On isolated umbilic points

  title={On isolated umbilic points},
  author={Brendan Guilfoyle},
  journal={arXiv: Differential Geometry},
  • B. Guilfoyle
  • Published 9 December 2018
  • Mathematics
  • arXiv: Differential Geometry
Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3-space. In particular, Riemannian metrics with a smooth strictly convex 2-sphere containing a single umbilic point are constructed explicitly, in contradiction with any direct extension of Caratheodory's conjecture. Additionally, a Riemannian metric with an embedded… 
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