  The gradient method for minimize a differentiable convex function on
Riemannian manifolds with lower bounded sectional curvature is analyzed in this
paper. The analysis of the method is presented with three different finite
procedures for determining the stepsize, namely, Lipschitz stepsize, adaptive
stepsize and Armijo's stepsize. The first procedure requires that the objective
function has Lipschitz continuous gradient, which is not necessary for the
other approaches. Convergence of the whole sequence to a minimizer, without any
level set boundedness assumption, is proved. Iteration-complexity bound for
functions with Lipschitz continuous gradient is also presented. Numerical
experiments are provided to illustrate the effectiveness of the method in this
new setting and certify the obtained theoretical results. In particular, we
consider the problem of finding the Riemannian center of mass and the so-called
Karcher's mean. Our numerical experiences indicate that the adaptive stepsize
is a promising scheme that is worth considering.
