Frames, Coframes, and Anholonomy

A frame is a smoothly varying basis of tangent vectors on an open set, and its dual coframe is a basis of one-forms. Unlike a coordinate basis, a perfectly smooth frame need not arise from coordinate lines. Its failure to do so is measured by commutators—or equivalently by exterior derivatives of the coframe—and is called anholonomy.

Required background. Pseudo-Riemannian Metrics and Signature supplies Lorentzian inner products, signature, and index operations; Affine Connections and Covariant Derivatives supplies the general connection and covariant-derivative framework. This page develops the moving-frame components used below.

Helpful background. Calculus and Linear Algebra for Relativity supplies changes of basis and bilinear forms; Exterior Algebra and Differential Forms supplies one-forms and wedge products, while The Exterior Derivative and the Poincare Lemma supplies $d$. Killing Fields and Isometry Groups motivates symmetry-adapted frames.

Local frames and dual coframes

On an $n$-manifold, a local frame $\{e_a\}_{a=0}^{n-1}$ on an open set $U$ consists of smooth vector fields that form a basis of $T_pM$ at every $p\in U$. The dual coframe $\{\theta^a\}$ is determined by

$$\theta^a(e_b)=\delta^a{}_b.$$

Relative to coordinates $x^\mu$,

$$e_a=e_a{}^\mu\partial_\mu, \qquad \theta^a=\theta^a{}_{\mu}dx^\mu,$$

with inverse relations $e_a{}^\mu\theta^b{}_{\mu}=\delta_a{}^b$ and $e_a{}^\mu\theta^a{}_{\nu}=\delta^\mu{}_{\nu}$. Frame indices label basis elements; coordinate indices label chart components. Neither kind is intrinsically “more physical” until a metric and an observer select an orthonormal frame.

Under a smooth change of local frame $e'_a=A^b{}_a e_b$, the coframe transforms by the inverse matrix. Tensor components transform pointwise in the familiar multilinear way, but the structure functions also acquire derivatives of $A(x)$. They are therefore coefficients attached to a particular moving frame, not the components of a tensor. The geometric statement that a specified frame is or is not integrable remains meaningful once the frame is fixed.

Carroll introduces noncoordinate bases and their transformation laws in Carroll 1997, §3, pp. 88–91, eqs. (3.114)–(3.123).

Commutators and structure functions

Coordinate vector fields commute, but a general frame satisfies

$$[e_a,e_b]=C^c{}_{ab}e_c, \qquad C^c{}_{ab}=-C^c{}_{ba}.$$

The coefficients $C^c{}_{ab}$ are smooth functions, not generally constants. They are sometimes called coefficients of anholonomy. Evaluate $d\theta^c$ on two frame vectors:

$$\begin{aligned} d\theta^c(e_a,e_b) &=e_a\!\left(\theta^c(e_b)\right) -e_b\!\left(\theta^c(e_a)\right) -\theta^c([e_a,e_b]) \\ &=-C^c{}_{ab}. \end{aligned}$$

Therefore

$$d\theta^c=-\frac12 C^c{}_{ab}\,\theta^a\wedge\theta^b.$$

The factor $1/2$ compensates for summing both antisymmetric index orders. This equation is an independent sign check on any commutator calculation.

A full frame is a coordinate frame locally exactly when all its vector fields commute. One direction is immediate from $[\partial_\mu,\partial_\nu]=0$; the other is the simultaneous straightening result for linearly independent commuting fields. A single nonzero $C^c{}_{ab}$ rules out coordinates whose coordinate vectors equal that frame, even though the frame remains a valid smooth basis.

This criterion concerns the entire basis. A subset of frame vectors may span an integrable distribution even when the full frame is anholonomic; Frobenius then provides coordinates only along the corresponding leaves. Conversely, rescaling one coordinate vector by a position-dependent function generally destroys commutativity with the others without making the underlying manifold less smooth.

Polar orthonormal frame

On the punctured Euclidean plane, write

$$ds^2=dr^2+r^2d\phi^2.$$

The orthonormal coframe and dual frame are

$$\theta^1=dr, \qquad \theta^2=r\,d\phi, \qquad e_1=\partial_r, \qquad e_2=\frac1r\partial_\phi.$$

Directly,

$$[e_1,e_2]=-\frac1r e_2,$$

so $C^2{}_{12}=-1/r$. The coframe calculation gives

$$d\theta^1=0, \qquad d\theta^2=dr\wedge d\phi =\frac1r\theta^1\wedge\theta^2,$$

which agrees with $d\theta^2=-C^2{}_{12}\theta^1\wedge\theta^2$. The frame is anholonomic even though the plane is flat. Anholonomy measures failure to be a coordinate basis, not curvature.

The coordinate $\phi$ and this coframe are not defined at $r=0$. That is a limitation of the chosen patch, not a curvature singularity. Cartesian coordinate vectors provide a smooth frame through the origin.

Local existence and global limitations

Every smooth manifold admits frames on sufficiently small coordinate neighborhoods. A single frame over the whole manifold exists only when the tangent bundle is trivial, a global topological condition called parallelizability. Frame calculations must therefore include transition functions when several patches are required.

Anholonomy is also distinct from torsion. The structure functions depend only on the chosen frame. Torsion compares the connection coefficients with those structure functions. A torsion-free connection can have nonzero coefficients in an anholonomic frame, as the polar example will show through Cartan’s first structure equation.

More explicitly, if $\nabla_{e_a}e_b=\Gamma^c{}_{ab}e_c$, then

$$T^c{}_{ab} =\Gamma^c{}_{ab}-\Gamma^c{}_{ba}-C^c{}_{ab}.$$

The antisymmetric part of the connection coefficients can therefore be nonzero even when $T^c{}_{ab}=0$; it must then reproduce the frame commutator. This component identity is the same content as Cartan’s first structure equation and supplies a second check on signs.

Nor is anholonomy curvature. The polar frame has nonzero $C^c{}_{ab}$ in a flat region, while a coordinate frame can have $C^c{}_{ab}=0$ in a curved region. Curvature measures the failure of covariant derivatives to commute, including connection terms, rather than the failure of the chosen basis vectors alone to commute.

Exercises

On the punctured plane, replace $e_2$ by $\widetilde e_2=f(r)\partial_\phi$ while keeping $e_1=\partial_r$. Compute the commutator and determine which nonzero functions $f$ make the pair commute.

Solution

Because $\partial_r$ and $\partial_\phi$ commute,

$$[e_1,\widetilde e_2] =f'(r)\partial_\phi =\frac{f'(r)}{f(r)}\widetilde e_2.$$

The frame commutes exactly when $f'(r)=0$, so $f$ is a nonzero constant on each connected region. The orthonormal choice $f=1/r$ is therefore necessarily anholonomic. This distinguishes the coordinate requirement from orthonormality: one can have either property here, but not both with $e_1=\partial_r$ and the angular direction normalized.

References

• Carroll, Sean M. Lecture Notes on General Relativity. arXiv:gr-qc/9712019 (1997), arXiv.