Geometric Control of Moving Parallel Transport in Riemannian Cucker--Smale Dynamics with Bonding Forces

What the math gives to ML

The paper provides a nonstandard geometric stability mechanism for consensus dynamics when states live on a curved manifold: velocity discrepancies must be compared by moving parallel transport, whose time derivative contains curvature-dependent Jacobi-field terms. A bonding potential confines pairwise states below the injectivity radius, while endpoint estimates control transport-variation errors and convert integrable dissipation into asymptotic alignment. This can transfer to manifold-valued neural layers, graph latent-state models, and Riemannian optimizer dynamics by replacing Euclidean consensus with transport-aware consensus plus an explicit distance barrier and curvature monitor.