  We give curvature-dependant convergence rates for the optimization of weakly
convex functions defined on a manifold of 1-bounded geometry via Riemannian
gradient descent and via the dynamic trivialization algorithm. In order to do
this, we give a tighter bound on the norm of the Hessian of the Riemannian
exponential than the previously known. We compute these bounds explicitly for
some manifolds commonly used in the optimization literature such as the special
orthogonal group and the real Grassmannian. Along the way, we present
self-contained proofs of fully general bounds on the norm of the differential
of the exponential map and certain cosine inequalities on manifolds, which are
commonly used in optimization on manifolds.
