Monatshefte für Mathematik

, Volume 146, Issue 2, pp 113–126 | Cite as

On Closed Weingarten Surfaces

  • Wolfgang Kühnel
  • Michael Steller


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. van-Brunt, B, Grant, K 1994Hyperbolic Weingarten surfaces.Math Proc Camb Philos Soc116489504Google Scholar
  2. van-Brunt, B, Grant, K 1996Potential applications of Weingarten surfaces in CAGD. I: Weingarten surfaces and surface shape investigation.Comput Aided Geom Des13569582CrossRefGoogle Scholar
  3. Chern, S-S 1945Some new characterizations of the Euclidean sphere.Duke Math J12279290CrossRefGoogle Scholar
  4. Hopf, H 1951Über Flächen mit einer Relation zwischen den Hauptkrümmungen.Math Nachr4232249Google Scholar
  5. Kapouleas, N 1991Compact constant mean curvature surfaces in Euclidean three-space.J Differ Geom33683715Google Scholar
  6. Kühnel, W 1981Zur inneren Krümmung der zweiten Grundform.Monatsh Math91241251CrossRefGoogle Scholar
  7. Kühnel W (2002) Differential Geometry, Curves – Surfaces – Manifolds. Providence, R.I.: Amer Math Soc (German edition: Differentialgeometrie, Wiesbaden, Vieweg 2003)Google Scholar
  8. Huang, X 1988Weingarten surfaces in three-dimensional spaces (Chinese).Acta Math Sin31332340Google Scholar
  9. Papantoniou, B 1984Classification of the surfaces of revolution whose principal curvatures are connected by the relation Ak1 + Bk2 = 0 where A or B is different from zero.Bull Calcutta Math Soc764956Google Scholar
  10. Voss, K 1959Über geschlossene Weingartensche Flächen.Math Annalen1384254CrossRefGoogle Scholar
  11. Weingarten, J 1861Über eine Klasse auf einander abwickelbarer Flächen.J Reine Angew Math59382393Google Scholar
  12. Wente, H 1986Counterexample to a conjecture of H. Hopf.Pacific J Math121193243Google Scholar

Copyright information

© Springer-Verlag/Wien 2005

Authors and Affiliations

  • Wolfgang Kühnel
    • 1
  • Michael Steller
    • 1
  1. 1.Universität StuttgartGermany

Personalised recommendations