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 :

$\displaystyle \int_M \kappa \, \mathrm d A + \int_{\partial M} k_g \, \mathrm d s = 2 \pi \chi\left({M}\right)$

where:

$\mathrm d A$ is the element of area of the surface
$\mathrm d s$ is the line element along $\partial M$
$\chi\left({M}\right)$ is the Euler characteristic of $M$.


Proof


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.