Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Loops Representing Elements of the Fundamental Group of a Complete
  Manifold with Nonnegative Ricci Curvature   This paper concerns complete noncompact manifolds with nonnegative Ricci
curvature. Roughly, we say that M has the loops to infinity property if given
any noncontractible closed curve, C, and given any compact set, K, there exists
a closed curve contained in M\K which is homotopic to C. The main theorems in
this paper are the following.
  Theorem I: If M has positive Ricci curvature then it has the loops to
infinity property.
  Theorem II: If M has nonnegative Ricci curvature then it either has the loops
to infinity property or it is isometric to a flat normal bundle over a compact
totally geodesic submanifold and its double cover is split.
  Theorem III: Let M be a complete riemannian manifold with the loops to
infinity property along some ray starting at a point, p. Let D containing p be
a precompact region with smooth boundary and S be any connected component of
the boundary containing a point, q, on the ray. Then the map from the
fundamental group of S based at q to the fundamental group of Cl(D) based at p
induced by the inclusion map is onto.

21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph