Geometry of measures on smoothly stratified metric spaces

Any measure $\mu$ on a CAT(k) space M that is stratified as a finite union of manifolds and has local exponential maps near the Fréchet mean $\bar{\mu }$ yields a continuous “tangential collapse” from the tangent cone of M at $\bar{\mu }$ to a vector space that preserves the Fréchet mean, restricts to an isometry on the “fluctuating cone” of directions in which the Fréchet mean can vary under perturbation of $\mu$, and preserves angles between arbitrary and fluctuating tangent vectors at the Fréchet mean.