Four-Dimensional Manifolds and Projective Structure

1. Algebra, Topology and Geometry. 1.1. Notation. 1.2. Groups. 1.3. Vector Spaces and Linear Transformations. 1.4. Dual Spaces and Bilinear Forms. 1.5. Eigen-structure, Jordan Canonical Forms and Segre Types. 1.6. Lie algebras. 1.7. Topology. 1.8. Euclidean Geometry. 2. Manifold Theory. 2.1. Manifolds. 2.2. The Manifold Topology. 2.3. Vectors, Tensors and their Associated Bundles. 2.4. Vector and Tensor Fields. 2.5. Derived Maps and Pullbacks. 2.6. Integral Curves of Vector Fields. 2.7. Submanifolds and Quotient Manifolds. 2.8. Distributions. 2.9. Linear Connections and Curvature. 2.10. Lie Groups and Lie Algebras. 2.11. The Exponential Map for G. 2.12. Covering Manifolds. 2.13. Holonomy Theory. 3. Four-Dimensional Manifolds. 3.1. Metrics on 4-dimensional Manifolds. 3.2. The Connection, the Curvature and Associated Tensors. 3.3. Algebraic Remarks, Bivectors and Duals. 3.4. The Positive Definite Case and Tensor Classification. 3.5. The Curvature and Weyl Conformal Tensor. 3.6. The Lie Algebra o(4). 3.7. The holonomy structure of (M,g). 3.8. Curvature and Metric. 3.9. Sectional Curvature. 3.10. The Ricci Flat Case. 4. Four-Dimensional Lorentz Manifolds. 4.1. Lorentz Tangent Space Geometry. 4.2. Classification of Second Order Tensors. 4.3. Bivectors in Lorentz Signature. 4.4. The Lorentz Algebra o(1,3) and Lorentz Group. 4.5. The Curvature and Weyl Conformal Tensors. 4.6. Curvature Structure. 4.7. Sectional Curvature. 4.8. The Ricci Flat (Vacuum) Case. 5. Four-Dimensional Manifolds of Neutral Signature. 5.1. Neutral Tangent Space Geometry. 5.2. Algebra and Geometry of Bivectors. 5.3. Classification of Symmetric Second Order Tensors. 5.4. Classification of Bivectors. 5.5. The Lie Algebra o(2,2). 5.6. The Curvature Tensor. 5.7. The Weyl Conformal Tensor I. 5.8. The Weyl Conformal Tensor II. 5.9. Curvature Structure. 5.10. Sectional Curvature. 5.11. The Ricci-Flat Case. 5.12. Algebraic Classification Revisited. 6. A Brief Discussion of Geometrical Symmetry. 6.1. Introduction. 6.2. The Lie Derivative. 6.3. Symmetries of the Metric Tensor. 6.4. Affine and Projective Symmetry. 6.5. Orbits and isotropy algebras for K(M). 7. Projective Relatedness. 7.1. Recurrence and Holonomy. 7.2. Projective Relatedness. 7.3. The Sinjukov Transformation. 7.4. Introduction of the Curvature Tensor. 7.5. Einstein Spaces. 7.6. Projective Relatedness and Geometrical Symmetry. 7.7. The 1−form ψ. 7.8. Projective Relatedness in 4-dimensional Manifolds.