Mathematical Notes

, Volume 58, Issue 2, pp 877–879 | Cite as

On bending of a convex surface to a convex surface with prescribed spherical image

  • A. V. Pogorelov
Article
  • 48 Downloads

Abstract

We prove the following theorem. LetF be a regular convex surface homeomorphic to the disk. Suppose the Gaussian curvature ofF is positive and the geodesic curvature of its boundary is positive as well. LetG be a convex domain on the unit sphere bounded by a smooth curve and strictly contained in a hemisphere. LetP be an arbitrary point on the boundary ofF andP* be an arbitrary point on the boundary ofG. If the area ofG is equal to the integral curvature of the surfaceF, then there exists a continuous bending of the surfaceF to a convex surfaceF′ such that the spherical image ofF′ coincides withG andP* is the image of the point inF′ corresponding to the pointP ∈ F under the isometry.

Keywords

Unit Sphere Arbitrary Point Smooth Curve Gaussian Curvature Integral Curvature 

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References

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    S. P. Olovyanishnikov, “On bending infinite convex surfaces,”Matem. Sbornik [Math. USSR-Sb.],18, No. 3 (1946).Google Scholar
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    A. D. Aleksandrov, “The smoothness of a convex surface with bounded Gaussian curvature,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],36, No. 7 (1942).Google Scholar
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    A. V. Pogorelov,The Outer Geometry of Convex Surfaces [in Russian], Nauka, Moscow (1969).Google Scholar
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    A. D. Milka, “On the continuous bending of convex surfaces,”Ukrainskii Geometricheskii Sbornik, No. 13 (1973).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • A. V. Pogorelov
    • 1
  1. 1.Physical-Technical Institute of Low TemperaturesUkrainian National Academy of SciencesKharkov

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