# Question on the Gauss-Bonnet theorem

According to Wikipedia the global Gauss-Bonnet theorem concludes,

$$\int_{M}KdA + \int_{\partial M}k_{g}ds=2\pi\chi(M)$$.

The lecture notes I am using however does not have the integral with the geodesic curvature but they seem to have the same hypotheses. Here is the proofs from the notes;

Either Wikipedia is wrong or the notes must be wrong at some point. In particular if a geodesic triangulation always is possible for any compact surface then there would never be a geodesic curvature integral imo.

The notes can be found at http://www.matematik.lu.se/matematiklu/personal/sigma/Gauss.pdf

Both are correct but your lecture notes cover only a subset of the cases the wikipedia formula covers. Namely you lecture notes assume that $$M$$ is a surface without boundary, whereas wikipedia allows surfaces with smooth boundary.
• To me it looks like the boundary belongs to $M$ but vanish since it is parametrised by a geodesic, somthing I find hard to belive work unless the boundary is a geodesic to begin with. If it is compact and in $\mathbb{R}^n$ it is closed which means that the boundary belongs to the set. Compactness is needed to have finite cover. – Lobsided Aug 13 at 14:28