  In this short note, exploits of constructions of $\mathcal{F}$-structures
coupled with technology developed by Cheeger-Gromov and Paternain-Petean are
seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe
invariant and to study collapsing with bounded sectional curvature on
inequivalent smooth structures and inequivalent PL-structures within a fixed
homeomorphism class. We compute these fundamental Riemannian invariants for
every high-dimensional smooth manifold on the homeomorphism class of any smooth
manifold that admits a Riemannian metric of zero sectional curvature. This
includes all exotic and all fake tori of dimension greater than four. We
observe that the minimal volume is not an invariant of the smooth structures,
yet the Yamabe invariant does discern the standard smooth structure from all
the others. We also observe that the fundamental group places no restriction on
the vanishing of the minimal volume and collapse with bounded sectional
curvature for high-dimensional manifolds.
