Let $E\to X$ be a vector bundle with flat connection $\nabla$. Is there a canonical way to construct a vector subbundle of $E$ from the flat sections ($\nabla \sigma=0$)? More specifically, I would like to know if the flat sections on $U_i$ (where $\{U_i\}$ is a cover for $X$) form a locally free sheaf.
If this is not the case, are there any restrictions we can put on the vector bundle so that we have this property? For example, does this work if $E$ is a trivial bundle?
Best Answer
Locally free mean that locally we have $ \Gamma(U,E) \cong C^{\infty}(U)^r$ for some fixed $r$ ($r$ is the rank of the vector bundle).
However, flat sections $\mathcal F$ form a local system, i.e locally $\Gamma(U,\mathcal F) \cong \Bbb R^r$.
So you see this is not really the same, but however there is a bijection bewteen local system and vector bundles + flat connexions. In fact these are also equivalent to representations $\rho : \pi_1(X,x_0) \to GL(r,\Bbb {R})$.