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 Fr\échet mean $\bar\mu$ yields a continuous "tangential collapse" from the tangent cone of M at $\bar\mu$ to a vector space that preserves the Fr\échet mean, restricts to an isometry on the "fluctuating cone" of directions in which the Fr\échet mean can vary under perturbation of $\mu$, and preserves angles between arbitrary and fluctuating tangent vectors at the Fr\échet mean.