle \mathbb {R} }$-actions and complete vector fields. Similarly, given a symplectic manifold ${\displaystyle (M,\omega )}$, there is a bijective correspondence between symplectic actions (the induced diffeomorphisms are all symplectomorphisms) and complete symplectic vector fields.

A closely related idea is that of Hamiltonian vector fields. Given a symplectic manifold ${\displaystyle (M,\omega )}$, we say that ${\displaystyle X_{H}}$ is a Hamiltonian vector field if there exists a smooth function ${\displaystyle H:M\to \mathbb {R} }$ satisfying

${\displaystyle dH=\iota _{X_{H}}\omega }$

where the map ${\displaystyle \iota }$ is the interior product. This motivates the definition of a Hamiltonian group action as follows: If ${\displaystyle G}$ is a Lie group with Lie algebra ${\displaystyle {\mathfrak {g}}}$ and ${\displaystyle A:G\times M\to M}$ is a group action of ${\displaystyle G}$ on a smooth manifold ${\displaystyle M}$, then we say that ${\displaystyle A}$ is a Hamiltonian group action if there exists a moment map ${\displaystyle \mu :M\to {\mathfrak {g}}^{*}}$ such that for each: ${\displaystyle X\in {\mathfrak {g}}}$,

${\displaystyle d\mu ^{X}=\iota _{X^{\#}}\omega ,}$

where ${\displaystyle \mu ^{X}:M\to \mathbb {R} ,p\mapsto \langle \mu (p),X\rangle }$ and ${\displaystyle X^{\#}}$ is the fundamental vector field of ${\displaystyle X}$.