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

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Source of Name

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

Historical Note

Carl Friedrich Gauss was aware of the 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