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Dear Steve Huntsman, 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 $\zeta^{\Psi}\equiv\zeta:X\in\frak{g}\to \zeta_X\in\frak{X} (M)$ 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. $\zeta:\frak{g}\to\frak{X} (M)$ 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 $G$.

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 $G$.

Dear Steve Huntsman, 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 $\zeta^{\Psi}\equiv\zeta:X\in\frak{g}\to \zeta_X\in\frak{X} (M)$ 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. $\zeta:\frak{g}\to\frak{X} (M)$ 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 $t\in\maathbb{R}$.

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 $G$.

Dear Steve Huntsman, 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 $\zeta^{\Psi}\equiv\zeta:X\in\frak{g}\to \zeta_X\in\frak{X} (M)$ 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. $\zeta:\frak{g}\to\frak{X} (M)$ 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 $G$.

Dear Steve Huntsman, 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.

About your feeling of unsatisfaction with the definition, and your notional equation, I would say that effectively the fundamental ve

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 $\zeta:X\in\frak{g}\to \zeta_X\in\frak{X}(M)$ such that $(T\psi)\circ(0_M+X)=\zeta_X\circ\Psi$, $\zeta_X$ and $0_M+0_M$ are $\Psi$-related, for any $X\in\frak{g}$.(Above $0_M$ denoted the zero vectorfield on $M$.) Obviously the following property is 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. $\zeta:\frak{g}\to\frak{X}(M)$ is a Lie algebra homomorphism. 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 $\Psi$. Remark: In order to justify the definition of $\zeta$ we have only to remark that, for any $X\in\frak{g}$, the map $T\Psi\circ(0_M+X)$ is constant on the fibers of $\Psi$, thanks to the left invariance of $X$.

Dear Steve Huntsman, 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 $\zeta^{\Psi}\equiv\zeta:X\in\frak{g}\to \zeta_X\in\frak{X} (M)$ 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. $\zeta:\frak{g}\to\frak{X} (M)$ 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 $t\in\maathbb{R}$.

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 $G$.