In the learning of existing radial basis function neural networks-based methods, it is difficult to propagate errors back. This leads to an inconsistency between the learning and recognition task. This article proposes a geodesic basis function neural network with subclass extension learning (GBFNN-ScE). The geodesic basis function (GBF), which is defined here for the first time, uses the geodetic distance in the manifold as a measure to obtain the response of the sample with respect to the local center. To learn network parameters by back-propagating errors for the purpose of correct classification, a specific GBF based on a pruned gamma encoding cosine function is constructed. This function has a concise and explicit expression on the hyperspherical manifold, which is conducive to the realization of error back propagation. In the preprocessing layer, a sample unitization method with no loss of information, nonnegative unit hyperspherical crown (NUHC) mapping, is proposed. The sample can be mapped to the support set of the GBF. To alleviate the problem that one-hot encoding is not effective enough in the differential expression of data labels within a class, a subclass extension (ScE) learning strategy is proposed. The ScE learning strategy can help the learned network be more robust. For the working of GBFNN-ScE, the original sample is projected onto the support set of GBF through the NUHC mapping. Then the mapped samples are sent to the nonlinear computation units composed of GBFs in the hidden layer. Finally, the response obtained in the hidden layer is weighted by the learned weight to obtain the network output. This article theoretically proves that the separability of the data with ScE learning is stronger. The experimental results show that the proposed GBFNN-ScE has a better performance in recognition tasks than existing methods. The ablation experiments show that the ideas of the GBFNN-ScE contribute to the algorithm performance.