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Suppose that $X$ is a connected smooth manifold and $\Gamma$ is a group acting smoothly, freely, properly and discretely on $X$, so that $Y=X/\Gamma$ is another smooth manifold endowed with a covering map $\pi:X\rightarrow Y$.

Suppose that $G$ is a Lie group and that $\rho:\Gamma \rightarrow G$ is a group homomorphism. Then we can consider the quotient $E_\rho =(X\times G)/\Gamma$, where the action of $\Gamma$ on $G$ is given by composing $\rho$ with the adjoint action of $G$ on itself. $E_\rho$ is naturally a principal $G$-bundle over $Y$.

My question is if there exists a condition for a principal $G$-bundle $E$ on $Y$ to be isomorphic to an $E_\rho$, for some homomorphism $\rho:\Gamma \rightarrow G$.

This can be interpreted as a "relative" notion of a flat connection since, if $X$ is the universal covering space of $Y$ and $\Gamma=\pi_1(Y)$, then the condition for $E$ to be of the form $E_\rho$ is that $E$ admits a flat connection.

Moreover, the same question can be extended to the holomorphic category. For example, we can take $X$ a Riemann surface and $G=U(n)$. In that situation, if $X$ is the hyperbolic plane and $Y$ is a compact Riemann surface of genus $\geq 2$, the condition for a holomorphic Hermitian vector bundle $E$ to be of the form $E_\rho$ is that it is stable of degree $0$ (this is the Theorem of Narasimhan-Seshadri).

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  • $\begingroup$ This is equivalent to: $\endgroup$ Feb 22, 2021 at 17:08

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Your bundle is of the form $E_\rho$ if and only if it admits a flat connection whose holonomy $\rho:\pi_1(Y) \to G$ is trivial in restriction to $\pi_1(X)$.

Even without this additional assumption, I don't know a general answer to when a principal bundle admits a flat connection. There are necessary conditions (namely, the vanishing of some characteristic classes) but they are not sufficient in general.

For instance, the Milnor--Wood inequality states that principal $\mathrm{SL}(2,\mathbb R)$ bundles over a Riemann surface of genus $g$ admit a flat connection if and only if their reduction of structure group to $\mathrm{SO}(2)$ has Euler class between $1-g$ and $g-1$.

Also be careful that the stability condition for holomorphic vector bundle is not a condition for a holomorphic Hermitian vector bundle to be flat, it is a condition for a holomorphic vector bundle to admit a Hermitian metric whose Chern connection is flat.

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  • $\begingroup$ I agree with all your comments, but I'm happy with your first sentence. Have you got any reference for it? $\endgroup$
    – G. Gallego
    Feb 22, 2021 at 23:43

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