List of Formulas in Riemannian Geometry - Curvature Tensors - Ricci and Scalar Curvatures

Ricci and Scalar Curvatures

Ricci and scalar curvatures are contractions of the Riemann tensor. They simplify the Riemann tensor, but contain less information.

The Ricci curvature tensor is essentially the unique nontrivial way of contracting the Riemann tensor:

R_{ij}=R^ell{}_{iell j}=g^{ell m}R_{iell jm}=g^{ell m}R_{ell imj}
=frac{partialGamma^ell{}_{ij}}{partial x^ell} - frac{partialGamma^ell{}_{iell}}{partial x^j} + Gamma^ell{}_{ij} Gamma^m{}_{ell m} - Gamma^m{}_{iell}Gamma^ell_{jm}.

The Ricci tensor is symmetric.

By the contracting relations on the Christoffel symbols, we have

R_{ik}=frac{partialGamma^ell{}_{ik}}{partial x^ell} - Gamma^m{}_{iell}Gamma^ell{}_{km} - nabla_kleft(frac{partial}{partial x^i}left(logsqrt{|g|}right)right).

The scalar curvature is the trace of the Ricci curvature,

R=g^{ij}R_{ij}=g^{ij}g^{ell m}R_{iell jm}

The "gradient" of the scalar curvature follows from the Bianchi identity (proof):

that is,

Read more about this topic:  List Of Formulas In Riemannian Geometry, Curvature Tensors