The sub-Riemannian geometry of the Heisenberg group in its incarnation as the first jet space arises in a variety of physical and biological systems. Such geometries arise naturally in the study of optimal control and path planning where one is dealing with vehicles with limited degrees of freedom, such as wheeled vehicles.
Less obviously (and perhaps more surprisingly), this representation is also relevant to the study of the geometry of the first layer of the mammalian visual cortex V 1. In this section, we describe the role of contact geometries modeled by the Heisenberg group in a simple mechanical model of nonholonomic motion, and in the aforementioned model of the functional structure of the first layer of the visual cortex.
In the process of describing these examples, we will encounter a new sub-Riemannian space, the roto-translation groupRT which, while distinct from the
3.2. Applied models 41
Heisenberg group, is locally approximated by it. We first introduceRT and then discuss the approximation by H.
The roto-translation group, RT , is the group of Euclidean rotations and translations of the plane equipped with a particular sub-Riemannian metric. More precisely,RT is a three-dimensional topological manifold diffeomorphic to R2×S1 with coordinates (x, y, θ). We identify the vector fields
X1= cos θ ∂
∂x+ sin θ ∂
∂y, X2= ∂
∂θ and
X3= sin θ ∂
∂x − cos θ ∂
∂y.
Observe that, similarly to the Heisenberg group, [X1, X2] = X3. However, this is not the only nontrivial bracket, indeed [X3, X2] = −X1. Note also that ω = sin θ dx− cos θ dy is a contact form on RT whose kernel is spanned by X1and X2. We equipRT with a sub-Riemannian metric using the same construction as Section 2.2.3. First, we introduce an inner product, ·, ·RT on the subbundle of the tangent bundle generated by {X1, X2}. Then, if γ : I → RT is a path in RT so that γ is always in the span of{X1, X2}, we define
LengthRT(γ) =
I
γ(t), γ(t)1/2RT dt.
The induced sub-Riemannian distance is
dRT(p, q) = inf{LengthRT(γ) : γ(0) = p, γ(1) = q, γ∈ span{X1, X2}}.
WhileRT is certainly distinct from H, we emphasize that, locally, the two are equivalent. One way to see this is to invoke Darboux’s theorem: any contact form on a three-dimensional manifold is locally diffeomorphic to the standard contact form ω inR3(defined in (2.5)). Another way to see this is to use a weighted Taylor expansion adapted to the bracket structure to describe the vector fields locally.
We follow the second approach. First, some definitions:
Definition 3.1. InRT , let L1 be the set of linear combinations of{X1, X2} with smooth coefficients and letL2=L1+ [L1,L1]. Let Li(p) be the subspace of TpRT given by the collection of vectors X(p), X ∈ Li.
By the above remarks,L2= T (RT ) and L2(p) = TpRT .
Definition 3.2. If {x1, x2, x3} are local coordinates at p so that {dx1, dx2, dx3} form a basis of Tp∗RT adapted to the flag L1(p)⊂ L2(p), we define the weight of xi at p by
weight(xi) = min{i : ∂x1(p)∈ Li(p)}.
42 Chapter 3. Applications of Heisenberg Geometry
Moreover, we let
weight(∂xi) =−weight(xi).
This notion of weight captures the bracket structure of RT and formalizes the intuitive notion that X3 is a second order derivative as it arises as a bracket of two horizontal vector fields. We note that at (0, 0, 0), the standard coordinates (x, y, θ) have weights 1, 2 and 1 respectively. Using this weighting, we expand the vector fields, X1, X2and X3at (0, 0, 0):
Considering only the weight−1 terms in the expansions of X1, X2 and the weight
−2 term in the expansion of X3, we recover the presentation of the Heisenberg group in its matrix model (2.1). It is in this sense thatRT is locally modeled by H in a neighborhood of (0, 0, 0). A similar computation, left to the reader, provides the same result in a neighborhood of any point ofRT .
We remark that the roto-translation group goes by other names in the liter-ature, e.g., “group of planar Euclidean rigid motions”.
3.2.1 Nonholonomic path planning
To illustrate the use of the roto-translation group in the context of nonholonomic path planning and optimal control, we consider the simplest example of a wheeled vehicle: the unicycle. To model the motion and control of a unicycle in a planar region, we introduce standard (x, y) coordinates in the plane, together with an angular variable θ describing the deviation of the wheel from the x-axis. See Figure 3.1.
The state space S = R2 × S1 describes all possible configurations of the unicycle. If the operator pedals the unicycle forward from a point (x, y, θ) ∈ S without changing the angle of the wheel, the unicycle follows the parametric path
(x + t cos θ, y + t sin θ, θ).
Taking one derivative in t yields one of the allowable directions of instantaneous motion:
X1= cos θ ∂
∂x+ sin θ ∂
∂y.
3.2. Applied models 43
θ x y
Figure 3.1: Coordinates describing the unicycle.
As the operator can change the angle of the wheel at will, another direction of instantaneous motion is simply
X2= ∂
∂θ. To complete to a basis for TS we add the vector
X3= sin θ ∂
∂x − cos θ ∂
∂y.
Thus, we recover the roto-translation group as a model space for the motion of a unicycle.
We note that the physical model provides an interpretation of the vector field X3 since a unicycle cannot move instantaneously in this direction as it is perpendicular to the axis of the wheel. However, using a combination of angle rotation and forward motion, the operator can access any position in the plane.
This is reflected mathematically in the bracket relation [X1, X2] = X3.
In this context, sub-Riemannian metric geometry has a close connection with path planning. To plan an optimally efficient path between two points p, q in the state space, we must minimize the length of the path connecting p to q while moving only in allowable directions. In other words, optimal path planning is equivalent to the geodesic problem with respect to the metric dRT.
3.2.2 Geometry of the visual cortex
Neuralbiological research over the past few decades has greatly clarified the func-tional mechanisms of the first layer (V1) of the visual cortex. Early research showed that V1 contains a variety of types of cells, including the so-called “simple cells”.
44 Chapter 3. Applications of Heisenberg Geometry
They found that simple cells are sensitive to orientation specific brightness gradi-ents and are associated to a specific retinotopic field (i.e., a region of the retina).
Simple cells are arranged into columns of cells with shared orientation preference and these columns are further arranged into the so-called “hypercolumns” Each hypercolumn is a stack of columns of simple cells that are all associated to spe-cific spatial points on the retina but with with orientation preferences, or “tuning”, ranging over all possible angles. Figure 3.2 shows a schematic representation of the hypercolumn structure. In this figure, the circles represent a column of simple cells and the arrow in each circle represents the orientation tuning of that column. The
“horizontal” direction is one possible direction of motion between retinotopic fields.
Vertical
Horizontal
Hypercolumn of cells
Figure 3.2: The hypercolumn structure of V1.
Early assumptions – supported by some research – that cortical connectivity runs mostly vertically along the hypercolumns and is severely restricted in horizon-tal directions (between hypercolumns), were contradicted by later evidence show-ing “long range horizontal” connectivity in the cortex. This experimental evidence demonstrated the properties of a specific geometric structure in the first layer of the visual cortex associated to intracortical communication. We may mathemat-ically model this structure of the cortex using a sub-Riemannian structure. The retinotopic fields are modeled by two spatial directions which we will denote by (x, y)∈ R2. Ignoring the column structure in favor of the hypercolumn structure, we may describe each hypercolumn using a copy of S1. We model this situation using the roto-translation group, RT , as R2× S1 where each point (x, y, θ) rep-resents a column of cells associated to a point of retinal data (x, y) ∈ R2, all of which are attuned to the orientation given by the angle θ∈ S1.
The experimental evidence referred to above shows that horizontal connec-tions are made between points (x1, y1, θ1) and (x2, y2, θ2) if θ1 = θ2, and that