  We consider modified Weyl gravity where a Dirac spinor field is nonminimally
coupled to gravity. It is assumed that such modified gravity is some
approximation for the description of quantum gravitational effects related to
the gravitating spinor field. It is shown that such a theory contains solutions
for a class of metrics which are conformally equivalent to the Hopf metric on
the Hopf fibration. For this case, we obtain a full discrete spectrum of the
solutions and show that they can be related to the Hopf invariant on the Hopf
fibration. The expression for the spin operator in the Hopf coordinates is
obtained. It is demonstrated that this class of conformally equivalent metrics
contains: (a) a metric describing a toroidal wormhole without exotic matter;
(b) a cosmological solution with a bounce and inflation; and (c) a transition
with a change in metric signature. A physical discussion of the results is
given. \end{abstract}
