Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geodesics on path spaces and double category   Let $M$ be a Riemannian manifold and ${\mathcal P}M$ be the space of all
smooth paths on $M$. We describe geodesics on path space ${\mathcal P}M$.
Normal neighbourhood structure on ${\mathcal P}M$ has been discussed. We
identify paths on $M$ under "back-track" equivalence. Under this identification
we show that if $M$ is complete, then geodesics on path space yield a double
category.We gave a physical interpretation of this double category.

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The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con