 Deformation of Surfaces Preserving Principal CurvaturesShiing-shen Chern A chapter in Differential Geometry and Complex Analysis, 1985, pp 155-163 from Springer Abstract: Abstract The isometric deformation of surfaces preserving the principal curvatures was first studied by O. Bonnet in 1867. Bonnet restricted himself to the complex case, so that his surfaces are analytic, and the results are different from the real case. After the works of a number of mathematicians, W. C. Graustein took up the real case in 1924-, without completely settling the problem. An authoritative study of this problem was carried out by Elie Cartan in [2], using moving frames. Based on this work, we wish to prove the following: Theorem: The non-trivial families of isometric surfaces having the same principal curvatures are the following: 1) a family of surfaces of constant mean curvature; 2) a family of surfaces of non-constant mean curvature. Such surfaces depend on six arbitrary constants, and have the properties: a) they are W-surfaces; b) the metric $$d{s^2} = {\left( {gradH} \right)^2}d{s^2}/\left( {{H^2} - K} \right)$$ , where d s 2 is the metric of the surface and H and K are its mean curvature and Gaussian curvature respectively, has Gaussian curvature equal to — 1. Keywords: Harmonic Function; Gaussian Curvature; Principal Curvature; Principal Direction; Connection Form (search for similar items in EconPapers) There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-69828-6_10 Ordering information: This item can be ordered from DOI: 10.1007/978-3-642-69828-6_10 Access Statistics for this chapter More chapters in Springer Books from Springer |
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