Average signature of geodesic paths in compact Lie groups

Chong Liu; Shi Wang

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arXiv · 2411.06760

Average signature of geodesic paths in compact Lie groups

Chong Liu·Shi Wang

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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.

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Chong Liu, Shi Wang. 2024-11-11. Average signature of geodesic paths in compact Lie groups. https://arxiv.org/abs/2411.06760

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