I was tryng to undertand if the Gauss-Bonnet theorem works for the real projective plane. I know that it does work for non orientable surfaces as long as they can be embedded in $\mathbb{R}^3$ (e.g. the Mobius strip) but the real projective plane is not.
So the real question is: does it work with surfaces embedded in $\mathbb{R}^4$?
By studying the problem I thing I have found a way to embed a sphere $S^2$ in $\mathbb{R}^4$ and twist it in such a way the total curvature become greater then $2\pi\chi$ but my example is quite involved and now I am not sure anymore if it is correct.
I have found very little on this on the internet. Please help!
The Gauss Bonnet Theorem works even for surfaces not embedded anywhere!
The theorem is about surfaces $S$ that are endowed with a Riemann metric $$ds^2=\sum_{i,k}g_{ik}(u_1,u_2)\, du_i\, du_k\ . $$ These surfaces can be closed or have boundary curves.
Many surfaces we meet are a priori embedded in some ${\mathbb R}^n$, $n\geq2$, and they obtain their $ds^2$ from this embedding (the $E$, $F$, $G$ in the "old" books). This is the case in your examples. Another example would be the hyperbolic plane, realized as half plane $H:=\bigl\{z=x+iy\in{\mathbb C}\bigm| y>0\bigr\}$. Here the $ds^2$ is given by $$ds^2={1\over y^2}(dx^2+dy^2)\ .$$ The $dx^2+dy^2$ is just the euclidean $ds^2$ in the plane, but we have this additional scalar factor ${1\over y^2}$.
The Gauss Bonnet Theorem has various proofs. The geodesic curvature $\kappa_g$ can be measured "within" the surface, and no embedding or surface normal is needed. See, e.g., Wilhelm Klingenberg, A course in differential geometry, Springer Graduate Text (1978), pp. 138–141.