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Deep-learning architectures for classification problems involve the cross-entropy loss sometimes assisted with auxiliary loss functions like center loss, contrastive loss and triplet loss. These auxiliary loss functions facilitate better discrimination between the different classes of interest. However, recent studies hint at the fact that these loss functions do not take into account the intrinsic angular distribution exhibited by the low-level and high-level feature representations. This results in less compactness between samples from the same class and unclear boundary separations between data clusters of different classes. In this paper, we address this issue by proposing the use of geometric constraints, rooted in Riemannian geometry. Specifically, we propose Angular Margin Contrastive Loss (AMC-Loss), a new loss function to be used along with the traditional cross-entropy loss. The AMC-Loss employs the discriminative angular distance metric that is equivalent to geodesic distance on a hypersphere manifold such that it can serve a clear geometric interpretation. We demonstrate the effectiveness of AMC-Loss by providing quantitative and qualitative results. We find that although the proposed geometrically constrained loss-function improves quantitative results modestly, it has a qualitatively surprisingly beneficial effect on increasing the interpretability of deep-net decisions as seen by the visual explanations generated by techniques such as the Grad-CAM. Our code is available at https://github.com/hchoi71/AMC-Loss.