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Let $L$ be a closed orientable Lagrangian embedding in $\mathbb C^n$. Then $\chi(L) = 0$, where $\chi(L)$ denotes the Euler characteristic of $L$.

This fact is more or less stated in section 3.2 of


and is attributed to Whitney. Does anyone know where can I find the original proof by Whitney? Thanks.

Remark: The proof is quite simple, This is because the self-intersection number of any submanifold of $\mathbb C^n$ is clearly zero, but it is also equal to the Euler characteristic of the normal bundle, which for Lagrangian submanifolds is isomorphic to the cotangent bundle. (Quoted from

http://www.math.uni-hamburg.de/home/latschev/forschung/fukaya.pdf )

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