  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.
