  This paper studies the robust isolated calmness property of the KKT solution
mapping of a class of nonsmooth optimization problems on Riemannian manifold.
The manifold version of the Robinson constraint qualification, the strict
Robinson constraint qualification, and the second order conditions are defined
and discussed. We show that the robust isolated calmness of the KKT solution
mapping is equivalent to the M-SRCQ and M-SOSC conditions. Furthermore, under
the above two conditions, we show that the Riemannian augmented Lagrangian
method has a local linear convergence rate. Finally, we verify the proposed
conditions and demonstrate the convergence rate on two minimization problems
over the sphere and the manifold of fixed rank matrices.
