Singularities of Surfaces with Constant Negative Gauss Curvature

Abstract

In this chapter we shall be concerned with (open) surfaces and their imbeddings in E³. The definition of an open surface is identical with definition II, 1.1 except that condition 1) that S be compact is no longer true. We will show that a surface with constant negative Gauss curvature cannot be imbedded as a general (open) surface in E³ without singularities (in a sense to be defined below). The first proof of this was given by Hilbert (~ 1900) for analytic surfaces. Our proof works for C³ surfaces and the theorem is still true for C² surfaces. However, Kuiper has given a C¹ isometric imbedding of the hyperbolic plane in E³ without singularities. For details see N.H. Kuiper, on C¹-isometric Imbeddings I and II; Indagationes Mathematicae, Vol. 17 (1955) pp 545–556 and pp 683–689.