  This paper presents strong worst-case iteration and operation complexity
guarantees for Riemannian adaptive regularized Newton methods, a unified
framework encompassing both Riemannian adaptive regularization (RAR) methods
and Riemannian trust region (RTR) methods. We comprehensively characterize the
sources of approximation in second-order manifold optimization methods: the
objective function's smoothness, retraction's smoothness, and subproblem
solver's inexactness. Specifically, for a function with a $\mu$-H\"older
continuous Hessian, when equipped with a retraction featuring a $\nu$-H\"older
continuous differential and a $\theta$-inexact subproblem solver, both RTR and
RAR with $2+\alpha$ regularization (where $\alpha=\min\{\mu,\nu,\theta\}$)
locate an $(\epsilon,\epsilon^{\alpha/(1+\alpha)})$-approximate second-order
stationary point within at most $O(\epsilon^{-(2+\alpha)/(1+\alpha)})$
iterations and at most $\tilde{O}(\epsilon^{-(4+3\alpha)/(2(1+\alpha))})$
Hessian-vector products. These complexity results are novel and sharp, and
reduce to an iteration complexity of $O(\epsilon^{-3/2})$ and an operation
complexity of $\tilde{O}(\epsilon^{-7/4})$ when $\alpha=1$.
