  We define the notions of absolute average and median treatment effects as
causal estimands on general metric spaces such as Riemannian manifolds, propose
estimators using stratification, and prove several properties, including strong
consistency. In the process, we also demonstrate the strong consistency of the
weighted sample Fr\'echet means and geometric medians. Stratification allows
these estimators to be utilized beyond the narrow constraints of a completely
randomized experiment. After constructing confidence intervals using
bootstrapping, we outline how to use the proposed estimates to test Fisher's
sharp null hypothesis that the absolute average or median treatment effect is
zero. Empirical evidence for the strong consistency of the estimators and the
reasonable asymptotic coverage of the confidence intervals is provided through
simulations in both randomized experiments and observational study settings. We
also apply our methods to real data from an observational study to investigate
the causal relationship between Alzheimer's disease and the shape of the corpus
callosum, rejecting the aforementioned null hypotheses in cases where
conventional Euclidean methods fail to do so. Our proposed methods are more
generally applicable than past studies in dealing with general metric spaces.
