# The notion of a "relatively" flat connection

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).

• This is equivalent to: Feb 22, 2021 at 17:08

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 $$\pi_1(X)$$.
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 $$g-1$$.