# Principal fiber bundles and invariant differential forms

Let $$G$$ be a real Lie group with a right action on a smooth manifold $$X$$. Assume the action is free and proper. This implies there is a unique manifold structure on the quotient space $$G \backslash X$$ such that the quotient map $$\pi$$ is a submersion, and that $$\pi$$ is a principal fiber bundle with $$G$$ is a fiber.

In other words, locally, the quotient map looks like $$U \times G \rightarrow U, (u,g) \mapsto u$$ for sufficiently small open sets $$U \subset G \backslash X$$, with $$G$$ acting by $$(u,x).g = (u,xg)$$.

Let $$\omega \in \Omega^k(X \backslash G)$$ be a smooth differential $$k$$-form on $$X \backslash G$$. Then $$\omega$$ pulls back to a differential $$k$$-form $$\pi^{\ast}(\omega)$$ on $$X$$ which is $$G$$-invariant.

1 . Is $$\omega \mapsto \pi^{\ast}(\omega)$$ an injective map $$\Omega^k(G \backslash X) \rightarrow \Omega^k(X)$$?

2 . What is the image of $$\pi^{\ast}$$? Does it consist of all $$G$$-invariant differential $$k$$-forms on $$X$$?

If $$X$$ is a trivial principal fiber bundle, i.e. $$X = Y \times G$$ for a smooth manifold $$Y$$, then all this seems obvious, but I'm not sure if there are some complications that can arise if $$X$$ is merely covered by such things.

$$\omega\rightarrow \pi^*\omega$$ is linear. Suppose that $$\pi^*\omega=0$$, since $$\pi$$ is a submersion, for every $$x\in G/X, u_1,...,u_k\in T_x(G/X)$$ and $$y\in\pi^{-1}(x)$$, there exists $$v\in T_yX$$ such that $$d\pi_y(v_i)=u_i$$, $$\pi^*\omega_y(v_1,...,v_k)=\omega_x(u_1,...,u_k)=0$$ implies that $$\omega=0$$ and $$\omega\rightarrow \pi^*\omega$$ is injective.
Consider $$X\times G$$ and take any non zero $$1$$-form $$\beta$$ invariant on $$G$$ by the right translations. Write $$\alpha_{(x,g)}(u,v)=\beta_g(v)$$ is a form invariant by $$G$$. You cannot have $$\alpha=\pi^*\omega$$ since $$\pi^*\omega$$ vanishes on the fibre.
• The images of $\pi^*$ consists of all $G$-invariant forms which are horizontal in the sense that they vanish upon insertion of a single vertical vector field. – Andreas Cap Jan 7 at 11:30