  Chekanov showed that the Hofer norm on the Hamiltonian diffeomorphism group
of a geometrically bounded symplectic manifold induces a nondegenerate metric
on the orbit of any compact Lagrangian submanifold under the group. In this
paper we consider the orbits of more general submanifolds. We show that, for
the Chekanov-Hofer pseudometric on the orbit of a closed submanifold to be a
genuine metric, it is necessary for the submanifold to be coisotropic, and we
show that this condition is sufficient under various additional geometric
assumptions. At the other extreme, we show that the image of a generic closed
embedding with any codimension larger than one is "weightless," in the sense
that the Chekanov-Hofer pseudometric on its orbit vanishes identically. In
particular this yields examples of submanifolds which have zero displacement
energy but are not infinitesimally displaceable.
