  In this paper we present a cubic regularized Newton's method to minimize a
smooth function over a Riemannian manifold. The proposed algorithm is shown to
reach a second-order $\epsilon$-stationary point within
$\mathcal{O}(1/\epsilon^{\frac{3}{2}})$ iterations, under the condition that
the pullbacks are locally Lipschitz continuous, a condition that is shown to be
satisfied if the manifold is compact. Furthermore, we present a local
superlinear convergence result under some additional conditions.
