TITLE: What is a good way to think about a fundamental field on a principal G-bundle?
QUESTION [10 upvotes]: Let $\pi:P \rightarrow M$ be a principal $G$-bundle, and let $A \in \mathfrak{g}$, where the Lie algebra of $G$ is indicated. The fundamental field $A$# used to define connections is given by
$A$#$(p) := \frac{d}{dt}(\exp(At)p)|_{t=0}$.
$A$# is well defined since $e^{At}p$ can be regarded as a vector in $\pi^{-1}(\pi(p))$. Intuitively, I try to think of $A$# as the (vertical) direction of the displacement on the fiber generated by $A$.  
By the defining properties of principal bundles (in particular, the free action of $G$), we have $\{A$# $: A \in \mathfrak{g}\} \simeq \mathfrak{g} \simeq \mathcal{V}(p)$, where $\mathcal{V}(p)$ is the vertical subspace at $p$. Something like 
$A$#$(p) = \lim_t t^{-1}(e^{At}p - p) = A \cdot p$ 
would be (to me) a nice way of thinking about $A$#$(p)$, except that this is (at best) a formal equality. More precisely, $L_{\exp(At)}$ is the 1-parameter group of diffeomorphisms generated by $A$#, where $L_g$ denotes left multiplication by $g$.
My question: is there a better way of thinking about a fundamental field than (either the definition itself) or the notional equation above?

REPLY [2 votes]: I find that the notion of fundamental vector field is well defined not only for $G$-principal bundle but even for any $G$-manifold, i.e. a manifold with an action by a Lie group $G$.
About your notional equation, I would say that the fundamental vector fields effectively arise from an action of $\frak{g}$ on $M$. However some clarifications are needed.
Let $\Psi:M\times G\to M$ be a right action of a Lie group $G$ on a manifold $M$.
Let $\frak{g}$ be the Lie algebra of $G$, viewed as formed by the left invariant vectorfields on $G$.
Then there exists a unique map $\zeta^{\Psi}\equiv\zeta:X\in\frak{g} \mapsto $$\zeta_X\in \mathfrak{X}$$(M)$ such that $(T\psi)\circ(0_M+X)=\zeta_X\circ\Psi$, $\zeta_X$ and $0_M+X$ are $\Psi$-related, for any $X\in\frak{g}$.(Above $0_M$ denoted the zero vectorfield on $M$.)
For any $X\in\frak{g}$, the vector field $\zeta_X$ on $M$ is called the fundamental vectorfield corresponding to $X$ w.r.t. the right action $\Psi$.
The definition of $\zeta$ is well posed just because, for any $X\in\frak{g}$, the map $T\Psi\circ(0_M+X)$ is constant on the fibers of $\Psi$; and this holds being $\Psi$ a right action and $X$ a left invariant vectorfield. 
Obviously the following properties are satisfied:

$\zeta_{aX+bY}=a\zeta_X+b\zeta_Y,\zeta_{[X,Y]}=[\zeta_X,\zeta_Y]$, for any $a,b\in\mathbb{R}$, and $X,Y\in\frak{g}$,  i.e. $\zeta:\mathfrak{g} \to \mathfrak{X} (M)$ is a Lie algebra homomorphism;
$\zeta_X$ is complete and its $t$-time flow is $\Psi^{\exp{tX}}$, for any $X\in\frak{g}$ and $t\in\mathbb{R}$.

For an abstract Lie algebra $\frak{g}$, an action of $\frak{g}$ on a manifold $M$ is defined to be a Lie algebra homomorphism from $\frak{g}$ to $\frak{X}$$ (M)$.
In such a way for any right action $\Psi$ of a Lie group $G$ on $M$, we have that $\zeta^{\Psi}$ is an action on $M$ by the Lie algebra of $G$.