Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integral Liouville theorem in a complete Riemannian manifold   If the Killing vector field in a Riemannian manifold is the gradient of a
smooth real valued function, then it is called Killing potential. In this paper
we have deduced a necessary condition for the existence of Killing potential in
a complete Riemannian manifold. Yau proved the Liouville theorem of harmonic
function in a Riemannian manifold using gradient estimation and after that many
authors have generalized this concept and investigated various types of
Liouville theorems of harmonic functions. In this article we have also proved a
Liouville theorem in integral form of harmonic functions in a complete
Riemannian manifold. Finally we have studied the behaviour of harmonic
functions in a complete Riemannian manifold that satisfies some gradient Ricci
solition and showed that harmonic function is a constant multiple of distance
function along some geodesics.

27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph