Riemannian Manifold has Zero Gaussian Curvature iff Euclidean

Theorem

Let $M$ be a Riemannian manifold of dimension $2$.

The Gaussian curvature on $M$ is zero if and only if the Riemannian metric on $M$ is the same as the Euclidean metric.

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.

Historical Note

Bernhard Riemann demonstrated that a Riemannian Manifold has Zero Gaussian Curvature iff Euclidean in a posthumous paper on heat conduction.

Thus the Riemann-Christoffel tensor measures how much a Riemannian manifold deviates from a Euclidean space.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)