Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Isolated Umbilic Points   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

35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph