2026/06/12 by Karl-Theodor Sturm
#math.DG #math.FA #math.MG #math.PR
We derive an explicit formula for the sectional curvature of the space \cal M(M) of finite measures on a Riemannian manifold M. The space \cal M(M) is equipped with the Hellinger-Kantorovich metric HK. Even in the case M=Rn, the curvature is comprised of two parts: the `lifted part' is negative, and the `twisted part' is positive. It will be analyzed in detail for the multidimensional torus. Our general approach to sectional curvature in geodesic spaces also leads to new insights into the curvature of the space P2(M) of probability measures on M equipped with the Kantorovich-Wasserstein metric W2.