  The derivative expnsion in the context of IIB string scattering compactified
on non-trivial K3 and other Calabi-Yau manifolds is formulated. The scattering
data in terms of automorphic functions can be inverted to find the these
metrics. The solutions are parameterized by the moduli information, and the
metrics may be found to any desired accuracy in derivatives. Metric information
to low orders in derivatives allows for a counting of curves inside the
manifold; in addition, the coefficients of these exponential terms via D-brane
wrappings are polynomials that may admit an invariant interpretation in
cohomology. An interesting case pertaining to M-theory compactifications is the
collection of seven-dimensional $G_2$ manifolds; they can also be obtained when
the moduli space degenerates into cases, such as a toroidal one or other limit
in which modular functions on the space are known. Note: this work was written
two years ago; the recipe without the explicit form of the scattering and
metrics is given.
