Consistent Curvature Approximation on Riemannian Shape Spaces

Abstract

We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time discretization of geodesic calculus presented in Rumpf and Wirth (2015), which just requires an approximation of the squared Riemannian distance that is typically easy to compute. First we obtain first order discrete covariant derivatives via a Schild's ladder type discretization of parallel transport. Second order discrete covariant derivatives are then computed as nested first order discrete covariant derivatives. These finally give rise to an approximation of the curvature tensor. First and second order consistency are proven for the approximations of the covariant derivative and the curvature tensor. The findings are experimentally validated on two-dimensional surfaces embedded in $\mathbb{R}^3$. Furthermore, as a proof of concept the method is applied to the shape space of triangular meshes, and discrete sectional curvature indicatrices are computed on low-dimensional vector bundles.