Two ways a direction can be “special”
A preferred direction can mean two very different things, and only one of them changes the geometry.
Slow (coefficient anisotropy). Transport across a magnetic field is suppressed relative to along it, $D_\perp \ll D_\parallel$. The metric is
\[g = \frac{1}{D_\parallel}\,\hat b\,\hat b + \frac{1}{D_\perp}\,(I - \hat b\,\hat b),\]an ordinary Riemannian metric — just a very eccentric one. You can still move in every direction; some are simply expensive. The reachable set from a point in “cost” $r$ is an ellipsoid whose every semi-axis is linear in $r$ (one long, the others short). Ball volume scales as $r^n$, so the homogeneous dimension is $Q = n$. However extreme the anisotropy, $Q$ never rises. The program measured this directly down to $D_\perp/D_\parallel = 10^{-6}$: still $Q = n = 3$.
Forbidden (exponent anisotropy). In a genuine sub-Riemannian structure the forbidden direction is not expensive — it is unavailable. You reach it only by a bracket of allowed moves (Appendix D1), and it fills in like $r^2$, not $r$. Its weight is 2, not 1, and $Q > n$. Different coefficient on the same power of $r$ is anisotropic-Riemannian; a different power is sub-Riemannian. That is the whole distinction, and the diagnostic is the exponent, never the coefficient.
The singular limit is a trap, not a shortcut
It is tempting to think that pushing $D_\perp \to 0$ — making the cross-field direction truly impossible — produces the sub-Riemannian structure. It produces the opposite. In the limit the distribution has rank 1: you may only move along the field line. A rank-1 distribution is integrable (Frobenius’ theorem): its horizontal curves are trapped on the one-dimensional field lines and cannot reach off them at all. There is no bracket, no Chow condition, no connectivity — the space fragments into disjoint field lines. Suppressing a direction to zero does not forbid-and-bracket it; it severs it.
Chow–Rashevskii vs Frobenius: the exact fork
The two outcomes are separated by a clean theorem about the distribution $\mathcal D$ (the allowed directions):
- Frobenius (integrable). If the brackets $[\mathcal D, \mathcal D]$ stay inside $\mathcal D$, the space foliates into leaves and horizontal curves never leave their leaf. Rank-1, or any involutive distribution. No sub-Riemannian geometry.
- Chow–Rashevskii (bracket-generating). If iterated brackets of $\mathcal D$ eventually span the whole tangent space, then any two points are joined by a horizontal curve, and the sub-Riemannian distance is finite and genuine. This is the case the program lives in — and by Appendix D1 it holds for the magnetic flux lift exactly where the curvature is nonzero.
So “does the field forbid a direction that brackets can nonetheless reach?” is not a vague intuition; it is the question of whether $\mathcal D$ is bracket-generating, decided by whether the curvature is nonzero.
The diagnostic a caustic can counterfeit
The clean signature of sub-Riemannian structure is $Q > n$. But there is a subtlety the program learned the hard way, and it is worth stating because it is exactly the kind of error that produces a false discovery.
$Q > n$ at a single point is not sufficient. A Lagrangian fold — the generic caustic of any smooth map, including gravitational structure formation, which has no sub-Riemannian structure at all — compresses one direction so that the image of a small ball extends like $r^2$ there. Measured pointwise, the reach estimator then returns an exponent sum of $n+1$ — numerically identical to a genuine contact structure’s $Q$, even though no sub-Riemannian homogeneous dimension exists there at all. Stated precisely, this is estimator confounding: a coincidence of measured scaling exponents, not an equality of geometric invariants — a Lagrangian caustic does not acquire a homogeneous dimension at its fold. The ADE-universality of caustics resurfaces at the level of the measured exponent.
The fix this program uses is measure-theoretic — stated for what it is:
The program’s operational screen (a working rule, not a standard theorem). A bracket-generating distribution has $Q > n$ on a set of full measure — the weights are a property of the distribution, present almost everywhere — while a Lagrangian catastrophe inflates the estimator’s exponent only on its caustic, a codimension-1 null set. The screen therefore demands full-measure prevalence before reading $Q > n$ as sub-Riemannian. No converse is claimed: full-measure $Q > n$ is used as a screen against caustic counterfeits, not as a characterisation of sub-Riemannian geometry.
The program confirmed both sides: a Zel’dovich gravitational flow gives $Q > n$ at $0\%$ of sampled points (folds are measure-zero), while a genuine magnetic contact structure gives $Q > n$ at $100\%$. The selection rule, made quantitative, is: not “is $Q > n$ somewhere?” but “is $Q > n$ almost everywhere?”
The rule, and its ledger
| System | Distribution | Bracket-generating? | $Q$ | Verdict |
|---|---|---|---|---|
| Anisotropic transport $D_\perp\ll D_\parallel$ | full rank, eccentric | n/a (Riemannian) | $= n$ | ✗ slow, not forbidden |
| $D_\perp \to 0$ | rank 1 | no (Frobenius) | — | ✗ integrable, severed |
| Gravitational flow | no distribution | — | $=n$ a.e. | ✗ a flow, and folds fake $Q{>}n$ on a null set |
| Magnetic field, $\mathbf B\neq0$ | rank $d$ flux lift | yes (curvature $\neq 0$) | $=d+2$ a.e. | ✓ |
| Rotating frame | flux lift, curvature $2\boldsymbol\omega$ | yes | $=d+2$ a.e. | ✓ |
Everything the series builds sits in the last two rows, and the discipline that keeps it honest is the refusal to confuse them with the first three.
References
- W.-L. Chow (1939). “Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung.” Math. Ann. 117, 98–105.
- R. Montgomery (2002). A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS. (Chow–Rashevskii, Frobenius, the Ball–Box theorem.)
- A. Bellaïche (1996). “The tangent space in sub-Riemannian geometry.” Progr. Math. 144, Birkhäuser. (Homogeneous dimension.)