We explore the use of a topological manifold, represented as a collection of charts, as the target space of neural network based representation learning tasks. This is achieved by a simple adjustment to the output of an encoder's network architecture plus the addition of a maximal mean discrepancy (MMD) based loss function for regularization. Most algorithms in representation and metric learning are easily adaptable to our framework and we demonstrate its effectiveness by adjusting SimCLR (for representation learning) and standard triplet loss training (for metric learning) to have manifold encoding spaces. Our experiments show that we obtain a substantial performance boost over the baseline for low dimensional encodings. In the case of triplet training, we also find, independent of the manifold setup, that the MMD loss alone (i.e. keeping a flat, euclidean target space but using an MMD loss to regularize it) increases performance over the baseline in the typical, high-dimensional Euclidean target spaces. Code for reproducing experiments is provided at https://github.com/ekorman/neurve .