| This dissertation contains two parts.;In the first part, 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 (CF. [Br2, BrGP]). For a class of even-dimensional complete LCF manifolds with integrable Q-curvature, we establish a Gauss-Bonnet-Chern inequality. As applications, finiteness theorems for certain classes of complete LCF manifolds are also proven. These are extensions of the classical results of Cohn-Vossen [CV] and Huber [H] in dimension two and those of Chang-Qing-Yang [CQY1, CQY2] in dimension four.;In the second part, we study the analytic torsion of manifolds. On a closed Riemannian manifold, the analytic torsion of the metric is equal to a topological invariant, the Reidemeister torsion, by the celebrated Cheeger-Muller theorem (Cf. [C, Mu1]. For a manifold with boundary, the difference between these two torsions is discussed in [LR], [L] and [V], when the metric near the boundary has a product structure. We prove a general formula for the difference between analytic torsion and Reidemeister torsion on a manifold with boundary. We find that an extra term appears in the non-product case. Interestingly, this term is precisely the transgression of the Pfaffian in the even dimensional case; while in the odd dimensional case, it is a term involving the second fundamental form of the boundary and the curvature tensor of the manifold. This part of the dissertation has appeared in a joint work with X. Dai [DF]. |
