  The neural manifold hypothesis postulates that the activity of a neural
population forms a low-dimensional manifold whose structure reflects that of
the encoded task variables. In this work, we combine topological deep
generative models and extrinsic Riemannian geometry to introduce a novel
approach for studying the structure of neural manifolds. This approach (i)
computes an explicit parameterization of the manifolds and (ii) estimates their
local extrinsic curvature--hence quantifying their shape within the neural
state space. Importantly, we prove that our methodology is invariant with
respect to transformations that do not bear meaningful neuroscience
information, such as permutation of the order in which neurons are recorded. We
show empirically that we correctly estimate the geometry of synthetic manifolds
generated from smooth deformations of circles, spheres, and tori, using
realistic noise levels. We additionally validate our methodology on simulated
and real neural data, and show that we recover geometric structure known to
exist in hippocampal place cells. We expect this approach to open new avenues
of inquiry into geometric neural correlates of perception and behavior.
