Curvature and halfbrackets, part I

Here is a little story about curvature and Lie brackets in the wider context of dilation structures and sub-riemannian Lie groups. The background of this post is provided by the following links:

  • Sub-riemannian geometry from intrinsic viewpoint, course notes, arXiv:1206.3093 [math.MG], especially section 2.5 “Curvdimension and curvature” and 12.3 “Coherent projections induce length dilation structures”,
  • Curvdimension and curvature of a metric profile part I, part II, part III,
  • Noncommutative Baker-Campbell-Haussdorf formula, the problem and a suggested solution.

We are in the frame of a metric space with dilations (X,d,\delta) (aka “dilation structure” or “dilatation structure”). I introduced these spaces under the name “dilatation structures” in the article Dilatation structures I. Fundamentals, arXiv:math/0608536 [math.MG], but see the course notes for the most advanced formulation.  In particular regular sub-riemannian manifolds, riemannian manifolds and Lie groups with a left invariant distance induced by a completely non-integrable (i.e. generating) distribution are examples of such spaces.

There are two  problems, both without a clear solution yet, concerning this class of spaces:

  • Problem 1: how to define a good notion of curvature?  In particular, how to define an intrinsic notion of curvature for a regular sub-riemannian manifold, such that, for example, the curvature of a Carnot group is null (i.e. Carnot groups are flat)? On one side, I have a definition, the one about the curvature of a metric profile, which I think is good, but I have troubles computing it for particular examples. On the other side, all other notions of curvature, like for example those coming from the Wasserstein distance, fail spectacularly for sub-riemannian spaces.
  • Problem 2: for sub-riemannian Lie groups, or more general for groups with dilations  (i.e. left-invariant dilation structures on a topological group), how to define a good generalization of a Lie bracket? As I explained in the posts about the noncommutative BCH formula, this problem comes from the two ways we may interpret the Lie bracket. The first way, classical, says that the Lie bracket is an object which measures the noncommutativity of the group operation. This is in fact a statement which applies not to the Lie bracket, but to the commutator. The Lie bracket itself is related to the second order variation of the commutator (i.e.  the commutator of two elements of the group, which are \varepsilon-small, equals \varepsilon^{2} times the Lie bracket of those elements plus terms of higher order in \varepsilon. The trouble with this definition of the Lie bracket is that if we replace the “usual” dilations associated to a Lie group, those coming from one-parameter subgroups, by  more general dilations then all the reasoning crushes in at least two places. The first place is that in the tangent space at the identity of the group we have a noncommutative (but nilpotent) addition operation instead of the commutative plus operation, i.e. a Carnot group instead of a vector space structure, which makes hard to understand what sense the BCH formula makes. The second place is that instead of  the \varepsilon^{2}  term, there is nothing clear about the “expansion” of the commutator with respect to \varepsilon in this more general case.  Another interpretation of the Lie bracket comes from the fact that a half of the Lie bracket  measures the speed of the deformation of the group operation by dilations, at \varepsilon = 0. So, in this  interpretation, the half of the Lie bracket appears as the first order variation of the deformed group operation. This can be generalized to the more general case of a group with dilations and it leads to the notion of a halfbracket.

My purpose is to explain a link between the curvature (defined as the curvature of a metric profile) and the halfbracket.

With the notations from dilation structures, we know that for any x, u, v sufficiently close and any \varepsilon > 0  sufficiently small we have the uniform estimate

d^{x}_{\varepsilon}(u,v) - d^{x}(u,v) = O(\varepsilon)

Here d^{x}(u,v)  is the distance between u and v, seen as elements of the tangent space at x. The quantity

d^{x}_{\varepsilon}(u,v)\frac{1}{\varepsilon} d(\delta^{x}_{\varepsilon} u , \delta^{x}_{\varepsilon} v)

is the deformation of the distance d by dilations \delta^{x}_{\varepsilon}, centered at x, of coefficient \varepsilon.

Let’s take, as an example, the case of a riemannian manifold with geodesic exponential \exp and dilations defined by:

\delta^{x}_{\varepsilon} \exp_{x} u = \exp_{x} (\varepsilon u).

In this case

d^{x}_{\varepsilon}(u,v) - d^{x}(u,v) = \varepsilon^{2} (M + O(\varepsilon))

where M is related to the sectional curvature at x (we suppose that u, v are not collinear).  This gives a curvdimension equal to 2 and also a notion of sectional curvature.

But in general all we can hope is an estimate of the form

d^{x}_{\varepsilon}(u,v) - d^{x}(u,v) = \varepsilon^{\alpha} (M + O(\varepsilon))

where \alpha > 0 is the curvdimension, and the trick is to have an estimate for the curvdimension. In the next post I shall use halfbrackets in order to estimate the curvdimension, for sub-riemannian Lie groups.

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5 thoughts on “Curvature and halfbrackets, part I”

    1. Yes, I know about this work, is very interesting, but is only half intrinsic. In my opinion, if I remember well, the problem of that approach is that somewhere they separate a certain quantity which is like a second order derivative into vertical and horizontal parts. That’s the non-intrinsic part, at least according to my definition of intrinsic.

  1. Sorry for the short post, in fact the draft was much longer and informative, but for a reason I cannot fathom, I get lots and lots of parsing errors for the latex expressions which look allright. Therefore at some point I just shortened the post and left the rest for a future one. Maybe a good subject for another future post (or posts) would be a critical enumeration of all proposed notions of curvature which I know about, along with comments concerning their applications in sub-riemannian geometry, what is right and what goes wrong. However, I am still rather reluctant to do so, I have to think more about the right manner which could stir some useful discussion. But it’s an interesting subject, one which could become hot, if for example theoretical physicists start to be interested in metric geometry and analysis.

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