Differential Constraints in Chaotic Flows on Curved Manifolds

The Lagrangian derivatives of finite-time Lyapunov exponents and the corresponding characteristic directions are shown to satisfy time-asymptotic differential constraints in chaotic flows. The constraints are valid for any metric tensor, and are realised with exponential accuracy in time. Some of these constraints were derived previously for chaotic systems on low-dimensional Euclidean spaces, by requiring that the Riemann curvature tensor vanish in Lagrangian coordinates. The new derivation applies in any number of dimensions, predicts the number of constraints for a given flow, and provides a rigorous convergence rate of the constraints.