Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fundamental theorem of hyperbolic geometry without the injectivity
  assumption   Let $\mathbb{H}^n$ be the $n-$dimensional hyperbolic space. It is well known
that, if $f: \mathbb{H}^n\to \mathbb{H}^n$ is a bijection that preserves
$r-$dimensional hyperplanes, then $f$ is an isometry. In this paper we make
neither injectivity nor $r-$hyperplane preserving assumptions on $f$ and prove
the following result:
  Suppose that $f: \mathbb{H}^n\to \mathbb{H}^n$ is a surjective map and maps
an $r-$hyperplane into an $r-$hyperplane, then $f$ is an isometry. The
Euclidean version was obtained by A. Chubarev and I. Pinelis in 1999 among
other things. Our proof is essentially different from their and the similar
problem arising in the spherical case is open.

12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas