  In this paper, we prove the following two results:
  First, we study a class of conformally invariant operators $P$ and their
related conformally invariant curvatures $Q$ on even-dimensional Riemannian
manifolds. When the manifold is locally conformally flat(LCF) and compact
without boundary, $Q$-curvature is naturally related to the integrand in the
classical Gauss-Bonnet-Chern formula, i.e., the Pfaffian curvature. For a class
of even-dimensional complete LCF manifolds with integrable $Q$% -curvature, we
establish a Gauss-Bonnet-Chern inequality.
  Second, a finiteness theorem for certain classes of complete LCF four-fold
with integrable Pfaffian curvature is also proven. This is an extension of the
classical results of Cohn-Vossen and Huber in dimension two. It also can be
viewed as a fully non-linear analogue of results of Chang-Qing-Yang in
dimension four.
