  We consider two natural problems arising in geometry which are equivalent to
the local solvability of specific equations of Monge-Ampere type. These two
problems are: the local isometric embedding problem for two-dimensional
Riemannian manifolds, and the problem of locally prescribed Gaussian curvature
for surfaces in R^3. We prove a general local existence result for a large
class of Monge-Ampere equations in the plane, and obtain as corollaries the
existence of regular solutions to both problems, in the case that the Gaussian
curvature vanishes to arbitrary finite order on a single smooth curve.
