Gauss-Bonnet Theorem

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.

Also see

• Chern-Gauss-Bonnet Theorem
• Riemann-Roch Theorem
• Atiyah-Singer Index Theorem

Sources

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