In general, I have seen that a consequence of the Gauss-Bonnet Theorem is the following:
Theorem. If S is a CONNECTED smooth compact oriented surface in $R^3$, then S is diffeomorphic to a $g$-tori for some $g=0,1,2,...$, and the characteristic of S is $\chi(S)=2(1-g)$.
My question is: what happens when we have a NON CONNECTED surface S?
For example, if $S=S_1\cup S_2$ for connected disjoint surfaces $S_1,S_2$, can we say that $\chi(S)=\chi(S_1)+\chi(S_2)$?
Can we obtain thus, surfaces (non-connected of course) with $\chi(S)> 2$?