Geometry of Complex Domains; a seminar conducted by Professors Oswald Veblen and John Von Neumann, 1935-36.

CONTENTS

Chapter I. SPINORS AND PROJECTIVE GEOMETRY
1. The Minkowski Space Represented by Hermitian Matrices
2. The Complex Projective Line
3. The Lorentz Group Isomorphic to the Quadric Group
4. The Projective Group in P_1 Isomorphic to the Proper Lorentz Group
5. The Antiprojective Group in P_1 Isomorphic to the Lorentz Group
6. Coordinate Transformations and Tensor Calculus
7. The Alternating Numerical Tensors
8. Dual Coordinates in R_3
9. The Spinor Caleulus in P_1
10. Transformations of Coordinates in P_1
11. Involutions in P_1
12. Antiinvolutions in P_1
13. Point-Place Reflections in R_3
14. Line Reflections in R_3
15. Factorization of the Fundamental Quadratic Form

Chapter II.
1. Underlying and Tangent Spaces
2. Spin and Gauge Spaces
3. Definition of Spinors
4. Gauge Transformations
5. Spinors of Weight -1/2
6. Spinors of Qther Weights
7. Spinors of Indices I≠0 and J=0
8. Differentiation of Spinors
9. Invariant Differential Equations
10. Dirac Equations
11. " " (Continued)
12. " " (Continued)
13. Current Vector

Chapter III.
1. Covariant Differentiation
2. The Transformation Law of Γ^A_Bj
3. Examples of Covariant Differentiation
4. Covariant Differentiation of Spinors with Tensor Indices
5. Relations between Γ^A_Bi and {i_jk}
6. " " " "
7. Extension to the General Theory of Relativity
8. Geodesic Spin Coordinates
9. Definition of the Covariant Derivative of a Spinor
10. Relations between the Curvature Tensor and the Curvature Spinor
11. Dirac Equations

Chapter IV. PROJECTIVE GEOMETRY OF (k-1)-DIMENSIONS
1. Definition of Projective Spaces
2. Hyperplanes
3. Coordinates of Linear Subspaces
4. Antiprojective Group
5. Matrix Notation
6. Commuting Transformations
7. Involutions
8. Anti-Involutions
9. Polarities
10. Antipolarities

Chapter V. LINEAR FAMILIES OF REFLECTIONS
1. Statement of the Problem
2. The Centered Euclidean Space E_m+1
3. Transformations of Linear Families of Involutions
4. Existence of Linear Families of Involutions
5. Equivalence of -/- sets
6. Algebraic Properties of -/- sets
7. Representation of Rotation Group in E_m+1 by Collineations in P_k-1
8. The Case k=2
9. Commutative and Anticommutative Involutions
10. Equivalence of Pairs of Anticommuting Involutions
11. The Reguli Determined by Two Anticommuting Involutions
12. Algebraic Discussion of the Involutions
13. Proof of Theorems (5.1) to (5.4)

Chapter VI. THE EXTENSION TO CORRELATIONS
1. The Dual Mapping of E_2v+1 onto Itself
2. Representation of Improper Orthogonal Transformations
3. The Invariant Polarity
4. Geometrical Properties of the Invariant Polarity
5. The (1-2) Matrix Representation of H_2v+1
6. Linear Families of Correlations
7. The Representation of H^+_2v+2 by Collineations in P_2^v-1


Chapter VII. TENSOR COORDINATES OF LINEAR SPACES
1. Introduction
2. Definition of the Coordinate Tensors
3. The Quadratic Identities
4. Joins and Intersections
5. The Quadratic Form
6. Linear Spaces on a Quadric

Chapter VIII. REPRESENTATION OF LINEAR SPACES ON A QUADRIC IN P_2v-1
1. The Projective Space P_2v-1
2. The Correspondence between Tensor Sets and Matrices
3. The Collineations Corresponding to Linear Spaces on the Quadric
4. Spaces in P_2^v-1 Determined by Spaces on the Quadric in P_2^v-1
5. Properties of R_B and N_B
6. Geometry of a Generalization of the Pluecker-Klein Correspondence
7. Collineation Representation of H^+_2v for v > 2
8. Matrix Representation of H^+_2v for v > 2
9. Representations of H^+_2v and H^+_4

Chapter IX. THE LORENTZ GROUPS
1. Definition of the Lorentz Groups
2. Definition and Spinor Representation of the Antiorthogonal Group A_m+1
3. The Invariant Antiinvolution and Antipolarity
4. Reality of the Spinor Representation Of L_2v+1;s
5. γ-Sets in which Each Matrix is Real or Pure Imaginary
6. Spatial and Temporal Signatures of Lorentz Matrices
7. The Invariant Spinors Associated with L_2v;s