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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 $\chi(S)> 2$?


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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).

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Reading the definition of $\chi(-)$ will also help you out. – Chris Gerig Jun 28 '12 at 1:11
I agree with you, but I am some confused because in the textbooks this topic is not covered. One can read in Diff. Geometry books that the word "connected" does not appear in Theromes like I've quoted. They say: every compact oriented surface is diffeomorphic to a g-tori. But I can see that the union of two disjoint spheres is not a g-tori, any g. – Singular Guy Jun 28 '12 at 1:38
This topic isn't covered because you can reduce everything to connected objects, and then take disjoint unions, so the matter is trivial. – Chris Gerig Jun 28 '12 at 2:58

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