Ricci Decomposition - Mathematical Definition

Mathematical Definition

Mathematically, the Ricci decomposition is the decomposition of the space of all tensors having the symmetries of the Riemann tensor into its irreducible representations for the action of the orthogonal group (Besse 1987, Chapter 1, §G). Let V be an n-dimensional vector space, equipped with a metric tensor (of possibly mixed signature). Here V is modeled on the cotangent space at a point, so that a curvature tensor R (with all indices lowered) is an element of the tensor product V⊗V⊗V⊗V. The curvature tensor is skew symmetric in its first and last two entries:

and obeys the interchange symmetry

for all x,y,z,w ∈ V∗. As a result R is an element of the subspace S2Λ2V, the second symmetric power of the second exterior power of V. A curvature tensor must also satisfy the Bianchi identity, meaning that it is in the kernel of the linear map

The space RV = ker b in S2Λ2V is the space of algebraic curvature tensors. The Ricci decomposition is the decomposition of this space into irreducible factors. The Ricci contraction mapping

is given by

This associates a symmetric 2-form to an algebraic curvature tensor. Conversely, given a pair of symmetric 2-forms h and k, the Kulkarni–Nomizu product of h and k

produces an algebraic curvature tensor.

If n > 4, then there is an orthogonal decomposition into (unique) irreducible subspaces

RV = SV ⊕ EV ⊕ CV

where

, where S2
0V is the space of trace-free symmetric 2-forms

The parts S, E, and C of the Ricci decomposition of a given Riemann tensor R are the orthogonal projections of R onto these invariant factors. In particular,

is an orthogonal decomposition in the sense that

This decomposition expresses the space of tensors with Riemann symmetries as a direct sum of the scalar submodule, the Ricci submodule, and Weyl submodule, respectively. Each of these modules is an irreducible representation for the orthogonal group (Singer & Thorpe 1968), and thus the Ricci decomposition is a special case of the splitting of a module for a semisimple Lie group into its irreducible factors. In dimension 4, the Weyl module decomposes further into a pair of irreducible factors for the special orthogonal group: the self-dual and antiself-dual parts W+ and W−.