The Bianchi Identities

The Bianchi identities are the compatibility relations obeyed by Levi–Civita curvature. The algebraic identity constrains the tensor at one point; the differential identity constrains how curvature can vary. Their contracted form forces the Einstein tensor to be divergence free, but the identities themselves are geometry, not gravitational dynamics.

Required background. Affine Connections and Covariant Derivatives supplies the general tensor-derivative framework; The Levi-Civita Connection supplies torsion freedom and metric compatibility; The Riemann Curvature Tensor supplies the curvature commutator and sign convention.

Helpful background. Calculus and Linear Algebra for Relativity supplies index notation and linear algebra for the contractions; Exterior Algebra and Differential Forms supplies alternating-form and wedge-product algebra used to organize cyclic antisymmetrization.

The algebraic identity

For a torsion-free connection, apply the Jacobi identity for vector-field brackets after replacing each bracket by a difference of covariant derivatives. The remaining second-derivative terms assemble into curvature and give

$$R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0.$$

For the Levi–Civita connection, lowering an index yields the equivalent component form

$$R_{a[bcd]}=0,$$

where antisymmetrization has unit weight. This is the first or algebraic Bianchi identity. Together with pair antisymmetry and pair exchange, it places $R$ in the space of algebraic curvature tensors. A direct proof from torsion freedom is given in Calegari 2022, Proposition 7.1.3, pp. 53–54.

The hypothesis matters. For a connection with torsion $T$, the cyclic curvature sum is balanced by $\nabla T$ and quadratic torsion terms; setting it to zero without those terms is generally false. Metric compatibility is additionally needed for the familiar fully lowered symmetries.

The differential identity

Choose normal coordinates at a point. Connection coefficients vanish there, although their derivatives need not. Substitute the coordinate expression for $R^a{}_{bcd}$ into

$$\nabla_eR^a{}_{bcd} +\nabla_cR^a{}_{bde} +\nabla_dR^a{}_{bec}.$$

At the chosen point, derivatives of the quadratic $\Gamma\Gamma$ terms vanish because each contains an undifferentiated $\Gamma$. The surviving second partial derivatives of $\Gamma$ cancel pairwise because partial derivatives commute. Since the result is tensorial, it vanishes in every chart:

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

This is the second or differential Bianchi identity. In an orthonormal frame, write the curvature two-form as

$$\Omega^a{}_b=d\omega^a{}_b+\omega^a{}_c\wedge\omega^c{}_b.$$

Its exterior covariant derivative satisfies

$$D\Omega^a{}_b =d\Omega^a{}_b+\omega^a{}_c\wedge\Omega^c{}_b -\Omega^a{}_c\wedge\omega^c{}_b=0.$$

The equality follows from $d^2=0$ and associativity of the matrix-valued wedge product. It holds for every bundle connection, including connections with torsion; translating it into covariant derivatives of tangent-bundle components then requires the corresponding torsion terms. The component derivation above and its contractions appear in Tong 2019, §3.4.1, pp. 130–132, PDF; §3.4.2, pp. 132–136 introduces the connection and curvature forms used here.

Contraction and the Einstein combination

Start with $\nabla_{[e}R_{ab]cd}=0$, expand the three terms, and contract one derivative index with one curvature index. Using the algebraic symmetries gives

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

For clarity, contract the first identity with $g^{bd}$. Its left side is $\nabla^aR_{ac}$, whereas its right side is $\nabla_cR-\nabla^bR_{cb}$. Moving the last term to the left yields

$$\nabla^aR_{ab}=\frac12\nabla_bR.$$

Because $\nabla g=0$,

$$\nabla^a\left(R_{ab}-\frac12Rg_{ab}\right)=0.$$

This contracted identity is exact in every dimension for a Levi–Civita connection. It does not say that $R_{ab}$ or $R$ is separately constant, and it does not invoke the Einstein equation.

Constant-curvature check and limitations

For

$$R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc})$$

with constant $K$, the differential identity holds immediately because $\nabla g=0$ and $\nabla K=0$. The contractions are $R_{ab}=(n-1)Kg_{ab}$ and $R=n(n-1)K$, so both sides of $\nabla^aR_{ab}=\tfrac12\nabla_bR$ vanish. More strongly, if the displayed isotropic form holds with $K=K(x)$ and $n>2$, the contracted identity forces $\nabla_bK=0$. Thus a connected manifold of pointwise constant sectional curvature has one constant $K$ in dimensions above two.

The Bianchi identities restrict allowed curvature data; they do not guarantee that an arbitrary algebraic curvature field comes from a global metric, nor do they classify that metric. Boundary conditions, topology, and field equations are additional structures.

There is also a useful hierarchy among the identities. The algebraic identity is pointwise and contains no derivative of curvature. It reduces the allowed algebraic components before any differential equation is considered. The second identity is a first-order differential compatibility condition. Its contraction discards information: $\nabla^aG_{ab}=0$ is a consequence of the full identity, but it cannot in general be expanded back into all components of $\nabla_{[e}R^a{}_{\lvert b\rvert cd]}=0$. Proving only the contracted statement is therefore not a proof of the differential Bianchi identity.

In form notation the identity $D\Omega=0$ resembles $dF=0$ for an abelian field strength, but the covariant derivative is essential. Curvature is endomorphism valued, and the connection terms implement its change under a frame rotation. Writing merely $d\Omega=0$ is generally false. This distinction becomes visible whenever curvature matrices at different points or in different frame directions fail to commute.

For practical component work, one can verify the identity at a point in normal coordinates. This eliminates undifferentiated Christoffel symbols but not curvature derivatives. The remaining cyclic cancellation checks the ordering of derivative indices. Afterward, contracting with the metric must use $\nabla g=0$; moving the metric through an ordinary partial derivative in a generic chart would omit connection terms. These two checkpoints distinguish an invariant proof from a coordinate coincidence.

Exercises

Let $R_{ab}=\lambda(x)g_{ab}$ on a connected $n$-manifold with $n>2$. Use the contracted Bianchi identity to show that $\lambda$ is constant.

Solution

The scalar curvature is $R=n\lambda$. Metric compatibility gives

$$\nabla^aR_{ab}=\nabla_b\lambda, \qquad \frac12\nabla_bR=\frac n2\nabla_b\lambda.$$

Their equality implies $(1-n/2)\nabla_b\lambda=0$. Since $n>2$, $\nabla_b\lambda=0$, and connectedness makes $\lambda$ constant.

References

• Danny Calegari, Introduction to Differential Geometry, preliminary version, May 26, 2022, University of Chicago. PDF.
• David Tong, General Relativity, University of Cambridge Part III Mathematical Tripos lecture notes, 2019. PDF.