This study provides a computable, direct, and mathematically rigorous approximation to the differential geometry of class manifolds for high-dimensional data, along with non-linear projections from input space onto these class manifolds. The tools are applied to the setting of neural network image classifiers, where we generate novel, on-manifold data samples and implement a projected gradient descent algorithm for on-manifold adversarial training. The susceptibility of neural networks (NNs) to adversarial attack highlights the brittle nature of NN decision boundaries in input space. Introducing adversarial examples during training has been shown to reduce the susceptibility of NNs to adversarial attack; however, it has also been shown to reduce the accuracy of the classifier if the examples are not valid examples for that class. Realistic “on-manifold” examples have been previously generated from class manifolds in the latent space of an autoencoder. Our study explores these phenomena in a geometric and computational setting that is much closer to the raw, high-dimensional input space than what can be provided by VAE or other black box dimensionality reductions. We employ conformally invariant diffusion maps (CIDM) to approximate class manifolds in diffusion coordinates and develop the Nyström projection to project novel points onto class manifolds in this setting. On top of the manifold approximation, we leverage the spectral exterior calculus (SEC) to determine geometric quantities such as tangent vectors of the manifold. We use these tools to obtain adversarial examples that reside on a class manifold, yet fool a classifier. These misclassifications then become explainable in terms of human-understandable manipulations within the data, by expressing the on-manifold adversary in the semantic basis on the manifold.