  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 surface containing an isolated umbilic point
of any index is presented, violating Hamburger's umbilic index bound.
  In both cases, it is shown that the metric can be made arbitrarily close to
the flat metric. A short video explaining the motivation and results of this
paper can be found at the following link: https://youtu.be/Wjja4PcMtxc
