I have been trying to learn about characteristic classes for months now, and every time I try a simple example something goes wrong. Any insights would be greatly appreciated.
I am trying to follow the procedure found in [Bott & Tu] in which the Euler class of an oriented, rank 2 bundle is constructed from the bundle transition maps. Here is what I have done so far:
Consider the two-sphere $S^2$ with the standard embedding in $\mathbb{R}^3$ as covered by two coordinate patches $U_1$ and $U_2$ which miss out the north and south poles respectively. And take $$\sigma_1 \colon U_1 \to \mathbb{R}^2 \quad ; \quad (x,y,z) \mapsto \frac{1}{1-z}(x,y)$$ $$\sigma_2 \colon U_2 \to \mathbb{R}^2 \quad ; \quad (x,y,z) \mapsto \frac{1}{1+z}(x,y)$$ as stereographic projection maps giving coordinate charts.
Then the tangent bundle of $S^2$ on the overlap $U_1 \cap U_2$ (which is everywhere except the two poles) should inherit two frames, given by $\{D_1(\sigma_1^{-1}),D_2(\sigma_1^{-1})\} $ and $ \{D_1(\sigma_2^{-1}),D_2(\sigma_2^{-1})\} $.
Now next what I have said is that, in order to find the bundle transition function which assigns to each point of $U_1 \cap U_2$ the transformation between these frames, it suffices by functoriality of the operator $D$ to find the push-forward of the coordinate transition map $\sigma_2 \circ \sigma_1^{-1} \colon \mathbb{R}^2 \to \mathbb{R}^2$ which I have calculated acts like $\sigma_2 \circ \sigma_1^{-1} \colon (a, b) \mapsto \frac{1}{a^2+b^2}(a,b) $
Then from this you can calculate $D(\sigma_2 \circ \sigma_1^{-1}) \colon (a,b) \mapsto \left[ {\begin{array}{cc} \frac{-a^2 + b^2}{(a^2 + b^2)^2} & \frac{-2ab}{(a^2+b^2)^2} \\ \frac{-2ab}{(a^2+b^2)^2} & \frac{a^2 - b^2}{(a^2 + b^2)^2} \\ \end{array} } \right] $ and I think this gives bundle transition function for the tangent bundle of the sphere in my case.
The problem is that I don't understand how from here you are supposed to reduce the structure group from $GL(2,\mathbb{R})$ to $U(1)$, and hence obtain a real valued function assigning a rotation to each point in the overlap. How do I obtain a value, at each point $(a,b)\in \mathbb{R}^2$, for the "rotation" of the above matrix?