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Let $P\longrightarrow M$ be a $G$-principal bundles. How do I define an action of $G$ over $TP$? Furthermore how can I identify the space of sections $\Gamma(TP/G)$ with $\mathfrak{X}(M)^G$ where $\mathfrak{X}(M)^G$ denotes the set of all $G$-invariant vector fields? Thanks

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In a G-principal bundle, you've got an action of $G$ on $P$. Now we have to say what a tangent vector is. I like the rule that says it's an equivalence class of paths, something like this: A tangent vector $v$ at $m$ is set of all paths $\gamma$ in $P$ with $\gamma(0)$ in the fiber over $m$, and such that if $\gamma_1$ and $\gamma_2$ are both in the set, then $\gamma_1 = \gamma_2$ to first order (measured via any coordinate system at $m$).

Now I need to define an action of $G$ on tangent vectors. I'll follow my nose on this one:

Define $$ (g \cdot \gamma)(t) = g \cdot (\gamma(t)) $$

That defines a NEW path $g \cdot \gamma$. With a bit of work, you show that if two paths agree to first order before the action of $g$, then they do so after as well.

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