  The standard theory of General Relativity (GR) currently provides the most
reliable description of all gravitational events in Astrophysics and Cosmology.
However, current Astronomy allows measurements that contradict the predictions
of GR in some gravitational scenarios such as the accelerated expansion of the
Universe, the fast rotation of cluster of galaxies of the so strongly
deflection of light by gravitational lenses. This has led the scientific
community to propose modifications of the theory, and in particular to
introduce the existence of dark matter and dark energy. One of these modified
theories of gravitation proposes, from the same scheme that the geometrical
theory of GR, a so called space-time-matter (STM) theory within a manifold of
five dimensions where the resulting metric were able to measure adequately and
adjusted to the current observations the most massive gravitational events.
Here we show that the gravitational features of a 4D metric in the standard
space-time can be understood as an induced effect provided by a geometrical
generalization of GR into a 5D manifold. In particular the rotation
corresponding to an stationary and axisymmetric 4D metric is explained as a
manifestation of the existence of a new additional dimension of the manifold
which can be related with the rest mass. Furthermore the equivalence between a
description of gravity in 4 dimensions and the static vacuum solution
constructed in a 5D manifold could allow us to obtain stationary and axially
symmetric solutions not known so far. These solutions can provide metrics that
lead to determine those small observational corrections that are currently
related to the existence of dark matter, but that in reality could be
gravitational effects well explained within a geometrical theory generalizing
Einstein gravity.
