In this work, we consider the problem of global exponential synchronization of a network of identical inputaffine nonlinear time-varying systems. To this end, we tackle the problem with incremental stability tools. We propose sufficient metric-based conditions to design a distributed diffusive coupling feedback law in two frameworks. First, we consider the Euclidean scenario, where the network is assumed to be connected. Second, we tackle the Riemannian framework by assuming the presence of a leader for an undirected network. This allows considering more general systems with significant nonlinearities. In both scenarios, we propose two different control laws: a full-state feedback and a static output feedback controller. Then, we apply our design to several cases-studies. To conclude, we propose an algorithm based on deep neural networks (DNNs) to practically implement such controllers.