  This paper connects nonpositive sectional curvature of a Riemannian manifold
with the displacement convexity of the variance functional on the space $P(M)$
of probability measures over $M$. We show that $M$ has nonpositive sectional
curvature and has trivial topology (i.e, is homeomorphic to $\mathbf{R}^n$) if
and only if the variance functional on $P(M)$ is displacement convex. This is
followed by a Jensen type inequality for the variance functional with respect
to Wasserstein barycenters, as well as by a result comparing the variance of
the Wasserstein and linear barycenters of a probability measure on $P(M)$ (that
is, an element of $P(P(M))$). These results are applied to invariant measures
under isometry group actions, giving a comparison for the variance functional
between the Wasserstein projection and the $L^2$ projection to the set of
invariant measures.
