Distributions on a manifold
Fix a smooth $n$-manifold $M$. A rank-$k$ distribution $\Delta$ on $M$ is a smooth assignment
\[p \;\longmapsto\; \Delta_p \;\subset\; T_p M, \qquad \dim \Delta_p = k,\]of a $k$-plane in the tangent space at every point. Equivalently, $\Delta$ is a rank-$k$ sub-bundle of $TM$. Locally one specifies $\Delta$ by giving $k$ pointwise-independent vector fields $X_1, \ldots, X_k$ that span $\Delta$ — but the distribution itself is the plane field, not the choice of frame.
Two extreme examples we will keep in mind:
- $M = \mathbb R^3$, $\Delta = \mathrm{span}(\partial_x, \partial_y)$. At every point the plane is the horizontal $xy$-plane. Curves tangent to $\Delta$ keep their $z$ fixed.
- $M = \mathbb R^3$, $\Delta = \ker(dz - y\,dx) = \mathrm{span}(\partial_x + y\,\partial_z,\, \partial_y)$. At each point the plane is tilted by the amount $y$. Curves tangent to $\Delta$ can change $z$ — but only via a precise interplay between $x$- and $y$-motion.
A curve $\gamma : [0, T] \to M$ is horizontal (or admissible) if $\dot\gamma(t) \in \Delta_{\gamma(t)}$ for all $t$. Horizontal curves are the only legal trajectories in the V1 model: at every moment the curve’s velocity must lie in the 2-plane the cortex can move through (forward + rotate).
Frobenius’s theorem: when is $\Delta$ integrable?
A rank-$k$ distribution is integrable if through every point passes a $k$-dimensional submanifold $L \subset M$ — an integral submanifold — with $T_p L = \Delta_p$ for every $p \in L$. When integrable, the integral submanifolds foliate $M$.
The clean test is purely algebraic:
Easy direction. Suppose $\Delta$ integrates, with leaves $L$. Two vector fields $X, Y \in \Delta$ are tangent to $L$. Their flows preserve $L$ (because the leaf is integral). Their commutator’s flow does too, so $[X, Y]$ is tangent to $L$, i.e. in $\Delta$.
Hard direction. If $[\Delta, \Delta] \subseteq \Delta$, choose a frame $X_1, \ldots, X_k$ for $\Delta$. By rectifying flows, find local coordinates in which $X_i = \partial_{x_i}$ for $i \leq k$. The integral submanifolds are the slices $x_{k+1}, \ldots, x_n = \text{const}$. The bracket-closure ensures the rectifying coordinates are consistent.
Worked test for the V1 distribution. $X_1 = \cos\theta\,\partial_x + \sin\theta\,\partial_y$, $X_2 = \partial_\theta$ (Appendix A1 §3). Compute $[X_1, X_2]$ in coordinates:
\([X_1, X_2] \;=\; (\cos\theta\,\partial_x + \sin\theta\,\partial_y)\partial_\theta - \partial_\theta(\cos\theta\,\partial_x + \sin\theta\,\partial_y)\) \(\quad \;=\; -(-\sin\theta)\,\partial_x - (\cos\theta)\,\partial_y \;=\; \sin\theta\,\partial_x - \cos\theta\,\partial_y \;=\; -X_3,\)
and $X_3 \notin \mathrm{span}{X_1, X_2}$ (its $\partial_x, \partial_y$ coefficients are not proportional to $X_1$’s). So $[X_1, X_2] \notin \mathcal H$ and Frobenius fails — the V1 distribution is not integrable.
Geometrically: if it had integrated, the cortex would foliate into 2-dimensional sheets, and you could never reach a sideways-displaced state from your starting one by horizontal motion. The model state space would be a disconnected stack. But it is not — and the failure of Frobenius is exactly what makes contour completion possible.
Bracket-generating distributions and Chow–Rashevskii
The dual case to Frobenius:
Sketch. Bracket-generating means iterated brackets of the frame fields span $T_p M$. Concatenating short flows along a frame field $X_i$ for times $\pm \varepsilon$ in the pattern of Appendix A1’s Figure A1.2 produces a net displacement of order $\varepsilon^2$ in the bracket direction, of order $\varepsilon^3$ in iterated-bracket directions, etc. The subtlety a careful proof must face: the endpoint map \((t_1, \ldots, t_n) \mapsto \Phi^{X_{i_1}}_{t_1} \circ \cdots \circ \Phi^{X_{i_n}}_{t_n}(p)\) is degenerate at the origin — its differential there only spans the distribution itself, and the bracket directions appear at second order. One therefore applies the inverse/rank argument at suitable nonzero parameter values (or invokes the Orbit Theorem); see Montgomery §2 or Agrachev–Barilari–Boscain Ch. 3 for the full proof. Compose enough hops and you can reach any point.
For SE(2) the Hörmander condition is satisfied after one bracket layer (step 2). We computed $[X_1, X_2] = -X_3$, and $X_1, X_2, X_3$ already span $T_g\mathrm{SE}(2)$ (three linearly independent fields). No iterated brackets beyond $[X_1,X_2]$ are needed. This is the minimal bracket-generating contact case and is what makes the sub-Riemannian (SR) Carnot–Carathéodory distance well-behaved on $\mathrm{SE}(2)$.
Contact structures on 3-manifolds
A contact form on a $(2n+1)$-manifold $M$ is a 1-form $\alpha$ with
\[\alpha \wedge (d\alpha)^n \;\neq\; 0 \quad\text{everywhere}.\]A contact structure is the rank-$2n$ distribution $\xi = \ker \alpha$. The condition $\alpha \wedge (d\alpha)^n \neq 0$ is the strongest possible non-integrability — exactly the opposite of Frobenius’s vanishing condition.
For $n = 1$ (3-manifolds, our case):
\[\alpha \wedge d\alpha \;\neq\; 0.\]The standard contact structure on $\mathbb R^3$. $\alpha_0 = dz - y\,dx$. Then $d\alpha_0 = -dy \wedge dx = dx \wedge dy$, and $\alpha_0 \wedge d\alpha_0 = dz \wedge dx \wedge dy \neq 0$. Contact, with kernel $\xi_0 = \mathrm{span}(\partial_x + y\,\partial_z, \partial_y)$.
The SE(2) contact structure. $\alpha = \sin\theta\,dx - \cos\theta\,dy$. Compute $d\alpha = \cos\theta\,d\theta \wedge dx + \sin\theta\,d\theta \wedge dy$, hence
\(\alpha \wedge d\alpha \;=\; (\sin\theta\,dx - \cos\theta\,dy) \wedge (\cos\theta\,d\theta \wedge dx + \sin\theta\,d\theta \wedge dy)\) \(\quad \;=\; \sin^2\theta\,dx \wedge d\theta \wedge dy - \cos^2\theta\,dy \wedge d\theta \wedge dx \;=\; -dx \wedge dy \wedge d\theta \;\neq\; 0.\)
Contact. The kernel $\xi = \mathrm{span}{X_1, X_2}$ is exactly the V1 horizontal distribution $\mathcal H$ of the Frobenius test above.
Darboux’s theorem (contact version): every contact structure on a 3-manifold is locally isomorphic to the standard one $(\mathbb R^3, \alpha_0)$. So the V1 contact structure and the engineering “rolling penny” contact structure are the same locally — a fact Petitot exploited to import results from the latter into V1 modelling.
Contact versus Martinet: one vanishing coefficient changes the geometry
The contact condition must hold everywhere. The flat Martinet distribution shows what happens when it fails on a surface:
\[\alpha_M=dz-\tfrac12y^2\,dx,\qquad \Delta_M=\ker\alpha_M =\mathrm{span}\!\left\{X=\partial_x+\tfrac12y^2\partial_z, Y=\partial_y\right\}.\]Its brackets are
\[[X,Y]=-y\,\partial_z,\qquad [Y,[X,Y]]=-\partial_z.\]Away from $y=0$, the first bracket recovers the missing $z$ direction, so the growth vector is $(2,3)$ and the distribution is contact. On the Martinet surface $\Sigma={y=0}$, that bracket vanishes and one more bracket is required: the growth vector is $(2,2,3)$. The weights at $\Sigma$ are $(1,1,3)$ and their sum is $Q=5$, rather than the contact value $Q=4$.
The magnetic-flux article meets this normal form at a transverse zero of a planar scalar field: $B=-y$ and $A=\tfrac12y^2dx$ make the fibre coordinate $\varphi$ play the role of $z$. Thus “the field vanishes” becomes “the contact bracket loses rank,” and the first field derivative appears in the next bracket.
Why Chow’s theorem matters for vision
The Chow theorem is what makes a shortest-path model of modal completion on $\mathrm{SE}(2)$ mathematically well posed. If the model distribution were integrable, the brain would be unable to bridge two oriented edges that do not lie on the same $\theta = \text{const}$ leaf — vision would split into orientation strata. Frobenius would have shut the door, and Petitot’s model would predict no perception of contour completion.
Bracket-generation says the opposite: any source/target state pair can be linked by a horizontal path, and the length of the shortest such path is the SR distance. Petitot’s programme proposes this geometry as a model of association and completion; whether cortical dynamics literally minimises this homogeneous distance is a separate empirical question.
Carnot–Carathéodory distance
Once a sub-Riemannian metric $\langle\cdot,\cdot\rangle_p$ on $\Delta_p$ is fixed, the horizontal length of an admissible curve is
\[L_{\mathrm{SR}}(\gamma) \;:=\; \int_0^T \sqrt{\langle\dot\gamma, \dot\gamma\rangle_{\gamma(t)}}\,dt,\]and the sub-Riemannian distance $d_{\mathrm{SR}}(p, q)$ is the infimum over all admissible curves from $p$ to $q$. Chow’s theorem ensures this infimum is finite (the set of admissible curves is non-empty).
For the V1 metric of Part 1 with frame ${X_1, X_2}$ orthonormal, an admissible $\gamma$ writes as $\dot\gamma = u_1(t) X_1 + u_2(t) X_2$ and
\[L_{\mathrm{SR}}(\gamma) \;=\; \int_0^T \sqrt{u_1^2 + u_2^2}\,dt.\]Pinning $u_1=1$ makes $s$ plane arc length and $\kappa=\dot\theta$. At fixed length, the quadratic control energy $\tfrac12\int(1+\kappa^2)\,ds$ differs by a constant from Euler’s bending energy $\tfrac12\int\kappa^2\,ds$. The free SR extremals are a different (cuspidal) family sharing the same pendulum core. Appendix A3 runs both through the Pontryagin Maximum Principle and keeps them apart.
The sub-Riemannian Hopf–Rinow theorem shows that on a connected, complete bracket-generating SR manifold the infimum is attained: a length-minimising geodesic exists between any two points. For $\mathrm{SE}(2)$ it means every pair of lifted model states is connected by an actual shortest horizontal curve — not just an approachable one.
Connection to the elliptic project
The reachability figure above uses the same flow concatenations as the
parking trajectory in
the dubins-back-wheel example: a piecewise concatenation of
$X_1$ (“forward + reverse”) and $X_2$ (“steer”) flows. The parking
trajectory is what Chow’s theorem looks like with finite $\varepsilon$.
Shrinking the loop isolates the bracket displacement; it is a local-control
construction, not an optimisation algorithm and not an elastica limit.
Both are computed by the same SE(2) ordinary-differential-equation (ODE)
integrator that lives in
elliptic-core.js on the elliptic site and is reused here.
Code
# Test the V1 distribution for Frobenius integrability.
# Computes [X1, X2] symbolically and checks if it's in span{X1, X2}.
import sympy as sp
x, y, th = sp.symbols('x y theta', real=True)
X1 = sp.Matrix([sp.cos(th), sp.sin(th), 0]) # cos θ ∂x + sin θ ∂y
X2 = sp.Matrix([0, 0, 1]) # ∂θ
def vf_bracket(X, Y, vars_):
# [X, Y]^i = X^j ∂_j Y^i - Y^j ∂_j X^i
out = sp.zeros(len(X), 1)
for i in range(len(X)):
s = 0
for j, v in enumerate(vars_):
s += X[j] * sp.diff(Y[i], v) - Y[j] * sp.diff(X[i], v)
out[i] = sp.simplify(s)
return out
br12 = vf_bracket(X1, X2, [x, y, th])
print("[X1, X2] =", br12.T)
# → [sin(θ), -cos(θ), 0] (this is -X3 from the text)
# In span{X1, X2}? Solve a·X1 + b·X2 = [X1, X2] for constants a, b
a, b = sp.symbols('a b')
sol = sp.solve(a * X1 + b * X2 - br12, [a, b], dict=True)
print("solution ([] means not in span):", sol)
# → [] — no solution; distribution is NOT integrable
# Contact-form check: α ∧ dα ≠ 0 for SE(2).
# For a 1-form α = αx dx + αy dy + αθ dθ on a 3-manifold,
# α ∧ dα = (α · curl α) · dx ∧ dy ∧ dθ,
# so the manifold is contact iff α · curl α never vanishes.
ax, ay, ath = sp.sin(th), -sp.cos(th), 0 # α = sin θ dx − cos θ dy
curl = (sp.diff(ath, y) - sp.diff(ay, th), # ∂y αθ − ∂θ αy
sp.diff(ax, th) - sp.diff(ath, x), # ∂θ αx − ∂x αθ
sp.diff(ay, x) - sp.diff(ax, y)) # ∂x αy − ∂y αx
vol_coeff = sp.simplify(ax * curl[0] + ay * curl[1] + ath * curl[2])
print("α ∧ dα coefficient:", vol_coeff)
# → -1, i.e. α ∧ dα = −dx ∧ dy ∧ dθ ≠ 0 (SE(2) is contact)
What we covered, and what comes next
A distribution is a smooth plane field; integrability is decided by Frobenius’s bracket-closure test; bracket-generation gives Chow’s reachability theorem; contact structures are the “maximally non-integrable” case in odd dimensions. The V1 distribution $\mathcal H = \mathrm{span}{X_1, X_2}$ on $\mathrm{SE}(2)$ is a contact structure, and Chow’s theorem guarantees any two V1 neurons can be linked by a horizontal curve.
Appendix A3 will turn the SR-length minimisation problem into a Hamiltonian system on $\mathfrak{se}(2)^{\ast}$ via the Pontryagin Maximum Principle — recovering the equations $\dot h_1 = h_2 h_3$ etc. that Part 2 §1 uses without proof.
References
- R. Montgomery (2002). A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS Mathematical Surveys 91. The textbook — Chapter 1 develops everything in this appendix.
- A. A. Agrachev, D. Barilari, U. Boscain (2019). A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge. Chapters 2 (distributions) and 3 (Chow's theorem).
- H. Geiges (2008). An Introduction to Contact Topology. Cambridge. The contact-geometry side, with Darboux's theorem.
- G. Citti & A. Sarti (2006). "A cortical based model of perceptual completion in the roto-translation space." J. Math. Imaging Vision 24(3): 307–326. The application of Chow + contact to V1.
- J. Petitot (2003). "The neurogeometry of pinwheels as a sub-Riemannian contact structure." Journal of Physiology–Paris 97: 265–309. Originator of the contact-geometric V1 model.
- Elliptic project — Dubins back wheel. Shows parking-style piecewise-horizontal reachability paths. Shrinking their commutator loops isolates the Lie-bracket direction; it does not by itself compute an SR minimiser or a pinned elastica.