  The mechanical properties of crystals on curved substrates mix elastic,
geometric and topological degrees of freedom. In order to elucidate the
properties of such crystals we formulate the low-energy effective action that
combines metric degrees of freedom with displacement fields and defects. We
propose dualities for elasticity coupled to curved geometry formulated in terms
of tensor gauge theories. We show that the metric degrees of freedom, evolving
akin to linearized gravity are mapped to tensors with three indices. When
coupled to crystals these degrees of freedom become gapped and, in the presence
of dislocations and disclinations, multivalued. The elastic degrees of freedom
remain gapless and mapped to symmetric gauge fields with two indices. In
analogy with elasticity on flat space formulation we assume that the trace of
the total quadrupole moment is conserved. In the dual formulation, topological
defects, which act as sources for the gauge fields, are fractons or excitations
with restricted mobility. This leads to generalized glide constraints that
restrict both displacement and gravitational defects.
