What this appendix covers

The third fingerprint leg is a single bit: does the geometry admit abnormal shortest paths? This appendix explains what abnormals are, why their existence is a topological property of the constraints alone, the corank criterion that decides it for the series' groups, and why — surprisingly — this coarse bit is the most noise-robust piece of the whole detector (experiment E2).

Two kinds of extremal

Shortest paths in sub-Riemannian geometry come from the Pontryagin Maximum Principle, which produces two families. Normal extremals are the ones you expect: solutions of the geodesic Hamiltonian $H = \tfrac12\sum_i h_i^2$, projecting to smooth locally-minimizing curves — these are what Appendices C3–C4 and the whole forward model deal with. But the Principle also allows abnormal extremals, for which the metric drops out of the equations entirely. An abnormal curve is a critical point of the endpoint map restricted to admissible controls: a path so constrained by the shape of the allowed directions that it is forced, regardless of how you measure length.

Abnormals are the delicate part of the subject. Whether a given abnormal curve is actually minimizing — and how regular such minimizers are — is a genuinely hard, still-open corner of the theory (Sard-type problems, the regularity of minimizers). That is why the series treats this leg as lower-confidence and never lets it hard-gate a verdict.

Existence is topological: the corank criterion

For our purposes the useful fact is coarse and robust: whether nontrivial abnormals exist at all is a topological property of the distribution, governed by its corank = ambient dimension − rank:

• Rank-2, corank-1 (contact) structures — Heisenberg, SE(2), and the other 3D contact groups — have no nontrivial abnormal minimizers. The distribution is “fat” enough that the endpoint map is a submersion; nothing is forced.
• Higher-corank structures — Engel (corank 2), Cartan (corank 3), SE(3) (corank 3) — do admit abnormals. There are directions in which the geometry is constrained enough to force particular curves.

So a single yes/no bit cleanly partitions the candidate list: ${$contact: Heisenberg, SE(2)$}$ versus ${$Engel, Cartan, SE(3)$}$. It is orthogonal to the within-class questions the growth vector and the moduli answer.

The trailer’s straight line

The cleanest concrete abnormal is the Engel one — the car with a trailer of Appendix C1. Drive dead straight, cab and trailer aligned and unmoving: that motion is an abnormal extremal. It owes its special status not to being shortest in any metric sense but to sitting on a singular stratum of the constraints — the configurations where the trailer angle cannot be independently steered. Ardentov & Sachkov (2017) located exactly these abnormal strata in the Engel cut locus; the analogous straight strata appear in Cartan.

Why the coarse bit is the robust one (experiment E2)

Here is the counter-intuitive payoff. The abnormal bit asks only how many directions can you drive? — the rank of the distribution against the ambient dimension. That is a far coarser question than resolving the full growth vector, which needs the fragile high-weight coordinates. So it survives noise that destroys the fine estimate. In the code (metric M4, src/growth.py), the bit is read from the count of weight-1 coordinates, and the experiments show the gap starkly: at a noise level where the full growth vector for Cartan is recovered 0% of the time, the abnormal bit is still correct 100% of the time (E2).

The fingerprint’s legs therefore have complementary noise profiles: when the growth vector collapses a high-step group to “unknown”, the abnormal bit still assigns it to the correct coarse class (“non-contact”). The Part 3 classifier uses exactly this fallback, which is why its class-level accuracy barely moves as its exact-group accuracy falls.

The honest caveat

The corank criterion tells you abnormals exist; it is not yet a detector of abnormal minimizers in caustic data. Building that — finding actual abnormal geodesics in an observed field, robustly, given the open regularity questions — is the genuinely hard piece the series flags as future work. The coarse existence bit is what is used today, and it is used only as corroboration and graceful fallback, never as a hard gate.

References

• A. Agrachev, D. Barilari & U. Boscain (2019). A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge University Press. (Abnormal extremals, PMP.)
• R. Montgomery (2002). A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS. (The first strictly abnormal minimizer.)
• A. A. Ardentov & Yu. L. Sachkov (2017). “Maxwell strata and cut locus in the sub-Riemannian problem on the Engel group.” Regul. Chaotic Dyn. 22, 909–936. arXiv:1710.00216.
• A. Belotto da Silva, A. Figalli, A. Parusiński & L. Rifford (2022). “Strong Sard conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures.” Invent. Math. 229, 395–448.