If a fibre bundle can be equipped with a flat connection then it must be necessarily trivial? Let us take for example a real line bundle $L\to M$ with base $M$. If $L$ can be equipped with a flat connection then $L=M\times\mathbb{R}$?


Thanks.



Fibre bundles and flat connections
No. Any local system (vector bundle with constant coefficient transition matrices) admits a flat connection. You may simply use $d$ in each coordinate of a local trivialization, and the fact that the transitions have zero derivative makes this well-defined. 


In fact, local systems are equivalent to representations of the fundamental group of the base. In your example, the Möbius bundle over $S^1$ admits a flat connection, since it arises from the sign representation of $\mathbb{Z}$ into $GL\_1(\mathbb{R})$.


