# nLab Riemann curvature

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Definition

For $\left(X,g\right)$ a Riemannian manifold let ${\nabla }_{g}$ be the corresponding Levi-Civita connection. The Riemann curvature ${R}_{g}$ of $\left(X,g\right)$ is the curvature ${F}_{{\nabla }_{g}}$ of ${\nabla }_{g}$:

${R}_{g}:={F}_{{\nabla }_{g}}\phantom{\rule{thinmathspace}{0ex}}.$R_g := F_{\nabla_g} \,.

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