Invariant connection on principal bundle

Question

Suppose I have a principal bundle, and some group G acting on the principal bundle. Is it always possible to find a G-invariant connection on the principle bundle? If G is compact, then I can imagine one can find such a connection by picking any connection to start with and then 'averaging' this connection over the group G. But what is G is not compact? Does an invariant connection still exist, in general? If not, what conditions do I need on G?

Answer 1

Maybe a partial answer is in H.C. Wang - (1957) On invariant connections over a principal fiber bundle, in my own notation below.

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Let $P\to M$ be a principal $G$-bundle, and let $K$ act transitively on $M$ on the left, so that $M$ is a $K$-homogeneous space. If we fix $x_0\in M$, then we can define an isomorphism $M\cong K/H$ where $H$ is the stabilizer of $x_0$ in $K$. A different point in $M$ gives a conjugate of $H$ in $K$.

We say that $P$ is $K$-homogeneous if the $K$ action on $M$ lifts to a left action on $P$ which commutes with the right $G$-action.

Now suppose $P$ is $K$-homogeneous, with $x_0$ and $H$ as above. Since $H$ fixes $x_0$ by definition, $H$ acts on the fiber $\pi^{-1}(x_0)$ (it may not be the identity on the fiber, it may shuffle things around).

Fix a point $p_0\in \pi^{-1}(x_0)$, and define the isotropy homomorphism $\lambda:H\to G$ by the following equation $$h.p_0 = p_0.\lambda(h)$$ A different choice of $p_0$ gives a $G$-conjugate of $\lambda$.

We call a connection $\omega$ $K$-invariant if $L_k^*\omega = \omega$ for all $k\in K$, where $L_k:P\to P$ is the left action.

Let $\mathfrak g,\mathfrak h,\mathfrak k$ denote the lie algebras of $G,H,K$ respectively.

Wang's Theorem classifies invariant connections on $K$-homogeneous principal $G$-bundles.

Theorem. The $K$-invariant connections $\omega$ on $P$ are in one-to-one correspondence with linear maps $\Lambda:\mathfrak k\to \mathfrak g$ such that

1. $\Lambda \circ \text{Ad}(h) = \text{Ad}(\lambda(h)) \circ \Lambda$, for all $h\in H$, and
2. $\Lambda\rvert_{\mathfrak h} = \mathrm d \lambda$.

The correspondence is, thinking of $\omega \in \Omega^1(P,\mathfrak g)$, $$\Lambda(v) := \omega_{p_0}(\tilde v)$$ where $\tilde v$ is the vector field on $P$ induced by $v\in \mathfrak k$.

The curvature can be computed in terms of $\Lambda$ $$F_{p_0}(\tilde v,\tilde w) = [\Lambda(v),\Lambda(w)] - \Lambda[v,w].$$

If $K$ doesn't act transitively, then you'll have the picture above for each orbit. For example, if the generic orbit has codimension one, then you have a one parameter family of linear maps $\Lambda(t)$, with some boundary conditions defined by behavior at the singular orbits (which you can calculate by Wang's theorem for the bigger stabilizers $H'$). Also there are smoothness conditions to worry about.