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Let $\pi \colon P \to M$ be a $G$-principal bundle. If $G$ is compact, we may lift any metric $m$ on $M$ to a $G$-invariant metric $\bar{m}$ on $P$ such that $\pi_\ast\bar{m} = m$. Doing so accounts for choosing a connection on $P$ and an $\mathrm{Ad}$-invariant inner-product on $\mathfrak{g}$ to put a metric on the vertical bundle of $P$. Finally one gets a metric on $P$ declaring the horizontal and vertical bundle to be orthogonal.

My question is:

What happens when $G$ is non-compact? Can we still lift arbitrary metrics on $M$ to $G$-invariant metrics on $P$? If so, is there any nice construction of the metric on $P$ as in the case when $G$ is compact?

On the one hand, denoting by $\xi_v$ the infinitesimal generator of the action one has that $$ (\text{d} \mu_g)_p(\xi_v(p)) = \xi_{\text{Ad}_{g^{-1}}(v)}(\mu_g(p))\qquad \forall g \in G, p \in P, v \in \mathfrak{g} $$ meaning that de construction above just works in the case when we can find an $\text{Ad}$-invariant inner-product on $\mathfrak{g}$, which just exists for (products of) compact and Euclidean Lie algebras.

On the other hand, a result of Palais says that if one has a proper $G$-action on a manifold $P$, there exists a metric $\bar{m}$ on $P$ that makes $\mu_G = \{\mu_g\mid g \in G\}$ into a closed subgroup of the isometries of $(M, \bar{m})$. In particular, for a principal bundle, this $G$-invariant metric descends into a metric $\pi_\ast\bar{m} = m$ on $M \cong P / G$. This means that, on a general principal bundle, there exist some metrics that can be lifted to the total space, regardless of $G$ being compact or not.

What is going on here? I have looked this up for quite some time now, but I have not been able to find a reference where they treat this, so any reference would be more than welcome as well.

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    $\begingroup$ I don't see where compactness of $G$ is needed. You need a metric on the base, $G$-invariant metric on the fiber, and a principal connection, and they can be put together in an obvious way to a Riemannian submersion metric on the total space. For a reference see Besse's "Einstein manifold", 9.59 and 9.54. Besse in turn refers to Vilms "Totally geodesic maps", projecteuclid.org/euclid.jdg/1214429276. $\endgroup$
    – Igor Belegradek
    Sep 23, 2020 at 17:31
  • $\begingroup$ How does one put together a $G$-invariant metric on the fiber when $G$ is non-compact? Using that theorem of Palais or is there a more explicit way of constructing it? If we have an inner-product on $\mathfrak{g}$ and we want to extend it to the whole $VP$, we would need it to be $\text{Ad}$-invariant by the formula above, but we can just do that on compact and Euclidean Lie algebras... $\endgroup$
    – Lezkus
    Sep 24, 2020 at 11:38
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    $\begingroup$ I don't what what you mean by "more explicit". The obvious strategy is to consider local $G$-invariant metrics obtained by averaging, and then glue on overlaps, and there are ways to glue it, and this is what Palais does. You may also wish to look at "Proper smooth G-manifolds have complete G-invariant Riemannian metrics" by Kankaanrinta, sciencedirect.com/science/article/pii/S0166864105000532. In Riemannian applications it often happens that an isometric action on the fiber is given. You usually need to construct a metric when you are trying to do topology, not geometry. $\endgroup$
    – Igor Belegradek
    Sep 24, 2020 at 12:31
  • $\begingroup$ By more explicit I meant as in the case when the group is compact, where one can be constructed via a bi-invariant metric. In any case, thank you so much for the clarification, all is clear now! $\endgroup$
    – Lezkus
    Sep 24, 2020 at 12:34
  • $\begingroup$ In the linked paper Kankaanrinta favors the global approach. The manifold is embedded equivariantly into a Hilbert space, and then one just takes the induced invariant Riemannian metric. $\endgroup$
    – Igor Belegradek
    Sep 24, 2020 at 12:37

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