Riemann Symmetries and Bianchi Identities

For Levi–Civita curvature, the pointwise symmetry package and the algebraic and differential Bianchi identities control which components are independent and how their derivatives are related. This record fixes the site convention, gives the useful contractions, and supplies dimensional checks. Full derivations and related curvature constructions are in Curvature, Holonomy, and Geometric Invariants.

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

Levi–Civita algebraic symmetries

Use

$$[\nabla_c,\nabla_d]V^a=R^a{}_{bcd}V^b, \qquad R_{bd}=R^a{}_{bad},$$

and lower the first curvature index with the metric. For the torsion-free, metric-compatible connection,

$$\begin{aligned} R_{abcd}&=-R_{bacd}, & R_{abcd}&=-R_{abdc},\\ R_{abcd}&=R_{cdab}, & R_{a[bcd]}&=0. \end{aligned}$$

Brackets have unit weight. Thus the last equation is equivalently

$$R_{abcd}+R_{acdb}+R_{adbc}=0.$$

The first two relations make curvature antisymmetric within each index pair, pair exchange makes it a symmetric bilinear form on $\Lambda^2T_pM$, and the cyclic relation is the algebraic, or first, Bianchi identity. The familiar fully lowered package uses both torsion freedom and metric compatibility; a general affine connection with torsion or nonmetricity obeys modified identities. The hypotheses and invariant curvature construction are developed in Wendl 2021–2022, §§25–31, printed pp. 212–278.

These four equations are a generating set, not four unrelated tests. For example, antisymmetry in the first pair follows only after the curvature endomorphism has been lowered with a compatible metric; it is not an automatic symmetry of $R^a{}_{bcd}$ for an arbitrary connection. Pair exchange likewise uses the Levi–Civita package. When permuting indices, first reduce within each pair, then exchange whole pairs, and only then apply the cyclic sum. That order avoids treating a Bianchi relation as an extra sign symmetry. After translating its curvature-operator order to the convention above, a concise proof of the full algebraic package is Calegari 2022, Proposition 7.1.3, printed pp. 53–54.

Differential and contracted identities

The differential, or second, Bianchi identity is

$$\nabla_eR^a{}_{bcd} +\nabla_cR^a{}_{bde} +\nabla_dR^a{}_{bec}=0,$$

or, with the excluded index shown explicitly,

$$\nabla_{[e}R^a{}_{\lvert b\rvert cd]}=0.$$

The antisymmetrization runs over $e,c,d$; the vertical bars exclude $b$. Expanding the three cyclic terms before contracting is the safest way to preserve the derivative order and the Ricci sign. This is a first-order compatibility relation, not a field equation. With the algebraic symmetries and $\nabla g=0$, one contraction gives

$$\nabla^aR_{abcd}=\nabla_cR_{db}-\nabla_dR_{cb},$$

and the next gives

$$\nabla^aR_{ab}=\frac12\nabla_bR, \qquad \nabla^aG_{ab}=0, \qquad G_{ab}=R_{ab}-\frac12Rg_{ab}.$$

The last equality follows geometrically and does not assume the Einstein equation. The uncontracted identity and this contraction chain appear, in the same convention, in Tong 2019, §3.4, printed pp. 130–132 and Carroll 1997, §3, Eqs. (3.128) and (3.133)–(3.137).

Contraction discards information. The divergence-free Einstein tensor is a consequence of the differential identity, but it cannot reconstruct every component of the uncontracted cyclic relation. It is therefore a useful check on a curvature calculation, not a replacement for testing the full second Bianchi identity when derivatives of Riemann are available. Neither identity asserts that Ricci or the scalar curvature is separately constant.

Component count and dimensional checks

Let $N=n(n-1)/2=\dim\Lambda^2T_pM$. Pair exchange leaves $N(N+1)/2$ components, and the algebraic Bianchi identity removes $\binom n4$. Therefore

$$N_R=\frac{N(N+1)}{2}-\binom n4 =\frac{n^2(n^2-1)}{12}.$$

This standard count is also recorded in Carroll 1997, §3, Eq. (3.120). It yields three immediate reconstruction checks:

• In two dimensions, $N_R=1$ and

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

• In three dimensions, $N_R=6$ and Ricci determines Riemann:

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

• In four dimensions, $N_R=20$: ten Weyl components, nine trace-free Ricci components, and one scalar component.

The count has a useful diagnostic meaning. Before the cyclic identity, curvature is a symmetric $N\times N$ matrix on two-forms. The removed $\binom n4$ combinations are precisely its totally antisymmetric four-index part. Thus there is no subtraction in dimensions two or three, whereas in four dimensions the single four-form component reduces the symmetric $6\times6$ count from twenty-one to twenty.

The displayed two- and three-dimensional formulas also test normalization: contracting the first and third indices must return the declared Ricci tensor, and a second contraction must return $R$. These counts are signature independent. As a sign check, constant sectional curvature $K$ has

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

A round sphere therefore has positive scalar curvature in this convention. Setting $K=0$ gives the flat check: every curvature component and every term in both Bianchi identities vanishes.

Translating curvature conventions

For a source using the constant switches

$$g'_{ab}=\sigma g_{ab}, \qquad R'^a{}_{bcd}=\kappa R^a{}_{bcd}, \qquad \sigma,\kappa\in\{-1,+1\},$$

the induced objects are

$$R'_{abcd}=\sigma\kappa R_{abcd}, \qquad R'_{ab}=\kappa R_{ab}, \qquad R'=\frac{\kappa}{\sigma}R, \qquad G'_{ab}=\kappa G_{ab}.$$

Every algebraic and Bianchi identity keeps its homogeneous form, but all contractions must be translated together. Changing only the displayed Riemann sign while retaining an unmodified Ricci or scalar formula produces an inconsistent package. Tong’s explicit component order and weighted brackets provide a compatible comparison point Tong 2019, §§2.3 and 3.2, printed pp. 67–75 and 100–117.

The metric switch must be constant; otherwise its Levi–Civita connection changes and this two-sign table no longer applies. A mere relabeling of indices or change of coordinates is not a curvature-sign switch. When a source declares only the opposite commutator order, set $\kappa=-1$ and leave $\sigma=+1$; when it also adopts the globally negated metric, apply both switches before comparing lowered components or scalar curvature.

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.