More generally, for any Lie group and any subspace $V$ of its Lie algebra (the tangent space at $1$) that generates the Lie algebra, there is a Carnot-Caratheodory metric that measures path lengths with along the left-invariant plane field which agrees with $V$, with respect to a left-invariant Riemannian metric defined on this plane field. These metrics are important in many real situations. Any non-abelian Lie group has many such subspaces.Just as in driving, where cars follow paths that have bounded curvature, one can further restrict to paths whose tangent vectors are in the left-invariant cone field that extends any cone in the Lie algebra that generates the Lie algebra, although the Lipschitz equivalence class of the resulting metric depends only on the linear span of the cone (assuming a finite dimensional Lie group).
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2 | Corrected grammar, tried to clarify some language. | ||
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A position Positions of the car are in 1-1 correspondence with the isometry isometries of the plane that takes take it from its initial parked position to its current a given position. Define a driving metric that is the minimum total number of turns of the drive shaft (~mean arclength of drive wheels) to get from one position to another. As every driver knows from parallel parking, the driving distance to move sideways by a small Euclidean distance $x$, annoyingly, does not go decrease to 0 linearly with Euclidean distancex: it is proportional to the square root of the Euclidean distance$\sqrt x$, because the front wheels need (approximately) to enclose an area proportional to the sideways distance. In the driving metric, the distance from $g*h*$ (to $h*g$,for example if $g = $ cramp the steering wheel to the left , then to the right, both while driving and drive forward 1 one unit) to while $h*g$ (h = $ cramp to the left, then steering wheel to the right ) and drive forward 1 unit is sometimes as much as $2( d(1,g)+d(1,h))$: in general, there's no there may not be a more efficient way to get from the first position $g*h$ to the second $h*g$ than to undo retrace: first , then do the second.inverse of $g*h$, then do $h*g*$.
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Yes, it's an easy and well-known fact that for a Lie group with a smooth Riemannian metric (which we may assume left-invariant) $$d([g,h], 1) = d(g*h, h*g) = O(d(1,g)*d(1,h)) . $$ This follows from differentiability, and from the observation that the commutator map $[,]: G \times G \rightarrow G$ maps the submanifolds $1 \times G$ and $G \times 1$ both to $1$, so the first derivative of the commutator is 0. The second derivative is given by the Lie bracket. However, any Lie group whose identity component is not abelian admits left-invariant path-metrics where this inequality fails, in particular, its Carnot-Caratheodory metrics. Consider, for instance, the group of isometries of the plane from the point of view of driving a car in a flat area. A position of the car are in 1-1 correspondence with the isometry of the plane that takes it from its initial parked position to its current position. Define a metric that is the minimum total number of turns of the drive shaft to get from one position to another. As every driver knows from parallel parking, the driving distance to move sideways by a small Euclidean does not go to 0 linearly with Euclidean distance: it is proportional to the square root of the Euclidean distance, because the front wheels need (approximately) to enclose an area proportional to the sideways distance. In the driving metric, the distance from $g*h*$ (cramp the steering wheel to the left, then to the right, both while driving forward 1 unit) to $h*g$ (cramp to the left, then to the right) is $2( d(1,g)+d(1,h))$: in general, there's no more efficient way to get from the first position to the second than to undo first, then do the second. More generally, for any Lie group and any subspace $V$ of its Lie algebra (the tangent space at $1$) that generates the Lie algebra, there is a Carnot-Caratheodory metric that measures path lengths with along the left-invariant plane field which agrees with $V$, with respect to a left-invariant Riemannian metric defined on this plane field. These metrics are important in many real situations. The question has at least as much to do with the metric as with the group. |
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