Sub - Riemannian geometry :

Take an n-dimensional manifold M . Endow it with a distribution, by which I mean a smooth linear subbundle D ⊂ TM of its tangent bundle TM . So, for x ∈ M , we have a k-plane Dx ⊂ TxM , and by letting x vary we obtain a smoothly varying family of k-planes on M . Put a smoothly varying family g of inner products on each k-plane. The data (M,D, g) is, by definition, a sub-Riemannian geometry. Take the viewpoint that we can explore M only by traveling along paths tangent to D. Call such paths horizontal. Since g measures lengths of horizontal (D) vectors, we can measure lengths of horizontal paths and formulate the sub-Riemannian geodesic problem: to find the shortest horizontal path connecting two given points. Gromov and his school use the term Carnot-Carathéodory geometry for what we call sub-Riemannian geometry. The sub-Riemannian analogues of Euclidean space are a class of Lie groups endowed with left-invariant sub-Riemannian metrics, christened Carnot groups by the Gromov school, homogeneous groups by Stein and his school, and their Lie algebras named symbol algebras by Tanaka’s school. The first nontrivial example is the 3-dimensional Heisenberg group and is the simplest non-Euclidean Carnot group. Its sub-Riemannian geodesic problem is essentially the isoperimetric problem. Take M to be standard 3-space R. Define D by the vanishing of the 1-form θ = dz− (1/2)(xdy−ydx); in other words D(x,y,z) is the 2-plane {(v1, v2, v3) : v3 − (1/2)(xv2 − yv1) = 0}. D can be visualized as a kind of continuous spiral staircase. See the accompanying Figure 1, inspired by the description on p. 40 of [1]. Along the z-axis, the 2-planes of D are parallel to the xy-plane. As we move out radially from the axis along any orthogonal ray, the 2-plane spins about the axis of the ray, rotating monotonically so that by the time we “reach” infinity, they have just rotated by 90 degrees, turning vertical. We have just described the standard contact structure on 3-space. The projection along the z-axis maps each plane linearly onto the xy-plane. Declaring these restricted projections to be isometries defines the family g of inner products on D. Consider a horizontal path γ connecting the origin to a point A units up along the z-axis. The projection c of γ to the xy-plane is closed, and by definition of g the subRiemannian length of γ equals the standard Euclidean length of c. An application of Stokes theorem to the equation A = (1/2) ∫