Differential Geometry in the Large pp 123-135 | Cite as

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

## 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 k_{1} and k_{2} of an ovaloid satisfy a relationship k_{2} = f(k_{1}) where f is a decreasing function, then the ovaloid is a sphere. Since K = k_{1}k_{2} 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.

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Lution Dian Alexan## Preview

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