Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Gauss map and second fundamental form of surfaces in a Lie group   In this article, we give the integrability conditions for the existence of an
isometric immersion from an orientable simply connected surface having
prescribed Gauss map and positive extrinsic curvature into some unimodular Lie
groups. In particular, we discuss the case when the Lie group is the euclidean
unit sphere $\mathbb{S}^3$ and establish a correspondence between simply
connected surfaces having extrinsic curvature $K$, $K$ different from 0 and -1,
immersed in $\mathbb{S}^3$ with simply connected surfaces having non-vanishing
extrinsic curvature immersed in the euclidean space $\mathbb{R}^3$. Moreover,
we show that a surface isometrically immersed in $\mathbb{S}^3$ has positive
constant extrinsic curvature if, and only if, its Gauss map is a harmonic map
into the Riemann sphere.

34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys