# at.algebraic topology – The judgement of a “relatively” flat connection retort

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## at.algebraic topology – The judgement of a “relatively” flat connection

Suppose that $$X$$ is a related gentle manifold and $$Gamma$$ is a bunch appearing easily, freely, correctly and discretely on $$X$$, in order that $$Y=X/Gamma$$ is one other gentle manifold endowed with a masking map $$pi:Xrightarrow Y$$.

Suppose that $$G$$ is a equivocate group and that $$rho:Gamma rightarrow G$$ is a bunch homomorphism. Then we are able to assume the quotient $$E_rho =(Xtimes G)/Gamma$$, the place the motion of $$Gamma$$ on $$G$$ is given by composing $$rho$$ with the adjoint motion of $$G$$ on itself. $$E_rho$$ is of course a principal $$G$$-bundle over $$Y$$.

My query is that if there exists a situation for a principal $$G$$-bundle $$E$$ on $$Y$$ to breathe isomorphic to an $$E_rho$$, for some homomorphism $$rho:Gamma rightarrow G$$.

This can breathe interpreted as a “relative” judgement of a flat connection since, if $$X$$ is the common masking area of $$Y$$ and $$Gamma=pi_1(Y)$$, then the situation for $$E$$ to breathe of the design $$E_rho$$ is that $$E$$ admits a flat connection.

moreover, the similar query can breathe prolonged to the holomorphic class. For occasion, we are able to maintain $$X$$ a Riemann floor and $$G=U(n)$$. In that standing, if $$X$$ is the hyperbolic aircraft and $$Y$$ is a compact Riemann floor of genus $$geq 2$$, the situation for a holomorphic Hermitian vector bundle $$E$$ to breathe of the design $$E_rho$$ is that it’s steady of diploma $$0$$ (that is the Theorem of Narasimhan-Seshadri).

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