  Weyl's conformal theory of gravity is an extension of Einstein's theory of
general relativity which associates metrics with 1-forms . In the case of
locally integrable (closed non-exact) 1-forms the spacetime manifolds are no
more simply connected. The Weil connections yield curvature tensors which
satisfy the basic properties of Riemann curvature tensors. The Ricci tensors
are symmetric, conformally invariant, and the Einstein tensors computed with
the Weyl connections implicate a cosmological term replacing the cosmological
constant by a function of spacetime, and a shear stress tensor. A toy model
based on the Schwarzschild metric is presented where the associated 1-form is
proportional to $d\varphi$ in Schwarzschild coordinates. This implies a
singularity on the whole z-axis and it generates a torque effect on geodesics.
According to initial conditions planar geodesics show almost constant
velocities independently of r. In the free case spin effects occur in the
neighbourhood of the singularity which are comparable to recent observations
concerning cosmic filaments.
