Some models of constant curvature in sub-Riemannian geometry
This is a reaction on Luca Rizzi’s talk last time (october 22nd).
1. GO spaces
V. Berestovskii has classified a class of sub-Riemannian manifolds which I now describe.
Definition 1 Say a Sub-Riemannian manifold is GO if every geodesic is an orbit of a one-parameter group of isometries.
This happens for a rather large class of Riemannian homogeneous spaces. Taking limits of such Riemannian homogeneous spaces provides examples of GO sub-Riemannian homogeneous spaces. Berestovskii has found examples which do not arise in this way.
Why are we interested in GO spaces ? Luca’s comparison theorem involves a notion of curvature (asociated to a point and geodesic) which is pretty complicated to compute. In the GO case, such curvature is constant along each geodesic.
Tangent cones of GO spaces are GO. Many step 2 Carnot groups are GO (not the most interesting, since curvature vanishes for Carnot groups).
Here is a less trivial example discovered by Berestovskii. Let be a Lie group with a bi-invariant pseudo-Riemannian metric (e.g. is semi-simple). Fix a left-invariant distribution generated by , where is a subalgebra. If is simple, is bracket-generating. Assume that metric is definite positive on . Then the corresponding sub-Riemannian metric is GO.
Question. Are these examples all 2-step or not ?
1.2. Proof that many 2-step Carnot groups are GO
Let be a basis of the distribution . Each defines a function on . Let
denote the Hamiltonian. Set
be there Poisson brackets. Then ‘s are constant along the motion. Indeed, there derivative is a Poisson bracket, which involves a third order Lie bracket, which vanishes, by assumption. The equations for geodesics are
This can be readily integrated : , then the horizontal projection
The matrix is skew-symmetric. Therefore it is a good candidate for an isometric automorphism, provided it lifts to an isomorphism, which happens often.
Question. Classify 2-step Carnot groups which are GO.
1.3. Proof that examples attached to simple Lie groups are GO
For any and , is an isometry. On computes that every geodesic has the form where , where and . So it is an orbit of a one-parameter group.
2. Structure of isometries
Capogna-Le Donne : sub-Riemannian isometries are smooth, they are determined by their derivative, they form compact Lie groups (on compact manifolds).
Question : is an isometry determined by its restriction to the distribution at one point ? Related question : on a Carnot group, is every isometry affine, i.e. translation times automorphism ?
Breuillard : doesn’t this follow from Capogna and Le Donne’s results ?
Pansu : generic 2-step Carnot groups with dimension vectors such that have no graded automorphisms but dilations, so no isometries fixing a point. Those cannot be GO.