Ricci decomposition

In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties. This decomposition is of fundamental importance in Riemannian- and pseudo-Riemannian geometry.

The decomposition is

The three pieces are:

Each piece possesses all the algebraic symmetries of the Riemann tensor itself, but has additional properties.

The decomposition can have different signs, depending on the Ricci curvature convention, and only makes sense if the dimension satisfies ${\displaystyle n>2}$.

The scalar part

is built using the scalar curvature ${\displaystyle R={R^{m}}_{m}}$, where ${\displaystyle R_{ab}={R^{c}}_{acb}}$ is the Ricci curvature, and a tensor constructed algebraically from the metric tensor ${\displaystyle g_{ab}}$,

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