# Closed Surfaces with Constant Gauss Curvature (Hilbert’s Method)— Generalizations and Problems — General Remarks on Weingarten Surfaces

• Heinz Hopf
Part of the Lecture Notes in Mathematics book series (LNM, volume 1000)

## Abstract

Our aim in this section is to prove that the spheres are 1) the only closed surfaces with constant Gauss curvature K, and 2) the only ovaloids with constant mean curvature H. We will actually prove the stronger result that if the principle curvatures k1 and k2 of an ovaloid satisfy a relationship k2 = f(k1) where f is a decreasing function, then the ovaloid is a sphere. Since K = k1k2 and $$H = \frac{1}{2}\left( {{k_1} + {k_2}} \right)$$ , the two results, 1) and 2) stated above will follows Irom this theorem. The difference in the formulation of 1) and 2) is 3ue to the fact that on any closed surface there are points where K>O (See II, 4.2). Therefore if K is constant, then K is a positive constant and hence by IV, 1.4, the surface already is an ovaloid. The problem of characterizing arbitrary closed surfaces for which H is constant is much more difficult. It will be considered in Chapter VI and VII.

### Keywords

Lution Dian Alexan