Relation between characteristic classes and fundamental forms of a surface

Question

I have an embedded oriented surface $M\subset\mathbb{R}^3$. Is there a way to express it's Euler/Pontryagin classes using first and second fundamental forms of $M$? I would really appreciate a reference.

Thanks.

Answer 1

Suppose that $S$ is a closed oriented Riemannian surface. Then the Gauss-Bonnet formula reads $$ \int_S K dA =2\pi \chi(S). $$ Thus, one recovers the Euler characteristic from the 1st and 2nd fundamental form (in the case when $S$ is embedded isometrically in the Euclidean 3-space $E^3$). If you want to recover the Euler class $e_S\in H^2(S)$, this can be done as well: $$ e_S= \left[\frac{1}{2\pi} KdA\right]. $$

For noncompact surfaces, I do not see any way to recover the Euler class (as an element of $H^2_c(S)$), since you can have an open surface realized as an open subset in a Euclidean plane sitting in $E^3$. In this case, the 2nd fundamental form is zero and, thus, all that you know is that you have a planar surface. Such a surface can have an arbitrary Euler characteristic $\le 1$.