The Penrose transform and its applications

• Main Author: David, Liana
• Published: University of Edinburgh 2001
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.649142

The smooth Penrose transforms is a map from smooth cohomology groups on a CR-manifold <i>Z</i> to kernels (or cokernels) of differential operators on smooth fields on another manifold <i>X</i> which parametrises a family of compact complex submanifolds of <i>Z</i>. In this thesis were present several extensions of the smooth Penrose transform. We construct Penrose transforms for compactly supported smooth cohomology and for distribution cohomology (with compact and non-compact supports). We apply these extended transforms to several concrete situations already existing in the literature. As a further application of our methods we consider the induced smooth Penrose transform an <i>X </i>with a point removed. As an application of twistor theory in differential geometry we construct an Atiyah-Ward Ansatz for 3 dimensional Einstein-Weyl spaces.