An elementary proof of a finite rigidity problem by infinitesimal rigidity methods

Author:
Edgar Kann

Journal:
Proc. Amer. Math. Soc. **60** (1976), 252-258

MSC:
Primary 53C45

DOI:
https://doi.org/10.1090/S0002-9939-1976-0420518-1

MathSciNet review:
0420518

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Abstract: Let two compact, isometric surfaces with boundary be given having positive gauss curvature. If the surfaces can be placed so that their normal spherical images lie in a compact subset of a hemisphere of the unit sphere and so that the isometry is the identity on the boundary then the isometry is the identity mapping.

The proof is elementary in the sense that no integral formulae or maximum principles for elliptic operators are needed.

An example is given of a surface satisfying the above hypotheses which is neither convex nor has a representation in the form .

**[1]**S. E. Cohn-Vossen,*Verbiegbarkeit von Flächen im Grossen*, Fortschritte Math.**1**(1936), 33-76.**[2]**N. W. Efimow,*Flächenverbiegung im Grossen*, Akademie-Verlag, Berlin, 1957 (German). MR**0105722****[3]**Edgar Kann,*A new method for infinitesimal rigidity of surfaces with 𝐾>0*, J. Differential Geometry**4**(1970), 5–12. MR**0259817****[4]**Detlef Laugwitz,*Differential and Riemannian geometry*, Translated by Fritz Steinhardt, Academic Press, New York-London, 1965. MR**0172184****[5]**A. V. Pogorelov,*Extrinsic geometry of convex surfaces*, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 35. MR**0346714**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0420518-1

Keywords:
Infinitesimal rigidity,
isometric surfaces,
congruent surfaces,
rotation vector

Article copyright:
© Copyright 1976
American Mathematical Society