Carathéodory Conjecture
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Conjecture claims that any convex, closed and three times differentiable surface in three dimensional Euclidean space admits at least two umbilic points. The claim has been noted not to admit any good mathematical motivation or any heuristic argument in its favour nor does its claim appear to have any impact on classical differential geometry. In the sense of the Conjecture, the spheroid with only two umbilic points and the sphere, all points of which are umbilic,…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Conjecture claims that any convex, closed and three times differentiable surface in three dimensional Euclidean space admits at least two umbilic points. The claim has been noted not to admit any good mathematical motivation or any heuristic argument in its favour nor does its claim appear to have any impact on classical differential geometry. 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 numbers of umbilics.