I need help determining the Euler class of this vector bundle $\phi:E\to X$.
The base space is the torus $X = \mathbb{R}^2/\mathbb{Z}^2$ and the fiber over each point, $f^{-1}(x) \simeq \mathbb{R}^2$. So it is a vector bundle of rank $2$.
The fundamental group of this torus is the $\pi_0(X) \simeq \mathbb{Z}^2$ and consider the map from $\pi_0(X) \to \text{GL}_2(\mathbb{R})$
- $(1,0) \to \frac{1}{\sqrt{5}}\left[ \begin{array}{rr} 2 & -1 \\ 1 & 2 \end{array} \right]$
- $(0,1) \to \;\;\,\frac{1}{5}\left[ \begin{array}{rr} 3 & -4 \\ 4 & 3 \end{array} \right]$
These are commuting matrices so this is a well-defined map. There should be a flat vector bundle with these properties.
How can I find a chart for this vector bundles? Can I just take a chart from the torus and lift it?
What is a formula for the flat conneciton $\nabla$ ?