Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Principal Mean Curvature Foliations on Surfaces immersed in ${\mathbb R}
  ^4$   Here are studied qualitative properties of the families of curves
--foliations-- on a surface immersed in ${\mathbb R}^4$, along which it bends
extremally in the direction of the mean normal curvature vector.
  Typical singularities and cycles are described, which provide sufficient
conditions, likely to be also necessary, for the structural stability of the
configuration of such foliations and their singularities, under small $C^3$
perturbations of the immersion.
  The conditions are expressed in terms of Darbouxian type of the normal and
umbilic singularities, the hyperbolicity of cycles, and the asymptotic behavior
of singularity separatrices and other typical curves of the foliations. They
extend those given by Gutierrez and Sotomayor in 1982 for principal foliations
and umbilic points of surfaces immersed in ${\mathbb R}^3$. Expressions for the
Darbouxian conditions and for the hyperbolicity, calculable in terms of the
derivatives of the immersion at singularities and cycles, are provided. The
connection of the present extension from ${\mathbb R}^3$ to ${\mathbb R}^4$to
other pertinent ones as well as some problems left open in this paper are
proposed at the end.

4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an