I guess I have to use that it is a closed surface (manifold) and therefore
its boundary is empty, so then I gotta use Gauss-Bonnet easily (but I just cannot think
of how to use it.)
i.e. $$\chi(\textit{Klein Bottle})=\frac{1}{2\pi}\int\int_{\textit{Klein Bottle}}K\,d\sigma$$
I was thinking on using that a Klein bottle is homeomorphic to 2 "glued" Möbius
strips, but how from then…?
P.S. this is from my first course on Differential Geometry, so talkings on higher-level courses wouldn't help me at all
Best Answer
This is one triangulation of the Klein bottle, with vertex $V$ and edges $a, b, c$:
Can you calculate the Euler characteristic from this?