Rocky Mountain Journal of Mathematics

A New Proof of Liebmann Classical Rigidity Theorem for Surfaces in Space Forms

Juan A. Aledo, Luis J. Alías, and Alfonso Romero

Full-text: Open access

Article information

Source
Rocky Mountain J. Math., Volume 35, Number 6 (2005), 1811-1824.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1181069618

Digital Object Identifier
doi:10.1216/rmjm/1181069618

Mathematical Reviews number (MathSciNet)
MR2210636

Zentralblatt MATH identifier
1100.53003

Subjects
Primary: 53A05: Surfaces in Euclidean space
Secondary: 53C45: Global surface theory (convex surfaces à la A. D. Aleksandrov)

Keywords
Rigidity theorem compact surface second fundamental form Gaussian curvature Ricci curvature scalar curvature Riemannian connection

Citation

Aledo, Juan A.; Alías, Luis J.; Romero, Alfonso. A New Proof of Liebmann Classical Rigidity Theorem for Surfaces in Space Forms. Rocky Mountain J. Math. 35 (2005), no. 6, 1811--1824. doi:10.1216/rmjm/1181069618. https://projecteuclid.org/euclid.rmjm/1181069618


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References

  • M. Becker and W. Kühnel, Hypersurfaces with constant inner curvature of the second fundamental form, and the non-rigidity of the sphere, Math. Z. 223 (1996), 693-708.
  • A.L. Besse, Einstein manifolds, Springer-Verlag, Berlin, 1987.
  • M.P. do Carmo, Differential geometry of curves and surfaces, Prentice Hall, New Jersey, 1976.
  • S.S. Chern, Some new characterizations of the Euclidean sphere, Duke Math. J. 12 (1945), 270-290.
  • L.P. Eisenhart, Riemannian geometry, 6th ed., Princeton Univ. Press, Princeton, 1966.
  • J. Hadamard, Sur certaines propriétés des trajectoires en dynamique, J. Math. Pures Appl. 3 (1897), 331-387.
  • D. Hilbert, Grundlagen der geometrie, 3rd ed., Leipzig, 1909.
  • W. Klingenberg, A course in differential geometry, Springer Verlag, New York, 1978.
  • D. Koutroufiotis, Two characteristic properties of the sphere, Proc. Amer. Math. Soc. 44 (1974), 176-178.
  • K. Leichtweiss, Convexity and differential geometry, in Handbook of convex geometry (P.M. Gruber and J.M. Wills, eds.), Elsevier, Amsterdam, 1993.
  • H. Liebmann, Eine neue eigenschaft der kugel, Nachr. Kgl. Ges. Wiss. Göttingen, Math.-Phys. Klasse (1899), 44-55.
  • S. Montiel and A. Ros, Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures, in Differential geometry, Pitman Monographs Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 279-296.
  • S. Montiel and A. Ros, Curvas y Superficies, Proyecto Sur, Granada, 1997.
  • B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
  • A. Ros, Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Differential Geom. 27 (1988), 215-220.
  • --------, Compact hypersurfaces with constant higher order mean curvatures, Rev. Mat. Iberomericana 3 (1987), 447-453.
  • R. Schneider, Zur affinen Differentialgeometrie im Grossen, I, Math. Z. 101 (1967), 375-406.
  • --------, Closed convex hypersurfaces with second fundamental form of constant curvature, Proc. Amer. Math. Soc. 35 (1972), 230-233.
  • U. Simon, Characterizations of the sphere by the curvature of the second fundamental form, Proc. Amer. Math. Soc. 55 (1976), 382-384.