Looking for Gauss-Bonnet counterexample

[Goklayeh]

Hello everybody! I was looking for a counterexample to Gauss-Bonnet Theorem, that is, a region $$R \subset \Sigma$$ (with $$\Sigma \subset \mathbb{R}^3$$ surface) such that $$\partial R$$ isn't union of closed piecewise regular curves and for which the Gauss Bonnet Theorem doesn't holds, i.e.
$$\iint_R{K \mathrm{d}\sigma} \ne 2\pi\chi(R) - \sum_{i=1}^{p}{\theta_i} - \int_{\partial R}{k_g(s) \mathrm{d}s}$$
Of course, if my question has sense, a definition of $$\chi(R)$$ should be provided for non-regular regions too... Thank you in advance!

 

[lavinia]

Hello everybody! I was looking for a counterexample to Gauss-Bonnet Theorem, that is, a region $$R \subset \Sigma$$ (with $$\Sigma \subset \mathbb{R}^3$$ surface) such that $$\partial R$$ isn't union of closed piecewise regular curves and for which the Gauss Bonnet Theorem doesn't holds, i.e.
$$\iint_R{K \mathrm{d}\sigma} \ne 2\pi\chi(R) - \sum_{i=1}^{p}{\theta_i} - \int_{\partial R}{k_g(s) \mathrm{d}s}$$
Of course, if my question has sense, a definition of $$\chi(R)$$ should be provided for non-regular regions too... Thank you in advance!

I am confused. If the boundary is not a union of finitely many piecewise regular curves then there is no notion of geodesic curvature and no notion of exterior angle.