Curvature and halfbrackets, part III

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.

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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 G 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 \delta^{x}_{\varepsilon} y = x \delta_{\varepsilon}(x^{-1}y) and group norm denoted by \| x \|) 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

Remarks:

  1. 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.
  2. 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.

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2. With the notations from the last post, I want to compute the quantities A, B, C. We already know that B is related to the curvature of G 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 B is controlled by A and C. But let’s see the expressions of these three quantities for sub-riemannian Lie groups.

I denote by d(u,v) the left invariant sub-riemannian distance, therefore we have d(u,v) = \| u^{-1}v\|.

Now, \rho_{\varepsilon}(x,u) = \| x^{-1} u \|_{\varepsilon} , where \varepsilon \| u \|_{\varepsilon} = \| \delta_{\varepsilon} u \|  by definition.  Notice also that \Delta^{x}_{\varepsilon}(u,v) = (\delta^{x}_{\varepsilon} u ) ((u^{-1} x) *_{\varepsilon} (x^{-1} v)), where  u *_{\varepsilon} v is the deformed group operation at scale \varepsilon, i.e. it is defined by the relation:

\delta_{\varepsilon} (u *_{\varepsilon} v) = (\delta_{\varepsilon} u) (\delta_{\varepsilon} v)

With all this, it follows that:

A_{\varepsilon}(x,u) = \rho_{\varepsilon}(x,u) - d^{x}(x,u) = \|x^{-1} u \|_{\varepsilon} - \| x^{-1} u \|_{0}

A_{\varepsilon}(\delta^{x}_{\varepsilon} u, \Delta^{x}_{\varepsilon}(u,v)) = \| (u^{-1} x) *_{\varepsilon} (x^{-1} v) \|_{\varepsilon} - \| (u^{-1} x) *_{\varepsilon} (x^{-1} v)\|_{0}.

A similar computation leads us to the expression for the curvature related quantity

B_{\varepsilon}(x,u,v) = d^{x}_{\varepsilon}(u,v) - d^{x}(u,v) = \| (u^{-1}x) *_{\varepsilon} (x^{-1} v)\|_{\varepsilon} - \| (u^{-1}x) *_{0} (x^{-1}v)\|_{0}.

Finally,

C_{\varepsilon}(x,u,v) = \|(u^{-1} x) *_{\varepsilon} (x^{-1} v)\|_{0} - \|(u^{-1}x) *_{0} (x^{-1}v)\|_{0}. This last quantity is controlled by a halfbracket, via a norm inequality.

The expressions of A, B, C make transparent that the curvature-related B is the sum of A and C. In the next post I shall use the length dilation structure of the sub-riemannian Lie group in order to show that A is controlled by C, which in turn is controlled by a norm of a halfbracket. Then I shall apply all this to SO(3), as an example.

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