<p>Outlier detection for manifold-valued functional data has garnered increasing research interest in machine learning communities due to its broad applications. However, the inherent nonlinear structures of manifolds render conventional linear operators ineffective, posing significant challenges for outlier detection on Riemannian manifolds. This paper develops a novel outlier detection framework for Riemannian manifold-valued functional data. First, we identify a clean subset of observations by estimating a robust Fréchet mean function and implementing a max-min clean subset selection procedure. Subsequently, we construct an outlyingness measure on the tangent space of the estimated robust Fréchet mean function, followed by the development of a new multiple hypothesis testing procedure that incorporates false discovery rate control. Numerical simulations demonstrate that our method exhibits superior accuracy in outlier detection compared to existing approaches that overlook the underlying nonlinear manifold structure. An application to real-world long-haul flight trajectories, which are modeled as curves on a two-dimensional sphere embedded in a three-dimensional Euclidean space, is also provided to illustrate the efficiency. By accounting for the Earth’s surface curvature instead of approximating trajectories as straight lines in flat space, our framework provides a more geometrically faithful solution for outlier detection in such manifold-valued functional data.</p>