Let $S$ be a compact submanifold of $X$ smooth manifold. I know that $T\_X|\_S=T\_S\oplus N\_{S/X}$ where $N\_{S/X}$ is the normal bundle. I have read that the euler class $e(N\_{S/X})$ corresponds (via integration over S, i suppose) to the self intersection number $S\cdot S$. I've thought about it, but i don't know how to prove it, also i can't find the proof in any book. Do you know something about it?



euler class of the normal bundle and self intersection number
I've just taught this in my graduate class. Check these [notes on intersection theory.](http://www.nd.edu/~lnicolae/TopicsFall2011-Intersect.pdf)


The result you want is contained in Thm. 4.7. Again, you need $2\dim S=\dim X$.


