  We consider various constructions of monotone Lagrangian submanifolds of $C
P^n, S^2\times S^2$, and quadric hypersurfaces of $C P^n$. In $S^2\times S^2$
and $C P^2$ we show that several different known constructions of exotic
monotone tori yield results that are Hamiltonian isotopic to each other, in
particular answering a question of Wu by showing that the monotone fiber of a
toric degeneration model of $C P^2$ is Hamiltonian isotopic to the Chekanov
torus. Generalizing our constructions to higher dimensions leads us to consider
monotone Lagrangian submanifolds (typically not tori) of quadrics and of $C
P^n$ which can be understood either in terms of the geodesic flow on $T^*S^n$
or in terms of the Biran circle bundle construction. Unlike previously-known
monotone Lagrangian submanifolds of closed simply connected symplectic
manifolds, many of our higher-dimensional Lagrangian submanifolds are provably
displaceable.
