  We investigate the possibility that the observed behavior of test particles
outside galaxies, which is usually explained by assuming the existence of dark
matter, is the result of the dynamical evolution of particles in a Weyl type
geometry, and its associated conformally invariant Weyl geometric quadratic
gravity. As a first step in our investigations we write down the simplest
possible conformally invariant gravitational action, constructed in Weyl
geometry, and containing the Weyl scalar, and the strength of the Weyl vector
only. By introducing an auxiliary scalar field, the theoretical model can be
reformulated in the Riemann geometry as scalar-vector-tensor theory, containing
a scalar field, and the Weyl vector, respectively. The field equations of the
theory are derived in the metric formalism, in the absence of matter. A
specific static, spherically symmetric model, in which the Weyl vector has only
a radial component, is considered. In this case, an exact analytic solution of
the gravitational field equations can be obtained. The behavior of the galactic
rotation curves is also considered in detail, and it is shown that an effective
geometric mass term, with an associated density profile, can also be
introduced. Three particular cases, corresponding to some specific functional
forms of the Weyl vector, are also investigated. A comparison of the model with
a selected sample of galactic rotation curves is also performed when an
explicit breaking of conformal invariance is introduced, which allows the fix
of the numerical values of the free parameters of the model. Our results show
that Weyl geometric models can be considered as a viable theoretical
alternative to the dark matter paradigm.
