flat trivial vector bundle

Question

Let $E=M\times V$ be a trivial vector bundle over $M$. If $\nabla$ is a flat linear connection on $E$, this implies that $(E,\nabla)$ is isomorphic to $(E,\nabla^{\mathrm{triv}})$??, where $\nabla^{\mathrm{triv}}$ is the trivial connection on $E$. In other words, except by isomorphism, every flat connection on trivial bundles must be trivial?

Answer 1

A flat connection on $E$ is defined by a foliation ${\cal F}$ of dimension $dim(V)$ whose leaves are transverse to the vertical foliation $x\times V$. Let $p:M\times V\rightarrow M$ be the canonical projection and $x_0\in M$. You can define $\Phi:M\times V\rightarrow M\times V$ such that $\Phi(z)=(p(z),u(z))$ where the intersection of ${\cal F}_z$ and $x_0\times V$ is $(x_0,u(z))$. $\Phi$ defines an isomorphism between the connection defined by $M\times y$ and the connection defined by ${\cal F}$.

Answer 2

No. For any choice of base point $p\in M$, a flat connection determines a holonomy representation $\pi_1(M,p) \to \operatorname{GL}(E_p)$ (by parallel translating around closed loops), and equivalent connections determine isomorphic representations. The flat connections associated with global frames have trivial holonomy, any nontrivial representation of $\pi_1$ will correspond to a flat connection that doesn't have a global parallel frame.