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arXiv · 2607.26748

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

Hyunjin Ahn·Woojoo Shim
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Abstract

We study a Cucker--Smale type system with bonding forces on complete Riemannian manifolds with uniformly bounded curvature. On general manifolds, the time variation of parallel transport between moving agents produces curvature-dependent terms, so the standard energy-dissipation argument does not directly yield asymptotic velocity alignment. The bonding energy confines all pairwise distances below the injectivity radius, providing global well-posedness and time integrability of the transported velocity discrepancies. To overcome the remaining geometric obstruction, we combine the variation formula for parallel transport with a uniform endpoint estimate for Jacobi fields along moving minimizing geodesics. This yields the uniform regularity needed to convert energy dissipation into asymptotic alignment. Under an energy-dependent injectivity condition and a positive communication bound on the dynamically relevant distance range, we establish asymptotic flocking. Numerical simulations illustrate the resulting dynamics in a nonconstant-sectional-curvature setting.

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Hyunjin Ahn, Woojoo Shim. 2026-07-29. Geometric Control of Moving Parallel Transport in Riemannian Cucker--Smale Dynamics with Bonding Forces. https://arxiv.org/abs/2607.26748

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