1st Edition

# Further Advances in Twistor Theory, Volume III Curved Twistor Spaces

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Although twistor theory originated as an approach to the unification of quantum theory and general relativity, twistor correspondences and their generalizations have provided powerful mathematical tools for studying problems in differential geometry, nonlinear equations, and representation theory. At the same time, the theory continues to offer promising new insights into the nature of quantum theory and gravitation.

Further Advances in Twistor Theory, Volume III: Curved Twistor Spaces is actually the fourth in a series of books compiling articles from Twistor Newsletter-a somewhat informal journal published periodically by the Oxford research group of Roger Penrose. Motivated both by questions in differential geometry and by the quest to find a twistor correspondence for general Ricci-flat space times, this volume explores deformed twistor spaces and their applications.

Articles from the world's leading researchers in this field-including Roger Penrose-have been written in an informal, easy-to-read style and arranged in four chapters, each supplemented by a detailed introduction. Collectively, they trace the development of the twistor programme over the last 20 years and provide an overview of its recent advances and current status.

The Nonlinear Graviton and Related Construction, L.H. Mason

The Good Cut Equation Revisited, K.P. Tod

Sparling-Tod Metric = 3D Eguchi Hanson, G. Burnett-Stuart

The Wave Equation Transfigured, C.R. LeBrun

Conformal Killing Vectors and Reduced Twistor Spaces, P.E. Jones

An Alternative Interpretation of some Nonlinear Graviton, P.E. Jones

H-Space from a Different Direction, C.N. Kazameh and E.T. Newman

Complex Quaternionic Kähler Maniforlds, M.G. Eastwood

A.L.E. Gravitational Instatons and the Icosahedron, P.B. Kronheimer

The Einstein Bundle of a Nonlinear Graviton, M.G. Eastwood

Example of Anti-Self-Dual Metrics, C.R. LeBrun

Some Quaternionically Equivalent Einstein Metrics, A.F. Swann

On he Topology of Quaternionic Manifolds, C.R. LeBrun

Homogeneity of Twistor Spaces, A.F. Swann

The Topology of Anti-Self-Dual 4-Manifolds, C.R. LeBrun

Metrics with SD Weyl Tensor from Painlevé-VI, K.P. Tod

Indefinite Conformally-ASD Metrics on S2 x S2, K.P. Tod

Cohomology of a Quaternionic Complex, R. Horan

Conformally Invariant Differential Operators on Spin Bundles, M.G. Eastwood

A Twistorial Construction of (1,1)-Geodesic Maps, P.Z. Kobak

Exceptional HyperKähler Reductions, P.Z. Kobak and A.F. Swann

A Nonlinear Graviton from the Sine-Gordon Equation, M. Dunajski

A Recursion Operator for ASD Vacuums and ZRM Fields on ASD Background, M. Dunaski and L.J. Mason

SPACES OF COMPLEX NULL GEODESICS

Introduction to Spaces of Complex Null Geodesic, L. Mason

Null Geodesics and Conformal Structures, C.R. LeBrun

Complex Null Geodesics in Dimension Three, C.R. LeBrun

Null Geodesics and Contact Structure, C.R. LeBrun

Heaven with a Cosmological Constant, C.R. LeBrun

Some Remakes on Non-Abelian Sheaf Cohomology, M.G. Eastwood

Formal Thickenings of Ambitwistors for Curved Space-Times, C.R. LeBrun

Superambitwistors, N.G. Eastwood

Formal Neighbourhoods, Supermanifolds and Relativised Algebras, R. Baston

Quaternionic Geometry and the Future Tube, C.R. LeBrun

Deformation of Ambitwistor Space and Vanishing Bach Tensors, R.H. Baston and L.J. Mason

Formal Neighbourhoods for Curved Ambitwistors, R.J. Baston and L.J. Mason

Towards and Ambitwistor Description of Gravity, J. Isenberg and P. Yasskin

HYPERSURFACE TWISTORS AND CAUCHY-RIEMANN STRUCTURES

Introduction to Hypersurface Twistors and Cauchy-Riemann Structure, L.J. Mason

A Review of Hypersuface Twistors, R.S. Ward

Twistor CR Manifolds, C.R. LeBrun

Twistor CR Structure and Initial Data, C.R. LeBrun

Visualizing Twistor CR Structures, C.R. LeBrun

The Twistor Theory of Hypersurfaces in Space-Time, G.A.J. Sparling

Twistors, Spinors, and the Einstein Vacuum Equations, G.A.J. Sparling

Einstein Vacuum Equations, G.A.J. Sparling

On Bryant's Condition for Holomorphic curves in CR-Spaces, R. Penrose

The Hill-Penrose-Sparling C.R.-Folds, M.G. Eastwood

The Structure and Evolution of Hypersurfaces Twistor Spaces, L.J. Mason

The Chern-Moser Connection for Hypersurface Twistor CR Manifolds, L.J. Mason

The constraint and Evolution Equations for Hypersurface CR Manifolds, L.J. Mason

A Characterization of Twistor CR Manifold, L.J. Mason

The Kähler Structure on Asymptotic Twistor Space, L.H. Mason

Twistor Cauchy-Riemann Manifolds for Algebraically Special Space-Times, L/H. Mason

Causal Relations and Linking in Twistor Space, R. Low

Hypersurface Twistors, L.H. Mason

A Twistorial Approach to the full Vacuum Equations, L.H. Mason and R. Penrose

A Note on Causal Relations and Twistor Space, R. Low

TOWARDS A TWISTOR DESCRIPTION OF GENERAL SPACE TIMES

Towards a Twistor Description of General Space-Times; Introductory Comments, R. Penrose

Remarks on the Sparling and Eguchi-Hanson (Googly?) Gravitons

A New Angle on the Googly Graviton, R. Penrose

Concerning a Fourier Contour Integral, R. Penrose

The Googly Maps for the Eguchi-Hanson/Sparling-Tod Graviton, P.R. Law

Physical Left-Right Symmetry and Googlies, R. Penrose

On the Geometry of Googly Maps, R. Penrose and P.R. Law

A Prosaic Approach to Googlies, A. Helfer

More on Googlies, A. Helfer

A Note on Sparling's 3-Form, r. Penrose

Remarks on Curved-Space Twistor Theory and Googlies, R. Penrose

Relative Cohomology, Googlies, and Deformations of I, R. Penrose

Is the Plebanski Viewpoint Relevant to the Googly Problem? G. Burnett-Stuart

Note on the Geometry of the Googly Mappings, P. Law

Exponentiating a Relative H2, R. Penrose

The Complex Structure of Deformed Twstor Space, P. Law

### Biography

St Peter’s College and the Mathematical Institute, Oxford, King’s College London, Instytut Matematyki, Uniwersytet Jagielloński Kraków, Center for Mathematical Sciences, Munich University of Technology, Munich

"… In summary, these articles contain many interesting facts and provocative ideas that do not otherwise appear in the published literature."

-Mathematical Reviews