Local and global aspects of conformal geometry

The present work consists of two separate results, each treating different aspects of the theory of conformal structures in Riemannian geometry.; Theorem A deals with local objects (differential operators) that exhibit invariance properties under conformal transformations. It should be seen as a chapter in the program initiated by Fefferman and Graham in their work on conformal invariants, [17], where they introduced the ambient metric and raised the problem of finding all local conformally invariant objects associated with a conformal manifold (Mn, [g n]).; Theorem B tackles a well-known open problem in geometry that originally arose in high energy physics. It is a partial result that confirms a conjecture of Deser and Schwimmer about "conformal anomalies" (which in our language will be "global conformal invariants"). These global invariants are integrals of polynomials in the curvature tensor and its covariant derivatives that remain invariant under conformal changes of the underlying metric. I prove this conjecture in Theorem 3.1.1 for the case where the polynomial depends only on the curvature tensor, without any covariant derivatives.; The only previous work that addresses the Deser-Schwimmer conjecture is the one by Branson, Gilkey and Pohjanpelto, [5], for the case of locally conformally flat metrics. Theorem B below is the first result for general metrics.; This second result relies on very little prior work. The main new method introduced here is the so-called "super divergence formula", proved in chapter 3. This tool is put to use in chapter 4, in order to prove Theorem B, the restricted version of the Deser-Schwimmer conjecture. In the future, I hope to apply the super divergence formula and demonstrate the full conjecture.