chapter  8
60 Pages

Grassmann Image of a Submanifold

In studying two-dimensional surfaces in three-dimensional Euclidean space, both local and global aspects, it is very useful to invoke the concepts of spherical image and spherical mapping. By Gauss's theorem the limit of the ratio of the spherical image area 6.w of the domain G in F 2 to the area 6.S of this domain while G contracts in F 2 to x is equal to the modulus of the Gaussian curvature at x :

lim ~w = IK(x) l. u S

An important role in the theory of minimal surfaces is played by the conformality of a spherical image. The first fundamental form of the spherical image dn2 is proportional to the first fundamental form ds2 of the surface: dn2 = IKI ds 2 . There are many important papers where the authors either use various properties of the spherical image or formulate theorems in terms of the spherical image. For instance, the metric of unit sphere dn2 has been used by Efimov as an ancillary metric on the surface of negative curvature.