  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.
