I've been on a topology kick, and I recently read on Wolfram that
- The only compact closed surfaces with Euler characteristic 0 are the Klein bottle and torus.
However, the reference provided stated it without proof. What makes these two surfaces in $\mathbb{R}^3$ the only two surfaces satisfying these given criteria? I suppose I'm not looking for a whole proof (although I would love to see one), only a sketch of one and references to further reading on the subject.
Best Answer
If you know the classification of surfaces and the formula $\chi(M \# N) = \chi(M) + \chi(N) - 2$ (valid for arbitrary connected sum of surfaces) then this follows immediately, as $\Sigma_g = \#^g T^2$ has $\chi(\Sigma_g) = 2 - 2g$, and $N_g = \#^g \Bbb{RP}^2$ has $\chi(N_g) = 2 - g$. Because all closed surfaces are diffeomorphic to exactly one of these, and $N_2 = K$, one concludes.
In fact it is not hard to prove the entire classification theorem if you know it for things with Euler characteristic zero by an inductive argument that shows you can safely add/remove handles. (Normally one might reduce it to classifying simply connected surfaces but there's no real additional difficulty caused by stopping at genus 1)