Surfaces with a parallel isoperimetric section

This announcement is a continuation of Chen [1] (also, Yau [3]). We shall present additional theorems relating surfaces in a space form with a parallel normal section. Let M be a surface in an m-dimensional Riemannian manifold R with the induced normal connection D. For a unit normal section £ on M (that is, a unit normal vector field of M in R\ let Aç be the second fundamental tensor with respect to £; if we have DC = 0 identically, then £ is called a parallel section; if the trace of A% is constant (respectively, zero), then £ is called an isoperimetric section (respectively, minimal section) on M ; if the determinant of A4 is nowhere zero, then £ is called a nondegenerate section; if Aç vanishes identically, then £ is called a geodesic section; and if ^ is not proportional to the identity transformation everywhere, then Ç is called a umbilical-free section.