Metrics, connections, and gluing theorems

the long version

In this book, the author's goal is to provide an introduction to some of the analytic underpinnings for the geometry of anti-self duality in 4-dimensions. Anti-self duality is rather special to 4-dimensions and the imposition of this condition on curvatures of connections on vector bundles and on curvatures of Riemannian metrics has resulted in some spectacular mathematics. The book reviews some basic geometry, but it is assumed that the reader has a general background in differential geometry (as would be obtained by reading a standard text on the subject). Some of the fundamental references include Atiyah, Hitchin and Singer, Freed and Uhlenbeck, Donaldson and Kronheimer, and Kronheimer and Mrowka. The last chapter contains open problems and conjectures.