What this appendix covers

The growth vector (Appendix C3) sorts groups into classes but cannot split those that share a tangent cone — Heisenberg and SE(2), both $(2,3)$. This appendix is the second fingerprint leg that does split them: not a single caustic germ, but the whole first conjugate locus from the base point — its shape, symmetry, and continuous moduli. It sets up the deviation statistic of Appendix C6.

Not one germ — the whole locus

Appendix C2 showed a single cusp is group-blind. The escape is to use the entire first conjugate locus seen from the base point $q_0$: the full set of first refocusing points over all geodesic directions. Three features of that set are structure-specific in a way no local germ is:

• its cusp count and how the folds are arranged;
• its symmetry group (dilations and rotations that map it to itself);
• its continuous moduli — real invariants that vary smoothly from group to group even when the cusp count and symmetry agree.

For the 3D contact structures of the series (Heisenberg, SE(2), and their relatives SL(2), SH(2), SU(2)) this locus has a specific and beautiful form.

The sub-Riemannian astroid

For a generic 3D contact structure the first conjugate locus, seen in the intrinsic dilation-adapted coordinates near the pole, is a four-cusped astroid — the classic star shape $x^{2/3} + y^{2/3} = a^{2/3}$, four sharp cusps joined by four fold arcs. This was computed by El-Alaoui, Gauthier & Kupka (1996) for small balls on $\mathbb{R}^3$, and the germ structure worked out by Agrachev, Charlot, Gauthier & Zakalyukin (2000); Bonnet, Gauthier & Rossi (2019) classified its generic singularities. The four cusps are $A_3$ germs (Appendix C2) — universal — but their arrangement and proportions are not.

The sub-Riemannian astroid. The first conjugate locus of a generic 3D contact structure: four cusps ($A_3$) joined by four folds. The symmetric shape (blue) is the flat/nilpotent Heisenberg reference; a curved structure like SE(2) deforms it (orange), and the size and shape of that deformation are the moduli $(\chi, \kappa)$. The germs are universal; the deformation is the fingerprint. Shape schematic; axes are dimensionless SR-normal coordinates at the pole.

The two invariants (χ, κ)

Agrachev & Barilari (2012) classified all left-invariant sub-Riemannian structures on 3D Lie groups by exactly two differential invariants, $\chi \ge 0$ and $\kappa$. They locate each group in a two-parameter family, with the flat model at the origin:

• $\chi = \kappa = 0$ — Heisenberg, the flat case. Its conjugate locus is maximally degenerate: rather than a non-degenerate astroid, the whole family of geodesics at a given momentum collapses to a single point, so the locus is just the central axis (proved in Part 2, $t_c = 2\pi/|w|$).
• $\kappa < 0$ — SE(2), the group of motions. It is curved; its conjugate locus is a genuine deformed astroid, and its geodesics are Euler elastica (Sachkov 2010).
• other signs and magnitudes — SU(2), SL(2), SH(2), the rest of the 3D contact family.

So the moduli $(\chi, \kappa)$ are the fingerprint that distinguishes structures with the same tangent cone: they are precisely what the growth vector throws away.

From shape to a number

Extracting $(\chi, \kappa)$ from a sampled locus is the goal, but the series’ working detector uses a scalar proxy that captures the same information for the Heisenberg/SE(2) split: the deviation of the conjugate locus from the flat reference, measured through the refocusing-time law. Heisenberg’s law is exactly $t_c = 2\pi/|w|$; SE(2) departs from it, increasingly at low momentum, giving a deviation $\delta \approx 0.14$ versus $0$. That statistic — the nilpotent deviation — is the subject of Appendix C6, and it is what breaks the alias in the Part 3 confusion matrix.

References

• El-H. Chakir El-Alaoui, J.-P. Gauthier & I. Kupka (1996). “Small sub-Riemannian balls on $\mathbb{R}^3$.” J. Dyn. Control Syst. 2, 359–421.
• A. Agrachev, G. Charlot, J.-P. Gauthier & V. Zakalyukin (2000). “On sub-Riemannian caustics and wave fronts for contact distributions in the three-space.” J. Dyn. Control Syst. 6, 365–395.
• A. Agrachev & D. Barilari (2012). “Sub-Riemannian structures on 3D Lie groups.” J. Dyn. Control Syst. 18, 21–44. arXiv:1007.4970.
• B. Bonnet, J.-P. Gauthier & F. Rossi (2019). “Generic singularities of the 3D-contact sub-Riemannian conjugate locus.” C. R. Acad. Sci. Paris 357, 542–549. arXiv:1812.01508.
• Yu. L. Sachkov (2010). “Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane.” ESAIM: COCV 16, 1018–1039. arXiv:0903.0727.