Riemann-Christoffel Tensor in Two Dimensions is Gaussian Curvature

Theorem

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

Then the Riemann-Christoffel tensor on $M$ reduces to the Gaussian curvature on $M$.

Proof

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Historical Note

Bernhard Riemann demonstrated that the Riemann-Christoffel tensor in a Riemannian manifold of $2$ dimensions is the same thing as the Gaussian curvature in a posthumous paper on heat conduction.

Sources

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