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

• Chapter 1: The nonlinear graviton and related constructions, III.1.1 The Nonlinear Graviton and Related Constructions, III.1.2 The Good Cut Equation Revisited, III.1.3 Sparling-Tod Metric = Eguchi-Hanson, III.1.4 The Wave Equation Transfigured, III.1.5 Conformal Killing Vectors and Reduced Twistor Spaces, III.1.6 An Alternative Interpretation of Some Nonlinear Gravitons, III.1.7 ℋ-Space from a Different Direction, III.1.8 Complex Quaternionic Kähler Manifolds, III.1.9 A.L.E. Gravitational Instantons and the Icosahedron, III.1.10 The Einstein Bundle of a Nonlinear Graviton, III.1.11 Examples of Anti-Self-Dual Metrics, III.1.12 Some Quaternionically Equivalent Einstein Metrics, III.1.13 On the Topology of Quaternionic Manifolds, III.1.14 Homogeneity of Twistor Spaces, III.1.15 The Topology of Anti-Self-Dual 4-Manifolds, III.1.16 Metrics with S.D. Weyl Tensor from Painlevé-VI, III.1.17 Indefinite Conformally-A.S.D. Metrics on S2 × S2, III.1.18 Cohomology of a Quaternionic Complex, III.1.19 Conformally Invariant Differential Operators on Spin Bundles, III.1.20 A Twistorial Construction of (1, 1)-Geodesic Maps, III.1.21 Exceptional Hyper-Kähler Reductions, III.1.22 A Nonlinear Graviton from the Sine-Gordon Equation, III.1.23 A Recursion Operator for A.S.D. Vacuums and ZRM Fields on A.S.D Backgrounds, Chapter 2: Spaces of complex null geodesies, III.2.1 Introduction to Spaces of Complex Null Geodesies, III.2.2 Null Geodesics and Conformai Structures, III.2.3 Complex Null Geodesics in Dimension Three, III.2.4 Null Geodesics and Contact Structures, III.2.5 Heaven with a Cosmological Constant, III.2.6 Some Remarks on Non-Abelian Sheaf Cohomology, III.2.7 Superstructure versus Formal Neighbourhoods, III.2.8 Formal Thickenings of Ambitwistors for Curved Space-Time, III.2.9 Deformations of Ambitwistor Space, III.2.10 Ambitwistors and Yang-Mills Fields in Self-Dual Space-Times, III.2.11 Superambitwistors, III.2.12 Formal Neighbourhoods, Supermanifolds and Relativised Algebras, III.2.13 Quaternionic Geometry and the Future Tube, III.2.14 Deformation of Ambitwistor Space and Vanishing Bach Tensors, III.2.15 Formal Neighbourhoods for Curved Ambitwistors, III.2.16 Towards an Ambitwistor Description of Gravity, Chapter 3: Hypersurface twistors and Cauchy-Riemann manifolds, III.3.1 Introduction to Hypersurface Twistors and Cauchy-Riemann Structures, III.3.2 A Review of Hypersurface Twistors, III.3.3 Twistor CR Manifolds, III.3.4 Twistor CR Structures and Initial Data, III.3.5 Visualizing Twistor CR Structures, III.3.6 The Twistor Theory of Hypersurfaces in Space-Time, III.3.7 Twistors, Spinors and the Einstein Vacuum Equations, III.3.8 Einstein Vacuum Equations, III.3.9 On Bryant's Condition for Holomorphic Curves in CR-Spaces, III.3.10 The Hill-Penrose-Sparling C.R.-Folds, III.3.11 The Structure and Evolution of Hypersurface Twistor Spaces, III.3.12 The Chern-Moser Connection for Hypersurface Twistor CR Manifolds, III.3.13 The Constraint and Evolution Equations for Hypersurface CR Manifolds, III.3.14 A Characterization of Twistor CR Manifolds, III.3.15 The Kähler Structure on Asymptotic Twistor Space, III.3.16 Twistor CR manifolds for Algebraically Special Space-Times, III.3.17 Causal Relations and Linking in Twistor Space, III.3.18 Hypersurface Twistors, III.3.19 A Twistorial Approach to the Full Vacuum Equations, III.3.20 A Note on Causal Relations and Twistor Space, Chapter 4: Towards a twistor description of general space-times, III.4.1 Towards a Twistor Description of General Space-Times; Introductory Comments, III.4.2 Remarks on the Sparling and Eguchi-Hanson (Googly?) Gravitons, III.4.3 A New Angle on the Googly Graviton, III.4.4 Concerning a Fourier Contour Integral, III.4.5 The Googly Maps for the Eguchi-Hanson/Sparling-Tod Graviton, III.4.6 Physical Left-Right Symmetry and Googlies, III.4.7 On the Geometry of Googly Maps, III.4.8 A Prosaic Approach to Googlies, III.4.9 More on Googlies, III.4.10 A Note on Sparling's 3-Form, III.4.11 Remarks on Curved-Space Twistor Theory and Googlies, III.4.12 Relative Cohomology, Googlies and Deformations of I, III.4.13 Is the Plebanski Viewpoint Relevant to the Googly Problem?, III.4.14 Note on the Geometry of the Googly Mappings, III.4.15 Exponentiating a Relative H2, III.4.16 The Complex Structure of Deformed Twistor Space, III.4.17 Local Twistor Transport at J+ : An Approach to the Googly, III.4.18 An Approach to a Coordinate Free Calculus at J, III.4.19 Twistor Theory for Vacuum Space-Time: A New Approach, III.4.20 Twistors as Charges for Spin 3/2 in Vacuum, III.4.21 Light Cone Cuts and Yang-Mills Holonomies: a New Approach, III.4.22 Twistor as Spin 3/2 Charges Continued: SL(3, ℂ) Bundles, III.4.23 The Most General (2,2) Self-Dual Vacuum: A Googly Approach, III.4.24 A Comment on the Preceding Article, III.4.25 Spin 3/2 Fields and Local Twistors, III.4.26 Another View of the Spin 3/2 Equation, III.4.27 The Bach Equations as an Exact Set of Spinor Fields, III.4.28 A Novel Approach to Quantum Gravity, III.4.29 Twistors and the Time-Irreversibility of State-Vector Reduction, III.4.30 Twistors and State-Vector Reduction, Bibliography, Index

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

• Chapter 1: The nonlinear graviton and related constructions, III.1.1 The Nonlinear Graviton and Related Constructions, III.1.2 The Good Cut Equation Revisited, III.1.3 Sparling-Tod Metric = Eguchi-Hanson, III.1.4 The Wave Equation Transfigured, III.1.5 Conformal Killing Vectors and Reduced Twistor Spaces, III.1.6 An Alternative Interpretation of Some Nonlinear Gravitons, III.1.7 ℋ-Space from a Different Direction, III.1.8 Complex Quaternionic Kähler Manifolds, III.1.9 A.L.E. Gravitational Instantons and the Icosahedron, III.1.10 The Einstein Bundle of a Nonlinear Graviton, III.1.11 Examples of Anti-Self-Dual Metrics, III.1.12 Some Quaternionically Equivalent Einstein Metrics, III.1.13 On the Topology of Quaternionic Manifolds, III.1.14 Homogeneity of Twistor Spaces, III.1.15 The Topology of Anti-Self-Dual 4-Manifolds, III.1.16 Metrics with S.D. Weyl Tensor from Painlevé-VI, III.1.17 Indefinite Conformally-A.S.D. Metrics on S2 × S2, III.1.18 Cohomology of a Quaternionic Complex, III.1.19 Conformally Invariant Differential Operators on Spin Bundles, III.1.20 A Twistorial Construction of (1, 1)-Geodesic Maps, III.1.21 Exceptional Hyper-Kähler Reductions, III.1.22 A Nonlinear Graviton from the Sine-Gordon Equation, III.1.23 A Recursion Operator for A.S.D. Vacuums and ZRM Fields on A.S.D Backgrounds, Chapter 2: Spaces of complex null geodesies, III.2.1 Introduction to Spaces of Complex Null Geodesies, III.2.2 Null Geodesics and Conformai Structures, III.2.3 Complex Null Geodesics in Dimension Three, III.2.4 Null Geodesics and Contact Structures, III.2.5 Heaven with a Cosmological Constant, III.2.6 Some Remarks on Non-Abelian Sheaf Cohomology, III.2.7 Superstructure versus Formal Neighbourhoods, III.2.8 Formal Thickenings of Ambitwistors for Curved Space-Time, III.2.9 Deformations of Ambitwistor Space, III.2.10 Ambitwistors and Yang-Mills Fields in Self-Dual Space-Times, III.2.11 Superambitwistors, III.2.12 Formal Neighbourhoods, Supermanifolds and Relativised Algebras, III.2.13 Quaternionic Geometry and the Future Tube, III.2.14 Deformation of Ambitwistor Space and Vanishing Bach Tensors, III.2.15 Formal Neighbourhoods for Curved Ambitwistors, III.2.16 Towards an Ambitwistor Description of Gravity, Chapter 3: Hypersurface twistors and Cauchy-Riemann manifolds, III.3.1 Introduction to Hypersurface Twistors and Cauchy-Riemann Structures, III.3.2 A Review of Hypersurface Twistors, III.3.3 Twistor CR Manifolds, III.3.4 Twistor CR Structures and Initial Data, III.3.5 Visualizing Twistor CR Structures, III.3.6 The Twistor Theory of Hypersurfaces in Space-Time, III.3.7 Twistors, Spinors and the Einstein Vacuum Equations, III.3.8 Einstein Vacuum Equations, III.3.9 On Bryant's Condition for Holomorphic Curves in CR-Spaces, III.3.10 The Hill-Penrose-Sparling C.R.-Folds, III.3.11 The Structure and Evolution of Hypersurface Twistor Spaces, III.3.12 The Chern-Moser Connection for Hypersurface Twistor CR Manifolds, III.3.13 The Constraint and Evolution Equations for Hypersurface CR Manifolds, III.3.14 A Characterization of Twistor CR Manifolds, III.3.15 The Kähler Structure on Asymptotic Twistor Space, III.3.16 Twistor CR manifolds for Algebraically Special Space-Times, III.3.17 Causal Relations and Linking in Twistor Space, III.3.18 Hypersurface Twistors, III.3.19 A Twistorial Approach to the Full Vacuum Equations, III.3.20 A Note on Causal Relations and Twistor Space, Chapter 4: Towards a twistor description of general space-times, III.4.1 Towards a Twistor Description of General Space-Times; Introductory Comments, III.4.2 Remarks on the Sparling and Eguchi-Hanson (Googly?) Gravitons, III.4.3 A New Angle on the Googly Graviton, III.4.4 Concerning a Fourier Contour Integral, III.4.5 The Googly Maps for the Eguchi-Hanson/Sparling-Tod Graviton, III.4.6 Physical Left-Right Symmetry and Googlies, III.4.7 On the Geometry of Googly Maps, III.4.8 A Prosaic Approach to Googlies, III.4.9 More on Googlies, III.4.10 A Note on Sparling's 3-Form, III.4.11 Remarks on Curved-Space Twistor Theory and Googlies, III.4.12 Relative Cohomology, Googlies and Deformations of I, III.4.13 Is the Plebanski Viewpoint Relevant to the Googly Problem?, III.4.14 Note on the Geometry of the Googly Mappings, III.4.15 Exponentiating a Relative H2, III.4.16 The Complex Structure of Deformed Twistor Space, III.4.17 Local Twistor Transport at J+ : An Approach to the Googly, III.4.18 An Approach to a Coordinate Free Calculus at J, III.4.19 Twistor Theory for Vacuum Space-Time: A New Approach, III.4.20 Twistors as Charges for Spin 3/2 in Vacuum, III.4.21 Light Cone Cuts and Yang-Mills Holonomies: a New Approach, III.4.22 Twistor as Spin 3/2 Charges Continued: SL(3, ℂ) Bundles, III.4.23 The Most General (2,2) Self-Dual Vacuum: A Googly Approach, III.4.24 A Comment on the Preceding Article, III.4.25 Spin 3/2 Fields and Local Twistors, III.4.26 Another View of the Spin 3/2 Equation, III.4.27 The Bach Equations as an Exact Set of Spinor Fields, III.4.28 A Novel Approach to Quantum Gravity, III.4.29 Twistors and the Time-Irreversibility of State-Vector Reduction, III.4.30 Twistors and State-Vector Reduction, Bibliography, Index

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

• Chapter 1: The nonlinear graviton and related constructions, III.1.1 The Nonlinear Graviton and Related Constructions, III.1.2 The Good Cut Equation Revisited, III.1.3 Sparling-Tod Metric = Eguchi-Hanson, III.1.4 The Wave Equation Transfigured, III.1.5 Conformal Killing Vectors and Reduced Twistor Spaces, III.1.6 An Alternative Interpretation of Some Nonlinear Gravitons, III.1.7 ℋ-Space from a Different Direction, III.1.8 Complex Quaternionic Kähler Manifolds, III.1.9 A.L.E. Gravitational Instantons and the Icosahedron, III.1.10 The Einstein Bundle of a Nonlinear Graviton, III.1.11 Examples of Anti-Self-Dual Metrics, III.1.12 Some Quaternionically Equivalent Einstein Metrics, III.1.13 On the Topology of Quaternionic Manifolds, III.1.14 Homogeneity of Twistor Spaces, III.1.15 The Topology of Anti-Self-Dual 4-Manifolds, III.1.16 Metrics with S.D. Weyl Tensor from Painlevé-VI, III.1.17 Indefinite Conformally-A.S.D. Metrics on S2 × S2, III.1.18 Cohomology of a Quaternionic Complex, III.1.19 Conformally Invariant Differential Operators on Spin Bundles, III.1.20 A Twistorial Construction of (1, 1)-Geodesic Maps, III.1.21 Exceptional Hyper-Kähler Reductions, III.1.22 A Nonlinear Graviton from the Sine-Gordon Equation, III.1.23 A Recursion Operator for A.S.D. Vacuums and ZRM Fields on A.S.D Backgrounds, Chapter 2: Spaces of complex null geodesies, III.2.1 Introduction to Spaces of Complex Null Geodesies, III.2.2 Null Geodesics and Conformai Structures, III.2.3 Complex Null Geodesics in Dimension Three, III.2.4 Null Geodesics and Contact Structures, III.2.5 Heaven with a Cosmological Constant, III.2.6 Some Remarks on Non-Abelian Sheaf Cohomology, III.2.7 Superstructure versus Formal Neighbourhoods, III.2.8 Formal Thickenings of Ambitwistors for Curved Space-Time, III.2.9 Deformations of Ambitwistor Space, III.2.10 Ambitwistors and Yang-Mills Fields in Self-Dual Space-Times, III.2.11 Superambitwistors, III.2.12 Formal Neighbourhoods, Supermanifolds and Relativised Algebras, III.2.13 Quaternionic Geometry and the Future Tube, III.2.14 Deformation of Ambitwistor Space and Vanishing Bach Tensors, III.2.15 Formal Neighbourhoods for Curved Ambitwistors, III.2.16 Towards an Ambitwistor Description of Gravity, Chapter 3: Hypersurface twistors and Cauchy-Riemann manifolds, III.3.1 Introduction to Hypersurface Twistors and Cauchy-Riemann Structures, III.3.2 A Review of Hypersurface Twistors, III.3.3 Twistor CR Manifolds, III.3.4 Twistor CR Structures and Initial Data, III.3.5 Visualizing Twistor CR Structures, III.3.6 The Twistor Theory of Hypersurfaces in Space-Time, III.3.7 Twistors, Spinors and the Einstein Vacuum Equations, III.3.8 Einstein Vacuum Equations, III.3.9 On Bryant's Condition for Holomorphic Curves in CR-Spaces, III.3.10 The Hill-Penrose-Sparling C.R.-Folds, III.3.11 The Structure and Evolution of Hypersurface Twistor Spaces, III.3.12 The Chern-Moser Connection for Hypersurface Twistor CR Manifolds, III.3.13 The Constraint and Evolution Equations for Hypersurface CR Manifolds, III.3.14 A Characterization of Twistor CR Manifolds, III.3.15 The Kähler Structure on Asymptotic Twistor Space, III.3.16 Twistor CR manifolds for Algebraically Special Space-Times, III.3.17 Causal Relations and Linking in Twistor Space, III.3.18 Hypersurface Twistors, III.3.19 A Twistorial Approach to the Full Vacuum Equations, III.3.20 A Note on Causal Relations and Twistor Space, Chapter 4: Towards a twistor description of general space-times, III.4.1 Towards a Twistor Description of General Space-Times; Introductory Comments, III.4.2 Remarks on the Sparling and Eguchi-Hanson (Googly?) Gravitons, III.4.3 A New Angle on the Googly Graviton, III.4.4 Concerning a Fourier Contour Integral, III.4.5 The Googly Maps for the Eguchi-Hanson/Sparling-Tod Graviton, III.4.6 Physical Left-Right Symmetry and Googlies, III.4.7 On the Geometry of Googly Maps, III.4.8 A Prosaic Approach to Googlies, III.4.9 More on Googlies, III.4.10 A Note on Sparling's 3-Form, III.4.11 Remarks on Curved-Space Twistor Theory and Googlies, III.4.12 Relative Cohomology, Googlies and Deformations of I, III.4.13 Is the Plebanski Viewpoint Relevant to the Googly Problem?, III.4.14 Note on the Geometry of the Googly Mappings, III.4.15 Exponentiating a Relative H2, III.4.16 The Complex Structure of Deformed Twistor Space, III.4.17 Local Twistor Transport at J+ : An Approach to the Googly, III.4.18 An Approach to a Coordinate Free Calculus at J, III.4.19 Twistor Theory for Vacuum Space-Time: A New Approach, III.4.20 Twistors as Charges for Spin 3/2 in Vacuum, III.4.21 Light Cone Cuts and Yang-Mills Holonomies: a New Approach, III.4.22 Twistor as Spin 3/2 Charges Continued: SL(3, ℂ) Bundles, III.4.23 The Most General (2,2) Self-Dual Vacuum: A Googly Approach, III.4.24 A Comment on the Preceding Article, III.4.25 Spin 3/2 Fields and Local Twistors, III.4.26 Another View of the Spin 3/2 Equation, III.4.27 The Bach Equations as an Exact Set of Spinor Fields, III.4.28 A Novel Approach to Quantum Gravity, III.4.29 Twistors and the Time-Irreversibility of State-Vector Reduction, III.4.30 Twistors and State-Vector Reduction, Bibliography, Index