We derive nonlinear stability results for numerical integrators on Riemannian manifolds, by imposing conditions on the ODE vector field and the step size that makes the numerical solution non-expansive whenever the exact solution is non-expansive over the same time step. Our model case is a geodesic version of the explicit Euler method. Precise bounds are obtained in the case of Riemannian manifolds of constant sectional curvature. The approach is based on a cocoercivity property of the vector field adapted to manifolds from Euclidean space. It allows us to compare the new results to the corresponding well-known results in flat spaces, and in general we find that a non-zero curvature will deteriorate the stability region of the geodesic Euler method. The step size bounds depend on the distance traveled over a step from the initial point. Numerical examples for spheres and hyperbolic 2-space confirm that the bounds are tight.

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