Gauss–Bonnet theorem

The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). It is named after Carl Friedrich Gauss who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published a special case in 1848.

Suppose ${\displaystyle M}$ is a compact two-dimensional Riemannian manifold with boundary ${\displaystyle \partial M}$. Let ${\displaystyle K}$ be the Gaussian curvature of ${\displaystyle M}$, and let ${\displaystyle k_{g}}$ be the geodesic curvature of ${\displaystyle \partial M}$. Then

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