  We propose a generalization of nonlinear stability of numerical one-step
integrators to Riemannian manifolds in the spirit of Butcher's notion of
B-stability. Taking inspiration from Simpson-Porco and Bullo, we introduce
non-expansive systems on such manifolds and define B-stability of integrators.
In this first exposition, we provide concrete results for a geodesic version of
the Implicit Euler (GIE) scheme. We prove that the GIE method is B-stable on
Riemannian manifolds with non-positive sectional curvature. We show through
numerical examples that the GIE method is expansive when applied to a certain
non-expansive vector field on the 2-sphere, and that the GIE method does not
necessarily possess a unique solution for large enough step sizes. Finally, we
derive a new improved global error estimate for general Lie group integrators.

Numerical Analysis;Numerical Analysis