nLab
Riemann curvature

Definition

For $(X,g)$ a Riemannian manifold let ${\nabla }_g$ be the corresponding Levi-Civita connection. The Riemann curvature $R_g$ of $(X,g)$ is the curvature $F_{{\nabla }_g}$ of ${\nabla }_g$:

$$R_g:=F_{{\nabla }_g}.$$R_g := F_{\nabla_g} \,.

Properties

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Related concepts

• torsion of a metric connection

• Poincare Lie algebra

• Ricci tensor?, scalar curvature

• gravity