  Let $D_F = \{(z_0, z) \in {\C}^{n} | |z_0|^2 < b, \|z\|^2 < F(|z_0|^2) \}$ be
a strongly pseudoconvex Hartogs domain endowed with the \K metric $g_F$
associated to the \K form $\omega_F = -\frac{i}{2} \partial \bar{\partial} \log
(F(|z_0|^2) - \|z\|^2)$.
  This paper contains several results on the Riemannian geometry of these
domains. In the first one we prove that if $D_F$ admits a non special geodesic
(see definition below) through the origin whose trace is a straight line then
$D_F$ is holomorphically isometric to an open subset of the complex hyperbolic
space. In the second theorem we prove that all the geodesics through the origin
of $D_F$ do not self-intersect, we find necessary and sufficient conditions on
$F$ for $D_F$ to be geodesically complete and we prove that $D_F$ is locally
irreducible as a Riemannian manifold. Finally, we compare the Bergman metric
$g_B$ and the metric $g_F$ in a bounded Hartogs domain and we prove that if
$g_B$ is a multiple of $g_F$, namely $g_B=\lambda g_F$, for some $\lambda\in
\R^+$, then $D_F$ is holomorphically isometric to an open subset of the complex
hyperbolic space.
