Isometric embeddings of surfaces with nonnegative curvature in R(3)

A classical problem in differential geometry is that of isometrically embedding surfaces in R$\sp3$. For a smooth metric with positive Gauss curvature, it is always possible to find a smooth isometric embedding. The case of nonnegative curvature is discussed in this thesis. In particular, the curvature is assumed to be zero at exactly one point and positive elsewhere. Here, there are examples of smooth metrics that have isometric embeddings but with little regularity. In particular, an example is given of an analytic metric with curvature zero at precisely one point which has no $C\sp3$ isometric embedding into R$\sp3$. Further, the obstruction to regularity is shown to be the presence of an umbilic point at precisely the point of zero curvature. Also, in this thesis, the question of the existence of isometric embeddings with curvature zero at one point is considered. It is shown that if the curvature is a convex function near the point of zero curvature, then there always exists a $C\sp{1,1}$ isometric embedding