  We study the convergence issue for the gradient algorithm (employing general
step sizes) for optimization problems on general Riemannian manifolds (without
curvature constraints). Under the assumption of the local
convexity/quasi-convexity (resp. weak sharp minima), local/global convergence
(resp. linear convergence) results are established. As an application, the
linear convergence properties of the gradient algorithm employing the constant
step sizes and the Armijo step sizes for finding the Riemannian $L^p$
($p\in[1,+\infty)$) centers of mass are explored, respectively, which in
particular extend and/or improve the corresponding results in
\cite{Afsari2013}.
