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

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.