  In this work, we consider the bilevel optimization problem on Riemannian
manifolds. We inspect the calculation of the hypergradient of such problems on
general manifolds and thus enable the utilization of gradient-based algorithms
to solve such problems. The calculation of the hypergradient requires utilizing
the notion of Riemannian cross-derivative and we inspect the properties and the
numerical calculations of Riemannian cross-derivatives. Algorithms in both
deterministic and stochastic settings, named respectively RieBO and RieSBO, are
proposed that include the existing Euclidean bilevel optimization algorithms as
special cases. Numerical experiments on robust optimization on Riemannian
manifolds are presented to show the applicability and efficiency of the
proposed methods.
