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 $\nabla_g$:

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

Properties

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Related concepts

curvature in Riemannian geometry
Riemann curvature
Ricci curvature
scalar curvature
sectional curvature
p-curvature

• torsion of a metric connection

• Weyl tensor

• Poincare Lie algebra

• Ricci tensor

• gravity

• higher order curvature corrections