  We prove two-sided inequalities for the $L^p$-norm of a pushforward or
pullback (with respect to an orientation-preserving diffeomorphism) on oriented
volume and Riemannian manifolds. For a function or density on a volume
manifold, these bounds depend only on the Jacobian determinant, which arises
through the change of variables theorem. For an arbitrary differential form on
a Riemannian manifold, however, these bounds are shown to depend on more
general "spectral" properties of the diffeomorphism, using an
appropriately-defined notion of singular values. These spectral terms
generalize the Jacobian determinant, which is recovered in the special cases of
functions and densities (i.e., bottom and top forms).
