Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Higher order parallel surfaces in three-dimensional homogeneous spaces   We give a full classification of higher order parallel surfaces in
three-dimensional homogeneous spaces with four-dimensional isometry group, i.e.
in the so-called Bianchi-Cartan-Vranceanu family. This gives a positive answer
to a conjecture formulated in 2002. As a partial result, we prove that totally
umbilical surfaces only exist if the space is a Riemannian product of a surface
of constant Gaussian curvature and the real line, and we give a local
parametrization of all totally umbilical surfaces.

3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg