Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below

Abstract

Abstract We show that in any infinitesimally Hilbertian 𝖢𝖣 * ( K , N ) $\mathsf {CD}^*(K,N)$ -space at almost every point there exists a Euclidean weak tangent, i.e., there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov–Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian 𝖢𝖣 * ( 0 , N ) $\mathsf {CD}^*(0,N)$ -spaces.