In the context of the connections on fibre bundle, I have found some difficulties trying to understand the fundamental vector field (my reference is Nakahara, but I’m having some problems with the chapters about fibre bundles, can someone suggest another book about this topic?).
It seems to me that the fundamental vector field is nothing more that an induced vector field, but induced by a right action instead of a left action of a Lie group on a manifold.

Let $M$ and $G$ be a smooth manifold and a Lie group respectively and suppose to have a left action of the group on the manifold $$
sigma_L:(p,g)in Mtimes Grightarrow sigma_L(p,g)equiv gp =q in M
$$
but, since $G$ is a Lie group, we can use the exponential map and write every element of the Lie group as $g=operatorname{exp}(tV)$, where $Vin T_eG$ is a generator of the group. So we can go from a flux on the group (the set of all the integral curves of the left invariant vector field associated to $V$) to a flux on the manifold, that we can see as the set of all the integral curves of a vector field on the manifold, called induced vector field $V^#$
$$
V^#(p)=left. frac{d}{dt}(operatorname{exp}(tV)p)right_{t=0}
$$
It’s also possible to prove that there is an antihomomorphism between $T_eG congmathfrak{g}$ and $mathfrak{X}(M)$. We can define another vector field on the manifold, the fundamental vector field, using $t$ instead of $t$ in the previous definition and in this case we have an homomorphism instead of an antihomomorphism. 
Let $(P,pi, M, G)$ be a principal bundle and, as such, there is a global right action of the structure group $G$ on the total space $P$. Since $P$ is a manifold and $G$ is a Lie group acting on $P$, we can follow the previous steps in order to define a fundamental vector field on the principal bundle (since the action is a right action, we have the fundamental vector field directly, with the plus sign in the definition). So we know that, evaluating a fundamental vector field in a point $uin P$
$$
V^#(u)=left. frac{d}{dt}(u operatorname{exp}(tV))right_{t=0} in T_uP
$$
but we also know that $pi(u)=pi(ug)=pin M forall g in G$,therefore the integral curve of the fundamental field with initial point $uin P$ is a curve contained entirely in the fiber $G_{pi(u)}$ so we can conclude that every vector $V^#(u)$ is a vector tangent to the principal bundle in the point $u$ which is also tangent to the fiber in that point.
At this point I have some problem with some statements:
 There is an homomorphism between $T_eG congmathfrak{g}$ and $V_uP$. Why the homomorphism is with $V_uP$ instead of the set of all the fundamental vector fields (like it is the “general” case, where the homomorphism is with $mathfrak{X}(M)$ and not with $T_pM$)?
 Then, when the horizontal subspace is defined, Nakahara says that $T_uP=V_uPoplus H_uP$ and states “a smooth vector field $X$ on $P$ is separated into smooth vector fields $X^Vin V_uP$ and $X^Hin H_uP$ as $X=X^V+X^H$“. Therefore $V_uP$ is the set of all the fundamental vector fields? I had understood that it was the set of all the vertical vectors in $uin P$, that you can get evaluating all the fundamental vector fields in the same point $u$, but from this statement it seems that an element of $V_uP$ is a vector field and not a vector.
Is there any difference between a fundamental vector field on a manifold and a fundamental vector field on a principal bundle?