Glidebending of general caps: an infinitesimal treatment
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- by Edgar Kann PDF
- Proc. Amer. Math. Soc. 84 (1982), 247-255 Request permission
Abstract:
We prove that two ${C^3}$ isometric surfaces $S’$ and $S$ with (not necessarily planar) boundary, immersed in ${E^3}$, with $K > 0$, having their spherical images in the same open hemisphere, and which are oriented so that the mean curvatures have the same sign, are congruent if they satisfy the glidebending boundary condition: Let $\{ n:e \cdot n > 0$, $e = {\text {const}}{\text {. vector\} }}$ be the open hemisphere. Then $e \cdot X’ = e \cdot X$ at corresponding boundary points, where $X’$ and $X$ are position vectors of $S’$ and $S$. The method uses the fact that the surface $\tfrac {1} {2}(X - X’)$ is an infinitesimal bending field for the mean surface $\tfrac {1} {2}(X + X’)$ and is elementary in that it uses the rotation vector of classical infinitesimal bending theory but no integral formulas, maximum principles for elliptic operators or index theorems. The surfaces considered need not have a simple projection on a plane, be convex in the large (an example is given) nor be simply connected. We use the method to prove the finite and infinitesimal rigidity of general caps and caps under glidebending.References
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S. E. Cohn-Vossen, Verbiegbarkeit von Flächen im Grossen, Fortschr. Math. 1 (1936), 33-76.
- N. W. Efimow, Flächenverbiegung im Grossen, Akademie-Verlag, Berlin, 1957 (German). MR 0105722
- Edgar Kann, An elementary proof of a finite rigidity problem by infinitesimal rigidity methods, Proc. Amer. Math. Soc. 60 (1976), 252–258 (1977). MR 420518, DOI 10.1090/S0002-9939-1976-0420518-1
- Edgar Kann, A new method for infinitesimal rigidity of surfaces with $K>0$, J. Differential Geometry 4 (1970), 5–12. MR 259817
- A. V. Pogorelov, Extrinsic geometry of convex surfaces, Translations of Mathematical Monographs, Vol. 35, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations. MR 0346714
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 84 (1982), 247-255
- MSC: Primary 53A05
- DOI: https://doi.org/10.1090/S0002-9939-1982-0637178-X
- MathSciNet review: 637178