Detecting communities on graphs has received significant interest in recent literature. Current state-of-the-art approaches tackle this problem by coupling Euclidean graph embedding with community detection. Considering the success of hyperbolic representations of graph-structured data in the last years, an ongoing challenge is to set up a hyperbolic approach to the community detection problem. The present paper meets this challenge by introducing a Riemannian geometry based framework for learning communities on graphs. The proposed methodology combines graph embedding on hyperbolic spaces with Riemannian K-means or Riemannian mixture models to perform community detection. The usefulness of this framework is illustrated through several experiments on generated community graphs and real-world social networks as well as comparisons with the most powerful baselines. The code implementing hyperbolic community embedding is available online https://www.github.com/tgeral68/HyperbolicGraphAndGMM.