Weyl-Ricci Decomposition

The Weyl–Ricci decomposition separates Levi–Civita curvature into its completely trace-free part, its trace-free Ricci part, and its scalar part. The coefficients depend on dimension, and the usual Schouten formula is unavailable in two dimensions. This record gives the reconstruction formulas and fast trace and limiting checks; full derivations are in Curvature, Holonomy, and Geometric Invariants.

Helpful background. Tensor Fields and Natural Operations supplies traces and index operations, while Affine Connections and Covariant Derivatives fixes Levi–Civita curvature.

Schouten and Weyl pieces

Let $n$ be the dimension and use $R_{bd}=R^a{}_{bad}$. For $n\ge3$, define the Schouten tensor

$$P_{ab}=\frac{1}{n-2} \left(R_{ab}-\frac{R}{2(n-1)}g_{ab}\right).$$

Its trace is $P^a{}_a=R/[2(n-1)]$. The denominator $n-2$ is structural: in two dimensions Ricci has no independent trace-free part, so this definition cannot be recovered by taking a formal limit. For $n\ge3$, subtracting the metric–Schouten piece from Riemann uniquely removes all traces while preserving the algebraic curvature symmetries.

For symmetric two-tensors, define the Kulkarni–Nomizu product by

$$(A\mathbin{\owedge}B)_{abcd} =A_{ac}B_{bd}-A_{ad}B_{bc}-A_{bc}B_{ad}+A_{bd}B_{ac}.$$

Then

$$R_{abcd}=C_{abcd}+(g\mathbin{\owedge}P)_{abcd},$$

where $C_{abcd}$ has the Riemann symmetries and every metric trace vanishes. Expanded in Ricci and scalar curvature,

$$\begin{aligned} R_{abcd}={}&C_{abcd} +\frac{1}{n-2} \left(g_{ac}R_{bd}-g_{ad}R_{bc}-g_{bc}R_{ad}+g_{bd}R_{ac}\right)\\ &-\frac{R}{(n-1)(n-2)} \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right). \end{aligned}$$

This normalization and the trace-free character of $C$ are given in Carroll 1997, §3, Eqs. (3.138)–(3.141); after translating its curvature-operator order, an arbitrary-dimensional derivation through the Schouten tensor is Calegari 2022, §7.2, printed pp. 54–55. The formula uses the same Riemann and Ricci signs as the site; reversing the mixed Riemann convention reverses $R_{abcd}$, $R_{ab}$, $R$, and $C_{abcd}$ as one package. Tong fixes the compatible component order in Tong 2019, §§3.2–3.4, printed pp. 100–132.

Trace-free Ricci split

For $n\ge2$, set

$$S_{ab}=R_{ab}-\frac{R}{n}g_{ab}, \qquad g^{ab}S_{ab}=0.$$

Substituting $R_{ab}=S_{ab}+Rg_{ab}/n$ into the Schouten formula gives, for $n\ge3$,

$$\begin{aligned} R_{abcd}={}&C_{abcd} +\frac{1}{n-2} \left(g_{ac}S_{bd}-g_{ad}S_{bc}-g_{bc}S_{ad}+g_{bd}S_{ac}\right)\\ &+\frac{R}{n(n-1)} \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right). \end{aligned}$$

This is a pointwise algebraic decomposition for a pseudo-Riemannian metric of any signature. It does not require a field equation, and it does not by itself assign radiative or matter content to a summand. The Levi–Civita symmetries on which the split rests, including pair exchange and the algebraic Bianchi identity, follow from the curvature construction in Wendl 2021–2022, §§25–31, printed pp. 212–278.

The two displayed decompositions are equivalent lookup forms. Use the Schouten form when conformal geometry or a compact reconstruction is central; use the $S_{ab}$ form when the independent component split must remain visible. In either form, the three summands are determined by traces of the same Riemann tensor. Choosing $C$, $S$, and $R$ independently without checking their curvature symmetries does not by itself produce the curvature of a metric.

Dimensions and component counts

The algebraically independent component counts are

$$\begin{aligned} N_R&=\frac{n^2(n^2-1)}{12},\\ N_S&=\frac{n(n+1)}{2}-1,\\ N_C&=\frac{n(n+1)(n+2)(n-3)}{12} \quad (n\ge3). \end{aligned}$$

The remaining one component is the scalar curvature. The formulas give the required low-dimensional cases:

• In $n=2$, the Schouten expression is singular and must not be used. Instead $S_{ab}=0$, $C_{abcd}=0$, and

  $$R_{abcd}=\frac{R}{2}(g_{ac}g_{bd}-g_{ad}g_{bc}).$$

• In $n=3$, $N_C=0$. Ricci’s five trace-free components plus its scalar determine all six Riemann components.

• In $n=4$, the split is $20=10+9+1$: ten Weyl, nine trace-free Ricci, and one scalar component.

In two dimensions the identity $R_{ab}=Rg_{ab}/2$ explains why $S$ vanishes. In three dimensions the vanishing of $C$ is algebraic, not an extra curvature condition. Only from four dimensions onward can a trace-free Weyl tensor remain after every Ricci contraction has vanished. The counts depend only on tensor symmetries, not on metric signature. Carroll’s Riemann count provides the unreduced total Carroll 1997, §3, Eq. (3.120).

Trace and curvature checks

Contracting the expanded formula with $g^{ac}$ must recover $R_{bd}$ exactly: the Weyl term vanishes, the $S$ term returns $S_{bd}$, and the scalar term returns $Rg_{bd}/n$. This is the quickest coefficient check.

More explicitly, trace freedom of $S$ gives

$$\begin{aligned} g^{ac}(g\mathbin{\owedge}S)_{abcd}&=(n-2)S_{bd},\\ g^{ac}(g_{ac}g_{bd}-g_{ad}g_{bc})&=(n-1)g_{bd}. \end{aligned}$$

These two contractions separately test the coefficients $1/(n-2)$ and $1/[n(n-1)]$. A wrong sign or denominator cannot be hidden by the Weyl term because its contraction is zero.

For constant sectional curvature,

$$R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc}), \qquad R_{ab}=(n-1)Kg_{ab}, \qquad R=n(n-1)K.$$

Hence $S_{ab}=0$ and $C_{abcd}=0$, and the scalar term reconstructs Riemann. Conversely, Ricci flatness gives $P_{ab}=0$ and therefore $R_{abcd}=C_{abcd}$. In dimensions four and above this can still describe curved geometry; in three dimensions, where $C$ vanishes identically, Ricci flatness implies full flatness. These tests distinguish “trace free” from “zero curvature.”

The constant-curvature check also fixes the scalar coefficient without referring to a special coordinate system: substitution must return exactly $K$ times the metric bivector. The Ricci-flat check probes the opposite extreme, where all trace terms disappear. Passing both checks is stronger than confirming the component count, because the count cannot detect an overall sign or a misplaced factor of two.

References

• Calegari, Danny. Introduction to Differential Geometry. Preliminary University of Chicago lecture notes, 2022. Open PDF.
• Carroll, Sean M. Lecture Notes on General Relativity. arXiv:gr-qc/9712019v1, 1997. Open PDF.
• Tong, David. General Relativity. University of Cambridge Part III Mathematical Tripos lecture notes, 2019. Official PDF.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022. Open PDF.