Monatshefte für Mathematik

, Volume 146, Issue 2, pp 113-126

First online:

On Closed Weingarten Surfaces

  • Wolfgang KühnelAffiliated withUniversität Stuttgart
  • , Michael StellerAffiliated withUniversität Stuttgart

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We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces of genus zero where F is a quadratic polynomial or F(λ) = cλ2n+1. This generalizes results by H. Hopf on the case where F is linear and the case of ellipsoids of revolution where F(λ) = cλ3.

2000 Mathematics Subject Classifications: 53A05, 53C40
Key words: Curvature diagram, surface of revolution, Hopf surface, Weingarten surface