Locally homogeneous connections on principal bundles over hyperbolic Riemann surfaces

Let $g$ be locally homogeneous (LH) Riemannian metric on a differentiable compact manifold $M$, and $K$ be a compact Lie group endowed with an $\mathrm {ad}$-invariant inner product on its Lie algebra $\mathfrak{k}$. A connection $A$ on a principal $K$-bundle $p:P\to M$ on $M$ is locally homogeneous if for any two points $x_1$, $x_2\in M$ there exists an isometry $\varphi:U_1\to U_2$ between open neighborhoods $U_i\ni x_i$ which sends $x_1$ to $x_2$ and admits a $\varphi$-covering bundle isomorphism preserving the connection $A$. This condition is invariant under the action of the automorphism group (gauge group) of the bundle, so the classification problem for LH connections leads to an interesting moduli problem: for fixed objects $(M,g,K)$ as above describe geometrically the moduli space of all LH connections on principal $K$-bundles on $M$ (up to bundle isomorphisms). Note that if $A$ is LH, then the associated connection metric $g_A$ on $P$ is locally homogenous, so it defines a geometric structure (in the sense of Thurston) on the total space of the bundle. Therefore this moduli problem is related to the classification of LH (geometric) Riemannian manifolds which admit a Riemannian submersion onto the given manifold $M$. Omitting the details, our moduli problem concerns the classification of geometric fibre bundles over a given geometric base. We develop a general method for describing moduli spaces of LH connections on a given base. Using our method we give explicit descriptions of these moduli spaces when the base manifold is a hyperbolic Riemann surface $(M,g)$ and $K\in\{S^1,PU(2)\}$. The case $K=S^1$ leads to a new construction of the moduli spaces of Yang-Mills $S^1$-connections on hyperbolic Riemann surfaces, and the case $K=PU(2)$ leads to a one-parameter family of compact, 5-dimensional geometric manifolds, which we study in detail.