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\'echet 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\'echet mean, restricts to an isometry on the "fluctuating cone" of directions in which the Fr\'echet mean can vary under perturbation of $\mu$, and preserves angles between arbitrary and fluctuating tangent vectors at the Fr\'echet mean.

翻译：任何在CAT(k)空间M上定义的测度μ，若该空间可分解为有限个流形的并集，且在其Fr\'echet均值$\bar\mu$附近具有局部指数映射，则存在从M在$\bar\mu$处的切锥到某个向量空间的连续"切向坍缩"映射。该映射保持Fr\'echet均值不变，在可随μ扰动变化的"波动锥"方向上限制为等距映射，并保持Fr\'echet均值处任意切向量与波动切向量之间的夹角。