Twistor transform of symmetries

• Main Author: Laffar, Darren Richard
• Published: University of Glasgow 2001
• Subjects: 510; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.368594

The twistor transform introduced by Penrose's fundamental articles ([28],[40],[19],[12]) encodes basic geometric and mathematical physics structures into holomorphic geometry. Differential equations get replaced by complex manifolds and holomorphic bundles over them with well defined properties. Hence direct constructions of such objects lead to constructions of various classes of solutions to basis equations of differential geometry and mathematical physics. The first successes of the twistor transformation method were associated with the self-dual Einstein and Yang-Mills equation ([29],[36],[3],[1],[2],[10]). Non-Self-dual Yang-Mills equations have been studied by Witten and Manin ([39],[14] and references therein). Deep interconnections between twistor theory and non-linear integrable equations have been unveiled by Mason, Singer and Sparling [23], [21] and [22]. More recently, twistor methods have been successfully used in the study of quaternionic Kahler and hyper-Kahler manifolds [4], [13], [17], [18], [25], [27], [32], [33] and [34].In 1997, Merkulov [24] has developed the twistor theory for general irreducible G-structures which was applied in 1999 by him and Schwachhofer to solve the long-standing holonomy problem [26].The main theme of our work is the study of symmetries of G-structures in the twistor theory context. The main result, Theorem (3.14), provides us with a surprisingly simple characterisation of Killing vector fields. This theorem establishes a one-to-one correspondence between Killing symmetry vectors and global sections of the dual contact line bundle on the associated twistor contact manifold (see Theorem (3.14) for a precise statement).This thesis is organised as follows. In the first chapter we provide a short introduction into the theory of G-structures, and explain our notation. The material is classical except Section (1.5) where we give a new characterisation of Killing vector fields. In Chapter 2 we explain Merkulov's work which associates to any irreducible G-structure a contact complex manifold (twistor space) and vice versa (via deformation theory). This mathematical set-up is one of the basic requirements of our study. Chapter 3 is the main part of our thesis where we prove our main Theorems. We start with the special case of conformal structures whose twistor (ambitwistor) spaces were understood long ago. The twistor characterisation of conformal Killing vectors is given by Theorem (3.6). Next we switch to the general case and prove in Section (3.4) the main Theorem (3.14). Finally, we apply this theorem to get a new twistor description of symmetries on quaternionic manifolds (Theorem (3.17)).