Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A knot characterization and 1-connected nonnegatively curved 4-manifolds
  with circle symmetry   We classify nonnegatively curved simply connected 4-manifolds with circle
symmetry up to equivariant diffeomorphisms. The main problem is rule out
knotted curves in the singular set of the orbit space. As an extension of this
work we classify all knots in S^3 which can be realized as an extremal set with
respect to an inner metric on S^3 which has nonnegative curvature in the
Alexandrov sense.

8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys