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This an introductory book on SelfDual Riemannian 4 Manifolds. SelfDual metrics are special type of metrics which provide solution to the "Optimal Metric" problem. Under a vanishing hypothesis, Donaldson and Friedman proved that the connected sum of two selfdual Riemannian 4Manifolds is again selfdual. Here we prove that the same result can be extended over to the positive scalar curvature case. The idea is to use Leray spectral sequence. Secondly we give an example of a 4manifold with b+ = 0 admitting a scalar?at antiselfdual metric. Finally we present an application of the Geometric Invariant Theory(GIT) for Toric Varieties to the EinsteinWeyl Geometry.
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