The dynamics of neuron populations commonly evolve on low-dimensional manifolds. Thus, we need methods that learn the dynamical processes over neural manifolds to infer interpretable and consistent latent representations. We introduce a representation learning method, MARBLE, which decomposes on-manifold dynamics into local flow fields and maps them into a common latent space using unsupervised geometric deep learning. In simulated nonlinear dynamical systems, recurrent neural networks and experimental single-neuron recordings from primates and rodents, we discover emergent low-dimensional latent representations that parametrize high-dimensional neural dynamics during gain modulation, decision-making and changes in the internal state. These representations are consistent across neural networks and animals, enabling the robust comparison of cognitive computations. Extensive benchmarking demonstrates state-of-the-art within- and across-animal decoding accuracy of MARBLE compared to current representation learning approaches, with minimal user input. Our results suggest that a manifold structure provides a powerful inductive bias to develop decoding algorithms and assimilate data across experiments.