I have questions about the definition of representation variety. In François Labourie's book "Lectures on representations of surface groups", [Section 3.5](https://math.unice.fr/~labourie/preprints/pdf/surfaces.pdf), the author gives four models of the representation variety. I am confused about the model using the language of vector bundles.


Definition 1 (representation variety):


A representation variety of $S$ is gauge equivalences of pairs ($G$-vector bundles $L$ over the surface $S$, flat $G$-connection on $L$).


Definition 2 (Gauge equivalence):


Two connections **on the same vector bundle** are said to be gauge equivalent if the can be connected using the pullback of some lift of the identity map.


What confused me is definition 2. In order to use gauge equivalence in the definition of representation variety, why do we need to restrict the definition to the same vector bundle?


In other words, can we say that every flat $\bf R$-vector bundle over the surface is trivial (where $\bf R$ is the real number field)?



Flat R-bundles on surfaces
Any flat $\mathbb{R}^{+}$-bundle is precisely given, as in Tsemo Aristide's answer, by a representation of the fundamental group of the surface into $\mathbb{R}^+$. As $\mathbb{R}^+$ is abelian, this descends to an element of $H\_1(S,\mathbb{R}^+)$, whose logarithm is a uniquely determined element of $H\_1(S,\mathbb{R})=\mathbb{R}^{2g}$ where $g$ is the genus (assuming $S$ is orientable). So there are many such bundles which are not trivial.


