# Riemannian Manifold has Zero Gaussian Curvature iff Euclidean

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## Contents

## 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 $\mathcal M$ is the same as the Euclidean metric.

## Proof

## 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 ($1826$ – $1866$)