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

## 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.