- Carathéodory conjecture
The Carathéodory conjecture is a mathematical
conjecture attributed toConstantin Carathéodory byHans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924, [1] . Other early referencesare the presentation [3] ofStefan Cohn-Vossen at the International Congress of Mathematicians in Bologna and the book [2] byWilhelm Blaschke . Carathéodory himself did publish research on the relatedlines of curvature but never committed the Conjecture into writing. In [1] ,J. E. Littlewood mentions the Conjecture as an example of a mathematical claim that is easy to state but difficult to prove.Dirk Struik describes in [5] the formal analogy of the Conjecture with theFour Vertex Theorem for plane curves. A modern reference for the Conjecture is the book [6] ofMarcel Berger .Mathematical content
The Conjecture claims that any convex, closed and three times differentiable surface in three dimensional
Euclidean space admits at least twoumbilic point s. The claim has been noted notto have any good mathematical motivation apart from the absence of counterexamples. In the sense of the conjecture, thespheroid with only two umbilic points and thesphere , all points of whichare umbilic, are examples of surfaces with minimal and maximal number of umbilics.Mathematical research on the conjecture
It has attracted substantial mathematical research but remains unproven.
ee also
*
Differential geometry of surfaces
*Second fundamental form
*Principal curvature References
[1] Sitzungsberichte der Berliner Mathematischen Gesellschaft, 210. Sitzung am 26. Maerz 1924, Dieterichsche Universitätsbuchdruckerei, Göttingen 1924
[2] W. Blaschke, Differentialgeometrie der Kreise und Kugeln, Vorlesungen ueber Differentialgeometrie, vol. 3, Grundlehren der mathematischen Wissenschaften XXIX, Springer, Berlin 1929
[3] S. Cohn-Vossen, Der Index eines Nabelpunktes im Netz der Kruemmungslinien, 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
[5] D. J. Struik, Differential Geometry in the large, Bull. Amer. Math. Soc. vol 37, number 2 (1931), 49 - 62
[6] M. Berger, A Panoramic View of Riemannian Geometry, Springer 2004
External links
* [http://www.w-volk.de/BMG/] Berliner Mathematische Gesellschaft
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