Notes of Erlend Grong’s Orsay lecture 27-06-2017

Asymptotic expansions of holonomy

Joint with Pierre Pansu.

1. Motivation

Given a connection on a principal bundle {G\rightarrow P\rightarrow M}, holonomy along a based loop {\gamma} of {M} is an element of {G} resulting from lifting horizontally {\gamma} to {P}. We look for an expression {F(\gamma)\in\mathfrak{g}} such that {\exp(F(\gamma))} is a good approximation of holonomy when {\gamma} is short,

\displaystyle  \begin{array}{rcl}  hol(\gamma)=\exp(F+O(\ell^r)). \end{array}

We want that {F(\gamma)} be simpler to compute than holonomy, and be related to curvature.

Hatton-Choset: motion of a snake with two joins. {M=S^1\times S^1}, {G=SE(2)}. Experimentalists have been led to choose the Coulomb gauge, and for {F(\gamma)} the integral over a disk spanning {\gamma} of curvature expressed in Coulomb gauge.

In this practical example, motions are tangent to a sub-bundle of the tangent bundle of {M}. Hence our interest in expansions which are particularly efficient on such curves. We call this setting sub-Riemannian.

Sub-Riemannian curvature is not easy to define. The obvious approach of using adapted connections on the tangent bundle is not illuminating.

2. Results

  1. Asymptotic, gauge-free formula in Euclidean space.
  2. Riemannian case not that different.
  3. Sub-Riemannian case suggests a notion of curvature.
  4. For certain sub-Riemannian structures,

2.1. Euclidean case

Dilations define radial fillings {disk(\gamma)} of loops. Use radial gauge (frame is parallel along rays through the origin). They turn out to be optimal. Using radial gauge, integrate curvature over radial filling. This defines

\displaystyle F(\gamma)=\int_{\mathrm{disk}(\gamma)}\Omega.

Say a differential form {\omega} has weight {\geq m} if dilates {\delta_s^*\omega} are {O(t^m)}. Use radial gauge to define weight of forms on {P}.

Theorem 1 If the curvature has weight {m}, then

\displaystyle  \begin{array}{rcl}  hol(\gamma)=\exp(F(\gamma)+O(\ell(\gamma)^2m). \end{array}

Furthermore, one can expand {F(\gamma)} in termes of Taylor’s expansion of curvature.

Since curvature has weight at least 2, one gets a 4-th order approximation.

2.2. Sub-Riemannian case

The flat sub-Riemannian case corresponds to Carnot groups, i.e. a Lie group whose Lie algebra has a gradation

\displaystyle  \begin{array}{rcl}  \mathfrak{n}=\mathfrak{n}_1\oplus\cdots\oplus \mathfrak{n}_r \end{array}

and is generated by {\mathfrak{n}_1}. Example: Heisenberg group.

Fix a norm on {\mathfrak{n}_1}. Left translates of {\mathfrak{n}_1} define a sub-Riemannian metric, for which dilations {\delta_t=t^i} on {\mathfrak{n}_i} are homothetic.

According to Le Donne, sub-Riemannian Carnot groups are characterized by being the only locally compact homogeneous geodesic metric spaces with homothetic homeos.

Carnot groups come with a left-invariant horizontal basis, we pick a connection on the tangent bundle which makes it parallel. It has torsion. We combine it with the principal bundle connection to define iterated covariant derivatives of curvature. We organize them according to weights adapted to the Lie algebra grading. The above theorem extends.

2.3. Horizontal holonomy

Since we are interested only in holonomy along horizontal loops, we have the freedon to change the connection outside the horizontal subbundle.

Chitour-Grong-Jean-Kokkonen: using this freedom, there are choices which minimize the curvature in the sense that as many components as possible vanish identically. This tends to increase the weight of curvature.

Example: on 3-dimensional Heisenberg group, the preferred connection has curvature which vanishes on the horizontal distribution, hence has weight {\geq 3} instead of 2. Above Theorem provides a 6-th order expansion, whose terms can be computed algebraically.

More generally, on free {k}-step nilpotent Lie groups, the curvature of a preferred connection has order at least {k+1}, whence a {2k+2}-th order expansion whose terms are linear in curvature (in fact, in the preferred curvature).

We expect to use it to refine the Euclidean expansion.

3. Question

What does this give in case of the two-joints snake? Requires to push computations further.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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