Gauss-Bonnet Theorem
Jump to navigation
Jump to search
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 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