  We show that the moduli space of positive Ricci curvature metrics on all the
total spaces of $S^7$-bundles over $S^8$ which are rational homology spheres
has infinitely many path components. Furthermore, we carry out the
diffeomorphism classification of quotients of Milnor spheres by a certain
involution and show that the moduli space of metrics of non-negative sectional
on them has infinitely many path components. Finally, a diffeomorphism
finiteness result is obtained on quotients of Shimada spheres by the same type
of involution and we show that for the types that can be expressed by an
infinite family of manifolds, the moduli space of positive Ricci curvature
metrics has infinitely many path components.
