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Average signature of geodesic paths in compact Lie groups

Chong Liu, Shi Wang

Abstract

For any compact 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 the trace spectrum of $\mathbb A(G)$ recovers certain geometric quantities of $G$, including the dimension, the diameter, the volume and the scalar curvature.

Average signature of geodesic paths in compact Lie groups

Abstract

For any compact Lie group , we introduce a novel notion of average signature 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 . We prove that the trace spectrum of recovers certain geometric quantities of , including the dimension, the diameter, the volume and the scalar curvature.

Paper Structure

This paper contains 5 sections, 17 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Signature on Lie groups
  3. The average signature
  4. Trace spectrum of the average signature
  5. The product structure

Key Result

Theorem 1.4

Let $G$ be a compact Lie group with a bi-invariant Riemannian metric. Denote $\overline{\mu}$ the normalized Haar measure of $G$ such that $\overline{\mu}(G)=1$. Given the trace spectrum of the average signature $\mathop{\mathrm{tr}}\nolimits(\mathbb A(G))\in \mathbb R^\infty$, then we can recover in an explicit manner.

Theorems & Definitions (43)

  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 33 more