  We analyze gravitational theories with quadratic curvature terms, including
the case of conformally invariant Weyl gravity, motivated by the intention to
find a renormalizable theory of gravity in the ultraviolet region, yet yielding
general relativity at long distances. In the Hamiltonian formulation of Weyl
gravity, the number of local constraints is equal to the number of unstable
directions in phase space, which in principle could be sufficient for
eliminating the unstable degrees of freedom in the full nonlinear theory. All
the other theories of quadratic type are unstable -- a problem appearing as
ghost modes in the linearized theory.
  We find that the full projection of the Weyl tensor onto a three-dimensional
hypersurface contains an additional fully traceless component, given by a
quadratic extrinsic curvature tensor. A certain inconsistency in the literature
is found and resolved: when the conformal invariance of Weyl gravity is broken
by a cosmological constant term, the theory becomes pathological, since a
constraint required by the Hamiltonian analysis imposes the determinant of the
metric of spacetime to be zero. In order to resolve this problem by restoring
the conformal invariance, we introduce a new scalar field that couples to the
curvature of spacetime, reminiscent of the introduction of vector fields for
ensuring the gauge invariance.
