In the paper, we study a class of useful non-convex minimax optimization problems on Riemanian manifolds and propose a class of Riemanian gradient descent ascent algorithms to solve these minimax problems. Specifically, we propose a new Riemannian gradient descent ascent (RGDA) algorithm for the deterministic minimax optimization. Moreover, we prove that the RGDA has a sample complexity of O(κ2ϵ−2) for finding an ϵ-stationary point of the nonconvex strongly-concave minimax problems, where κ denotes the condition number. At the same time, we introduce a Riemannian stochastic gradient descent ascent (RSGDA) algorithm for the stochastic minimax optimization. In the theoretical analysis, we prove that the RSGDA can achieve a sample complexity of O(κ4ϵ−4). To further reduce the sample complexity, we propose a novel momentum variance-reduced Riemannian stochastic gradient descent ascent (MVR-RSGDA) algorithm based on a new momentum variance-reduced technique of STORM. We prove that the MVR-RSGDA algorithm achieves a lower sample complexity of O~(κ4ϵ−3) without large batches, which reaches near the best known sample complexity for its Euclidean counterparts. Extensive experimental results on the robust deep neural networks training over Stiefel manifold demonstrate the efficiency of our proposed algorithms.