  We investigated the possibility of the homogeneous and isotropic cosmological
solution in Weyl geometry, which differs from the Riemannian geometry by adding
the so called Weyl vector. The Weyl gravity is obtained by constructing the
gravitational Lagrangian both to be quadratic in curvatures and conformal
invariant. It is found that such solution may exist provided there exists the
direct interaction between the Weyl vector and the matter fields. Assuming the
matter Lagrangian is that of the perfect fluid, we found how such an
interaction can be implemented. Due to the existence of quadratic curvature
terms and the direct interaction the perfect fluid particles may be created
straight from the vacuum, and we found the expression for the rate of their
production which appeared to be conformal invariant. In the case of creating
the universe ``from nothing'' in the vacuum state, we investigated the problem,
whether this vacuum may persist or not. It is shown that the vacuum may persist
with respect to producing the non-dust matter (with positive pressure), but
cannot resist to producing the dust particles. These particles, being
non-interactive, may be considered as the candidates for dark matter.
