  The Riemannian Augmented Lagrangian Method (RALM), a recently proposed
algorithm for nonsmooth optimization problems on Riemannian manifolds, has
consistently exhibited high efficiency as evidenced in prior studies
\cite{ZBDZ21,ZBD22}. It often demonstrates a rapid local linear convergence
rate. However, a comprehensive local convergence analysis of the RALM under
more realistic assumptions, notably without the imposition of the uniqueness
assumption on the multiplier, remains an uncharted territory. In this paper, we
introduce the manifold variational sufficient condition and demonstrate that
its strong version is equivalent to the manifold strong second-order sufficient
condition (M-SSOSC) in certain circumstances. Critically, we construct a local
dual problem based on this condition and implement the Euclidean proximal point
algorithm, which leads to the establishment of the linear convergence rate of
the RALM. Moreover, we illustrate that under suitable assumptions, the M-SSOSC
is equivalent to the nonsingularity of the generalized Hessian of the augmented
Lagrangian function, which is an essential attribute for the semismooth
Newton-type methods.
