# Non connected surfaces and Gauss Bonnet Theorem

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 $\chi(S)> 2$?

Thanks

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If $S$ is not connected, then it will be a disjoint union of connected surfaces, each one a $g_i$-tori by your quoted theorem (where $g_i$ can vary).
Reading the definition of $\chi(-)$ will also help you out. –  Chris Gerig Jun 28 '12 at 1:11