# Gauss-Bonnet Theorem

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## Theorem

Let $M$ be a compact $2$-dimensional Riemannian manifold with boundary $\partial M$.

Let $\kappa$ be the Gaussian curvature of $M$.

Let $k_g$ be the geodesic curvature of $\partial M$.

Then:

- $\ds \int_M \kappa \rd A + \int_{\partial M} k_g \rd s = 2 \pi \map \chi M$

where:

- $\d A$ is the element of area of the surface
- $\d s$ is the line element along $\partial M$
- $\map \chi M$ is the Euler characteristic of $M$.

## Proof

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## Source of Name

This entry was named for Carl Friedrich Gauss and Pierre Ossian Bonnet.

## Historical Note

Carl Friedrich Gauss was aware of the Gauss-Bonnet Theorem but never published it.

It was Pierre Ossian Bonnet who first published, in $1848$, a special case.

## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 1$: What Is Curvature? Surfaces in Space