# 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.

## Motivation

Important to applications in mathematics and physics is the notion of a flow on a manifold. In particular, if $M$ is a smooth manifold and $X$ is a smooth vector field, one is interested in finding integral curves to $X$. More precisely, given $p\in M$ one is interested in curves $\gamma _{p}:\mathbb {R} \to M$ such that:
    $\gamma _{p}'(t)=X_{\gamma _{p}(t)},\qquad \gamma _{p}(0)=p,$
for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If $X$ is furthermore a complete vector field, then the flow of $X$, defined as the collection of all integral curves for $X$, is a diffeomorphism of $M$. The flow $\phi _{X}:\mathbb {R} \times M\to M$ given by $\phi _{X}(t,p)=\gamma _{p}(t)$ is in fact an action of the additive Lie group $(\mathbb {R} ,+)$ on $M$.
Conversely, every smooth action $A:\mathbb {R} \times M\to M$ defines a complete vector field $X$ via the equation:
    $X_{p}=\left.{\frac {d}{dt}}\right|_{t=0}A(t,p).$
It is then a simple result that there is a bijective correspondence between $\mathbb {R}$ actions on $M$ and complete vector fields on $M$.
In the language of flow theory, the vector field $X$ is called the infinitesimal generator. 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 $M$.

## Definition

Let $G$ be a Lie group with corresponding Lie algebra ${\mathfrak {g}}$. Furthermore, let $M$ be a smooth manifold endowed with a smooth action $A:G\times M\to M$. Denote the map $A_{p}:G\to M$ such that $A_{p}(g)=A(g,p)$, called the orbit map of $A$ corresponding to $p$. For $X\in {\mathfrak {g}}$, the fundamental vector field $X^{\#}$ corresponding to $X$ is any of the following equivalent definitions:
* $X_{p}^{\#}=d_{e}A_{p}(X)$
* $X_{p}^{\#}=d_{(e,p)}A\left(X,0_{T_{p}M}\right)$
* $X_{p}^{\#}=\left.{\frac {d}{dt}}\right|_{t=0}A\left(\exp(tX),p\right)$
where $d$ is the differential of a smooth map and $0_{T_{p}M}$ is the zero vector in the vector space $T_{p}M$.
The map ${\mathfrak {g}}\to \Gamma (TM),X\mapsto -X^{\#}$ can then be shown to be a Lie algebra homomorphism.

## Applications
### Lie groups

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

### Hamiltonian group actions

In the motivation, it was shown that there is a bijective correspondence between smooth $\mathbb {R}$ actions and complete vector fields. Similarly, 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 $(M,\omega )$, we say that $X_{H}$ is a Hamiltonian vector field if there exists a smooth function $H:M\to \mathbb {R}$ satisfying
    $dH=\iota _{X_{H}}\omega$
where the map $\iota$ is the interior product. This motivates the definition of a Hamiltonian group action as follows: If $G$ is a Lie group with Lie algebra ${\mathfrak {g}}$ and $A:G\times M\to M$ is a group action of $G$ on a smooth manifold $M$, then we say that $A$ is a Hamiltonian group action if there exists a moment map $\mu :M\to {\mathfrak {g}}^{*}$ such that for each: $X\in {\mathfrak {g}}$,
    $d\mu ^{X}=\iota _{X^{\#}}\omega ,$
where $\mu ^{X}:M\to \mathbb {R} ,p\mapsto \langle \mu (p),X\rangle$ and $X^{\#}$ is the fundamental vector field of $X$.