Average signature of geodesic paths in compact Lie groups

TL;DR

This paper introduces a new concept called the average signature for geodesic paths in compact Lie groups, enabling the recovery of key geometric properties from these signatures.

Contribution

It defines the average signature in compact Lie groups and demonstrates how it can be used to determine geometric quantities like dimension, diameter, volume, and scalar curvature.

Findings

Average signature encodes geometric information of the Lie group.

Using the average signature and trace, one can recover the group's dimension and diameter.

The method applies to all compact connected Lie groups.

Abstract

For any compact connected Lie group $G$, we introduce a novel notion of average signature $\mathbb A(G)$ valued in its tensor Lie algebra, by taking the average value of the signature of the unique length-minimizing geodesics between all pairs of generic points in $G$. we prove that using the average signature together with the trace operation with respect to the given bi-invariant Riemannian metric on $G$, one can recover certain geometric quantities of $G$, including the dimension, the diameter, the volume and the scalar curvature.