Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Riemannian manifold of all Riemannian metrics   The space of all Riemannian metrics on a smooth second countable finite
dimensional manifold is itself a smooth manifold modeled on the space of
symmetric (0,2)-tensor fields with compact support. It carries a canonical
Riemannian metric which is invariant under the action of the diffeomorphism
group. We determine its geodesics, exponential mapping, curvature, and Jacobi
fields in a very explicit manner.

12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn