I continue from “Curvature and halfbrackets, part II“. This post is dedicated to the application of the previously introduced notions to the case of a sub-riemannian Lie group.
1. I start with the definition of a sub-riemannian Lie group. If you look in the literature, the first reference to “sub-riemannian Lie groups” which I am aware about is the series Sub-riemannian geometry and Lie groups arXiv:math/0210189, part II arXiv:math/0307342 , part III arXiv:math/0407099 . However, that work predates the introduction of dilation structures, therefore there is a need to properly define this object within the actual state of matters.
Definition 1. A sub-riemannian Lie group is a locally compact topological group with the following supplementary structure:
- together with the dilation structure coming from it’s one-parameter groups (by the Montgomery-Zippin construction), it has a group norm which induce a tempered dilation structure,
- it has a left-invariant dilation structure (with dilations and group norm denoted by ) which, paired with the tempered dilation structure mentioned previously, it satisfies the hypothesis of “Sub-riemannian geometry from intrinsic viewpoint” Theorem 12.9, arXiv:1206.3093
- there is no assumption on the tempered group norm to come from a Riemannian left-invariant distance on the group. For this reason, some people use the name sub-finsler arXiv:1204.1613 instead of sub-riemannian, but I believe this is not a serious distinction, because the structure of a scalar product which induces the distance is simply not needed for understanding sub-riemannian Lie groups.
- by Theorem 12.9, it follows that the left-invariant field of dilations induces a length dilation structure. I shall use this further. Length dilation structures are maybe a more useful object than simply dilation structures, because they explain how the length functional behaves at different scales, which is a much more detailed information about the microscopic structure of a length metric space than just the information about how the distance behaves at different scales.
This definition looks a bit mysterious, unless you read the course notes cited inside the definition. Probably, when I shall find the interest to pursue it, it would be really useful to just apply, step by step, the constructions from arXiv:1206.3093 to sub-riemannian Lie groups.
2. With the notations from the last post, I want to compute the quantities . We already know that is related to the curvature of with respect to it’s sub-riemannian (sub-finsler if you like it more) distance, as introduced previously via metric profiles. We also know that is controlled by and . But let’s see the expressions of these three quantities for sub-riemannian Lie groups.
I denote by the left invariant sub-riemannian distance, therefore we have .
Now, , where by definition. Notice also that , where is the deformed group operation at scale , i.e. it is defined by the relation:
With all this, it follows that:
A similar computation leads us to the expression for the curvature related quantity
. This last quantity is controlled by a halfbracket, via a norm inequality.
The expressions of make transparent that the curvature-related is the sum of and . In the next post I shall use the length dilation structure of the sub-riemannian Lie group in order to show that is controlled by , which in turn is controlled by a norm of a halfbracket. Then I shall apply all this to , as an example.