The goal of this paper is to learn the differential geometry of pose image manifolds for 3D objects. Indexed by the rotation group SO(3), a pose manifold constitutes images of a 3D object from all viewing angles. Learning geometry implies computing geodesics, intrinsic statistics (means, etc), and curvatures on estimated manifolds. As these goals are unattainable in the huge image space, we perform dimension reduction that is geometry preserving and invertible. This paper introduces two distinct concepts: (1) A Geometry-Preserving StyleGAN (GP-StyleGAN2) that maps training images to a low-dimensional latent space with two novel geometry-preserving terms. These terms penalize changes in pairwise distances between points and pairwise angles between tangent spaces under the map. (2) Densifying the estimated manifold in latent space using Euler’s Elasticae-based nonlinear interpolations between sparse data points. In contrast to the past findings, the latent pose manifolds are found to be distinctly nonlinear and similar in shape across objects. Incorporating these features results in superior performance in image interpolation, denoising, and computing image summaries when compared to state-of-the-art GANs and VAEs.