Geodesic Dynamics for Constrained State-Space Models on Riemannian Manifolds

We present a geodesic dynamics framework for discrete-time state evolution on the unit sphere SN−1 that maintains exact unit-norm constraints through Riemannian exponential mapping. Given an input sequence and an initial state, the method constructs trajectories by projecting inputs to tangent spaces and updating states along geodesics, incorporating temporal memory via approximate parallel transport of velocity directions. Unlike traditional approaches requiring post hoc normalization of linear updates, the geodesic formulation preserves ∥xt∥=1 to machine precision while eliminating explicit N×N transition matrices in favor of D×N input embeddings when the intrinsic input dimension D is much smaller than the ambient dimension N. The update corresponds to a first-order exponential integrator on the sphere. We establish local Lipschitz continuity of the exponential map on positively curved manifolds with careful treatment of basepoint dependence, derive perturbation bounds showing linear-to-exponential growth transitions via Grönwall-type estimates, and we prove third-order asymptotic equivalence with normalized linear systems under appropriate scaling. Numerical experiments on synthetic data validate exact norm preservation over extended time horizons, confirm theoretical perturbation growth predictions, and demonstrate the effectiveness of the temporal memory mechanism in reducing long-horizon prediction errors. The framework provides a principled geometric approach for applications requiring exact directional or compositional constraints.