Carathéodory conjecture
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The Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of Mathematical Society in Berlin in 1924, [1]. Other early references are the announcement [3] of Stefan Cohn-Vossen at the International Congress in Bologna in 1928 and the book [2] by Wilhelm Blaschke. Carathéodory himself did publish research on the related lines of curvature but never committed this conjecture into writing. In [1], J. E. Littlewood mentiones the conjecture as an example of a claim that is easy to state but difficult to prove.
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[edit] Mathematical content
The conjecture claims that a convex, closed and three times differentiable surface in Euclidean three-space admits at least two umbilic points. The conjecture has been noted not to have any good mathematical motivation apart from the absence of counterexamples. In the sense of the conjecture, the spheroid with only two umbilic points and the sphere, all points of which are umbilic, are examples of surfaces with minimal and maximal number of umbilics.
[edit] Mathematical research on the conjecture
It has attracted substantial mathematical research but remains unproven.
[edit] References
[1] Sitzungsberichte der Berliner Mathematischen Gesellschaft, 210. Sitzung am March 26, 1924, Dieterichsche Universitätsbuchdruckerei, Göttingen 1924
[2] W. Blaschke, Vorlesungen ueber Differentialgeometrie III, Differentialgeometrie der Kreise und Kugeln, Grundlehren XXIX, Springer, Berlin 1929
[3] S. Cohn-Vossen, Der Index eines Nabelpunktes im Netz der Kruemmingslinien, Proceedings of the International Congress of Mathematicians, vol II, Nicola Zanichelli Editore, Bologna 1929
[4] J. E. Littlewood, A mathematician's miscellany, Methuen & Co, London 1953
[edit] External links
- [1] Berliner Mathematische Gesellschaft
