Logarithmic Velocity Alignment on Riemannian Manifolds

Abstract

We introduce a logarithmic velocity alignment model for collective motion on Riemannian manifolds. The interaction law is defined by the covariant time derivative of logarithmic displacement vectors and therefore provides an intrinsic geometric analogue of pairwise velocity-difference coupling. Under two-sided nonpositive sectional-curvature bounds and a compatibility condition between the interaction kernel and the curvature-induced growth of logarithmic terms, we derive a kinetic-energy dissipation estimate and prove interaction-weighted asymptotic velocity alignment, assuming that the logarithmic interactions remain globally well-defined and that transported velocity discrepancies have uniformly controlled time variation. In hyperbolic space, both assumptions are verified directly from the geometry and the energy estimates, yielding an unconditional interaction-weighted alignment result within the stated class of global solutions.