  We consider a non-relativistic free scalar field theory with a type of
anisotropic scale invariance in which the number of coordinates "scaling like
time" is generically greater than one. We propose the Cartesian product of two
curved spaces, the metric of each space being parameterized by the other space,
as a notion of curved background to which the theory can be extended. We study
this type of geometries, and find a family of extensions of the theory to
curved backgrounds in which the anisotropic scale invariance is promoted to a
local, Weyl-type symmetry.
