Chern-Gauss-Bonnet proof

Question

I'm reading through Chern original proof as stated in Yin Li of the Chern-Gauss-Bonnet theorem, and I think I got almost everything except for the last step (Page 17).

I fail to see how passing to the limit allows to integrate over $SM_x$ by Poincaré-Hopf theorem, as stated at Page 7.

The same step in Chern's original paper (Page 6-7) is just as unclear to me.

In what form is the Poincaré-Hopf theorem being used?

Answer 1

Poincaré-Hopf says that $\chi(M)$ is the sum of the indices of a vector field (with isolated zeroes). The index at a zero $x$ is a local degree (say, of the map from the boundary of a small ball centered at $x$ to the unit sphere in $T_xM$). Then we compute the degree (at each zero $x$) by pulling back the volume form of the sphere and integrating. Now sum.