The neural manifold hypothesis postulates that the ac-tivity of a neural population forms a low-dimensional man-ifold whose structure reflects that of the encoded task vari-ables. In this work, we combine topological deep generativemodels and extrinsic Riemannian geometry to introduce anovel approach for studying the structure of neural mani-folds. This approach (i) computes an explicit parameteriza-tion of the manifolds and (ii) estimates their local extrinsiccurvature—hence quantifying their shape within the neuralstate space. Importantly, we prove that our methodology isinvariant with respect to transformations that do not bearmeaningful neuroscience information, such as permutationof the order in which neurons are recorded. We show empir-ically that we correctly estimate the geometry of syntheticmanifolds generated from smooth deformations of circles,spheres, and tori, using realistic noise levels. We addition-ally validate our methodology on simulated and real neuraldata, and show that we recover geometric structure knownto exist in hippocampal place cells. We expect this approachto open new avenues of inquiry into geometric neural cor-relates of perception and behavior, while providing a newmeans to compare representations in biological and artifi-cial neural systems.