Average signature of geodesic paths in compact Lie groups

Source abstract

Abstract For any compact connected 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 using the average signature together with the trace operation with respect to the given bi‐invariant Riemannian metric on , one can recover certain geometric quantities of , including the dimension, the diameter, the volume, and the scalar curvature.