Asymptotic expansion of holonomy

Erlend Grong; Pierre Pansu

doi:10.1515/forum-2017-0029

Article

Asymptotic expansion of holonomy

Erlend Grong and Pierre Pansu

Published/Copyright: June 30, 2018

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From the journal Forum Mathematicum Volume 30 Issue 6

Abstract

Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an asymptotic formula that is independent of choice of gauge. We also show how our results from sub-Riemannian geometry can give improved approximations for the case of studying expansions of holonomy of principal bundles over the Euclidean space.

Keywords: Asymptotic expansion of holonomy; radial gauge; sub-Riemannian geometry; horizontal holonomy

MSC 2010: 53C29; 41A99; 53C17

Communicated by Anna Wienhard

Funding source: Norges Forskningsråd

Award Identifier / Grant number: 249980/F20

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: project SRGI

Award Identifier / Grant number: ANR-15-CE40-0018

Funding source: Fonds National de la Recherche Luxembourg

Award Identifier / Grant number: AFR 4736116

Award Identifier / Grant number: OPEN Project GEOMREV

Funding statement: The first author is supported by the Fonds National de la Recherche Luxembourg (AFR 4736116 and OPEN Project GEOMREV) and by the Research Council of Norway (project number 249980/F20). The second author acknowledges support from ANR, project SRGI, ANR-15-CE40-0018.

A Approximation of ODEs in Lie groups

Let G be a connected Lie group with Lie algebra 𝔤. For any continuous curve [0,t1]→𝔤, t↦A⁢(t), consider the initial value problem on G:

a˙⁢(t)=A⁢(t)⋅a⁢(t),a⁢(0)=1.

Assume that for any t∈[0,t1] the curve Q⁢(t)=exp-1⁡a⁢(t) in 𝔤 is well defined. Note that if exp:𝔤→G is the group exponential, then

d⁢exp:TA⁢𝔤→Texp⁡A⁢G,B↦exp⁡(A)⋅f⁢(ad⁡A)⁢B,f⁢(z)=1-e-zz.

Hence,

f⁢(ad⁡Q⁢(t))⁢Q˙⁢(t)=A⁢(t),Q⁢(0)=0,

and

(A.1)Q˙⁢(t)=g⁢(ad⁡Q⁢(t))⁢A⁢(t),g⁢(z)=z1-e-z.

Consider 𝔤 as a subalgebra of 𝔤⁢𝔩⁢(q,ℂ) for some q>0 and let ∥⋅∥ be a Banach algebra norm on 𝔤⁢𝔩⁢(q,ℂ). Write

∥A∥L∞⁢[0,t]=sups∈[0,t]⁡∥A⁢(s)∥.

Lemma A.1.

Define constants B1 and B2 such that on the disk {z∈C:|z|≤π},

|g⁢(z)|≤B1,|g′⁢(z)|≤B2.

Define

∥ad∥=sup⁡{∥ad⁡(A1)⁢A2∥:∥A1∥=∥A2∥=1}.

Then there exists ε>0, depending on B1, B2 and ∥a⁢d∥, such that for any A:[0,T]→g with

t⁢∥A∥L∞⁢[0,t]<ε

we have

∥Q⁢(t)-∫0tA⁢(s)⁢𝑑s∥≤∥ad∥⁢t2⁢B221-∥ad∥⁢B2⁢ε⁢∥A∥L∞⁢[0,t]2.

Proof.

We want to estimate Q⁢(t) using the Picard iteration. For any t>0, write Mt=sups∈[0,t]⁡∥A⁢(s)∥. If q:[0,T]→𝔤 is an arbitrary continuous curve, then

𝒜⁢(q)⁢(t)=∫0tg⁢(ad⁡q⁢(s))⁢A⁢(s)⁢𝑑s.

Note that

∥𝒜⁢(q)∥L∞⁢[0,t]≤t⁢Mt⁢B1 whenever ⁢∥q∥L∞⁢[0,t]≤π∥ad∥,

and for any pair of curves q1 and q2 we have

∥𝒜⁢(q1-q2)∥L∞⁢[0,t]≤t⁢Mt⁢B2⁢∥ad∥⁢∥q1-q2∥L∞⁢[0,t].

Define B=max⁡{B1⁢∥ad∥/π,B2⁢∥ad∥} and assume that t⁢Mt<1B. Then it follows that Q⁢(t)=limn→∞⁡𝒜n⁢(0) is the solution of (A.1) and

∥Q⁢(t)-𝒜⁢(0)⁢(t)∥≤∥Q-𝒜⁢(0)∥L∞⁢[0,t]≤t⁢Mt⁢B2⁢∥ad∥⁢∥Q⁢(t)∥L∞⁢[0,t]≤t2⁢Mt2⁢B221-t⁢Mt⁢B2.

The result follows from choosing ε≤1B. ∎

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Received: 2017-02-09

Revised: 2018-02-26

Published Online: 2018-06-30

Published in Print: 2018-11-01

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Articles in the same Issue

https://doi.org/10.1515/forum-2017-0029

Keywords for this article

Asymptotic expansion of holonomy; radial gauge; sub-Riemannian geometry; horizontal holonomy