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.
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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 $GL_1(\mathbb{R})$.
answered Feb 6, 2015 at 18:05
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$\begingroup$ Thanks. Does this mean that a flat vector bundle with simply connected base is trivial? A vector bundle endowed with a flat connection always admits a local system? (constant transition functions) $\endgroup$ Commented Feb 10, 2015 at 13:02
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1$\begingroup$ To be precise, I am referencing the Riemann-Hilbert correspondence, which should hold in smooth and holomorphic, but not algebraic, categories. It says that here is an equivalence between vector bundles with connection and local systems, given by taking the flat sections to be a basis for the local system (this works by the existence and uniqueness of solutions for integrable differential equations). This answers yes to both questions. $\endgroup$ Commented Feb 11, 2015 at 4:28