  The five-dimensional (5D) Riemannian G\"odel-type manifolds are examined in
light of the equivalence problem techniques, as formulated by Cartan. The
necessary and sufficient conditions for local homogeneity of these 5D manifolds
are derived. The local equivalence of these homogeneous Riemannian manifolds is
studied. It is found that they are characterized by two essential parameters
$m^2$ and $\omega $: identical pairs $(m^2, \omega)$ correspond to locally
equivalent 5D manifolds. An irreducible set of isometrically nonequivalent 5D
locally homogeneous Riemannian G\"odel-type metrics are exhibited. A
classification of these manifolds based on the essential parameters is
presented, and the Killing vector fields as well as the corresponding Lie
algebra of each class are determined. It is shown that apart from the $(m^2= 4
\omega^2, \omega\not=0)$ and $(m^2\not=0, \omega = 0)$ classes the homogeneous
Riemannian G\"odel-type manifolds admit a seven-parameter maximal group of
isometry ($G_7$). The special class $(m^2= 4 \omega^2, \omega\not=0)$ and the
degenerated G\"odel-type class $(m^2\not=0, \omega=0)$ are shown to have a
$G_9$ as maximal group of motion. The breakdown of causality in these classes
of homogeneous G\"odel-type manifolds are also examined.
