Covariant-Derivative Commutators

A commutator is meaningful only after its connection, representation, and operator order are fixed. This card uses the site curvature convention and separates the invariant curvature action from the antisymmetric part of a covariant Hessian. The distinction is invisible for the Levi–Civita connection but essential when torsion is present.

Helpful background. Tensor Fields and Natural Operations supplies tensor slots and contractions; Affine Connections and Covariant Derivatives supplies connection coefficients and torsion.

Curvature action and the torsionful Hessian

Let $\nabla$ be an affine connection and define

$$\begin{aligned} T(X,Y)&=\nabla_XY-\nabla_YX-[X,Y],\\ \mathcal R(X,Y)A &=\nabla_X\nabla_YA-\nabla_Y\nabla_XA-\nabla_{[X,Y]}A. \end{aligned}$$

Here $A$ is a section of any tensor bundle carrying the induced connection. The operator $\mathcal R(X,Y)$ is $C^\infty$-linear in $X$, $Y$, and $A$; it is curvature and contains no derivative of $A$. In a coordinate basis, set $T^\rho{}_{\mu\nu}=\Gamma^\rho{}_{\mu\nu}-\Gamma^\rho{}_{\nu\mu}$ and

$$\mathcal R_{\mu\nu}V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma.$$

These definitions and their coordinate realization are developed in Wendl 2021–2022, §§16–21, pp. 134–183; §§25–31, pp. 212–278.

By contrast, the covariant Hessian includes its derivative-direction slot:

$$H_{\mu\nu}A =\nabla_{\partial_\mu}(\nabla_{\partial_\nu}A) -\nabla_{\nabla_{\partial_\mu}\partial_\nu}A.$$

Its antisymmetric part is

$$H_{\mu\nu}A-H_{\nu\mu}A =\mathcal R_{\mu\nu}A -T^\lambda{}_{\mu\nu}\nabla_\lambda A.$$

In particular, before imposing torsion freedom,

$$\begin{aligned} H_{\mu\nu}V^\rho-H_{\nu\mu}V^\rho &=R^\rho{}_{\sigma\mu\nu}V^\sigma -T^\lambda{}_{\mu\nu}\nabla_\lambda V^\rho,\\ H_{\mu\nu}\omega_\rho-H_{\nu\mu}\omega_\rho &=-R^\sigma{}_{\rho\mu\nu}\omega_\sigma -T^\lambda{}_{\mu\nu}\nabla_\lambda\omega_\rho. \end{aligned}$$

For an ordinary scalar $f$, $\mathcal R_{\mu\nu}f=0$, while

$$H_{\mu\nu}f-H_{\nu\mu}f =-T^\lambda{}_{\mu\nu}\partial_\lambda f.$$

Thus “covariant derivatives commute on scalars” refers to the curvature operator, or to a torsion-free Hessian—not to an arbitrary affine Hessian.

In a noncoordinate frame $e_a$, the bracket $[e_a,e_b]$ need not vanish. The bracket subtraction in $\mathcal R(e_a,e_b)$ removes this frame anholonomy, while the torsion components also contain the frame structure coefficients. A formula obtained by merely antisymmetrizing connection coefficients is therefore coordinate-basis specific. The invariant definitions above remain valid in every frame and make clear which derivative term belongs to torsion rather than curvature; this distinction is part of the affine-connection treatment in Wendl 2021–2022, §§16–21, pp. 134–183.

Levi–Civita actions on tensor slots

Now specialize to the torsion-free, metric-compatible Levi–Civita connection. The coordinate commutator and Hessian alternation coincide. With the adopted index order,

$$\begin{aligned} [\nabla_\mu,\nabla_\nu]V^\rho &=R^\rho{}_{\sigma\mu\nu}V^\sigma,\\ [\nabla_\mu,\nabla_\nu]\omega_\rho &=-R^\sigma{}_{\rho\mu\nu}\omega_\sigma. \end{aligned}$$

Every upper slot contributes a plus curvature action and every lower slot a minus action. For a tensor $A$ of type $(r,q)$,

$$\begin{aligned} ([\nabla_\mu,\nabla_\nu]A)^{\alpha_1\ldots\alpha_r}{}_{\beta_1\ldots\beta_q} ={}&\sum_{i=1}^{r}R^{\alpha_i}{}_{\lambda\mu\nu} A^{\alpha_1\ldots\lambda\ldots\alpha_r}{}_{\beta_1\ldots\beta_q}\\ &-\sum_{j=1}^{q}R^\lambda{}_{\beta_j\mu\nu} A^{\alpha_1\ldots\alpha_r}{}_{\beta_1\ldots\lambda\ldots\beta_q}. \end{aligned}$$

The sign follows immediately by commuting the scalar contraction $\omega_\rho V^\rho$: the vector and covector curvature terms must cancel. See Tong 2019, §§3.2–3.4, pp. 100–132 and Carroll 1997, §3 for the compatible curvature convention.

Metric compatibility is consequential here. It lets one raise or lower a slot before or after differentiation and obtain the same result. For a connection with nonmetricity, the induced connection still acts by a plus curvature term on each contravariant slot and a minus term on each covariant slot, but lowering a vector formula with a nonparallel metric is not a valid shortcut. The slot-by-slot expression, not index typography alone, determines the action.

Tensor densities

Declare a weight-$w$ scalar density by $\phi'(x')=\lvert\det(\partial x/\partial x')\rvert^w\phi(x)$ and use

$$\nabla_\mu\phi =\partial_\mu\phi-w\Gamma^\rho{}_{\mu\rho}\phi.$$

The invariant curvature of this one-dimensional density bundle is

$$\mathcal R_{\mu\nu}\phi =-wR^\rho{}_{\rho\mu\nu}\phi.$$

A tensor density receives this trace term in addition to the upper- and lower-slot actions above; its Hessian alternation also receives the torsion derivative term. For Levi–Civita, $\Gamma^\rho{}_{\mu\rho}=\partial_\mu\log\sqrt{\lvert g\rvert}$, so $R^\rho{}_{\rho\mu\nu}=0$. The density curvature therefore vanishes, a useful trace check consistent with the connection and volume formulas in Wendl 2021–2022, §§9–13, pp. 69–118.

The weight convention must accompany the formula. Replacing the inverse-coordinate Jacobian in the transformation law by the forward Jacobian reverses the meaning of $w$ and therefore reverses the connection trace term. At $w=0$ the density formula must reduce to the ordinary scalar result. Neither the vanishing Levi–Civita trace nor this weight-zero check permits discarding the trace for an arbitrary density connection.

Spin and gauge representations

On a spin spacetime, choose an orthonormal frame with $\{\gamma^a,\gamma^b\}=2\eta^{ab}$ and declare

$$\gamma^{ab}=\frac12[\gamma^a,\gamma^b], \qquad \nabla_\mu\psi =\partial_\mu\psi+\frac14\omega_{\mu ab}\gamma^{ab}\psi.$$

For the Levi–Civita lift and $R_{ab\mu\nu}=\eta_{ac}R^c{}_{b\mu\nu}$,

$$[\nabla_\mu,\nabla_\nu]\psi =\frac14R_{ab\mu\nu}\gamma^{ab}\psi.$$

The factor and sign belong to the declared Clifford and curvature conventions as one package; Wernli 2019, §§2.3.1–2.3.4, pp. 60–72 gives the spin-connection construction and curvature action.

For a principal connection in a local representation, suppress $\rho_*$ on the potential and choose

$$D=d+A, \qquad F=dA+A\wedge A.$$

Then an associated-bundle section $s$ obeys

$$[D_\mu,D_\nu]s=\rho_*(F_{\mu\nu})s.$$

Under $A'=g^{-1}Ag+g^{-1}dg$, curvature transforms homogeneously, $F'=g^{-1}Fg$. This is the decisive gauge check. A convention $D=d-A$, or a switch between Hermitian and anti-Hermitian generators, changes visible signs; translate the derivative, quadratic term, and commutator together. See Gaiter 2024, §5.2, pp. 27–31.

If a spinor also carries a gauge representation, the product-bundle connection is the sum of the spin and gauge connections. Because the two representation factors commute, its curvature action is the sum of the displayed spin-curvature term and $\rho_*(F_{\mu\nu})$; there is no additional mixed curvature term. This assumes that both sign packages have already been fixed as above, as in the corresponding constructions of Wernli 2019, §§2.3.1–2.3.4, pp. 60–72 and Gaiter 2024, §5.2, pp. 27–31.

Discriminating checks

Four checks localize most mistakes. First, contracting a vector with a covector must make their curvature actions cancel; failure identifies a lower-slot sign error. Second, a flat connection written in curvilinear coordinates or a rotating frame may have nonzero coefficients, but its invariant curvature and every representation curvature must still vanish. Third, the Levi–Civita density trace vanishes even when individual Christoffel symbols do not. Fourth, a gauge transformation must leave the commutator covariant by conjugation; an inhomogeneous derivative term in the transformed curvature exposes a missing $A\wedge A$ term or an inconsistent sign convention.

These are local identities. They do not imply that flat curvature eliminates global holonomy, that a parallel spinor exists merely because one algebraic contraction vanishes, or that torsion and gauge curvature are dynamically related. Each such conclusion requires additional global or physical hypotheses.

Detailed constructions and checks are in Connections, Covariant Differentiation, and Parallel Transport, Spin Connections and Spinor Curvature, and Bundle Connections and Characteristic Classes.

References

• Carroll, Sean M. Lecture Notes on General Relativity. arXiv:gr-qc/9712019v1, 1997, §3. Open PDF.
• Gaiter, Jacob. Characteristic Classes, Principal Bundles, and Curvature. University of Texas at Austin, 2024, §5.2, pp. 27–31. Open PDF.
• Tong, David. General Relativity. Cambridge Part III lecture notes, 2019, §§3.2–3.4, pp. 100–132. Official PDF.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022, §§9–13, pp. 69–118; §§16–21, pp. 134–183; §§25–31, pp. 212–278. Course PDF.
• Wernli, Konstantin. Lecture Notes on Spin Geometry. arXiv:1911.09766v1, 2019, §§2.3.1–2.3.4, pp. 60–72. Open PDF.