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We introduce a Markov chain for sampling from the uniform distribution on a Riemannian manifold $mathcal{M}$, which we call the $textit{geodesic walk}$. We prove that the mixing time of this walk on any manifold with positive sectional curvature $C_{x}(u,v)$ bounded both above and below by $0 u003c mathfrak{m}_{2} leq C_{x}(u,v) leq mathfrak{M}_2 u003c infty$ is $mathcal{O}^*left(frac{mathfrak{M}_2}{mathfrak{m}_2}right)$. In particular, this bound on the mixing time does not depend explicitly on the dimension of the manifold. In the special case that $mathcal{M}$ is the boundary of a convex body, we give an explicit and computationally tractable algorithm for approximating the exact geodesic walk. As a consequence, we obtain an algorithm for sampling uniformly from the surface of a convex body that has running time bounded solely in terms of the curvature of the body.

Rapid mixing of geodesic walks on manifolds with positive curvature