Riemann curvature (changes) in nLab

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Definition

For (X,g)(X,g) a Riemannian manifold let ∇ g\nabla_{g} be the corresponding Levi-Civita connection. The Riemann curvature R gR_g of (X,g)(X,g) is the curvature F ∇ gF_{\nabla_g} of ∇ g\nabla_g:

R g:=F ∇ g. R_g := F_{\nabla_g} \,.

Properties

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• torsion of a metric connection

• Weyl tensor

• Poincare Lie algebra

• Ricci tensor, scalar curvature

• gravity

• higher order curvature corrections

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