To be tractable and robust to data noise, existing metric learning algorithms commonly rely on PCA as a pre-processing step. How can we know, however, that PCA, or any other specific dimensionality reduction technique, is the method of choice for the problem at hand? The answer is simple: We cannot! To address this issue, in this paper, we develop a Riemannian framework to jointly learn a mapping performing dimensionality reduction and a metric in the induced space. Our experiments evidence that, while we directly work on high-dimensional features, our approach yields competitive runtimes with and higher accuracy than state-of-the-art metric learning algorithms.