Discusses differential invariants of generalized spaces, including various discoveries in the field by Levi-Civita, Weyl, and the author himself, and theories of Schouten, Veblen, Eisenhart and others.
$N$-Dimensional Spaces: 1 Space. Coordinates; 2 Affine connection; 3 Affine geometry of paths; 4 Projective geometry of paths; 5 Riemann or metric space; 6 Space of distant parallelism; 7 Conformal space; 8 Weyl space. Gauge; 9 Transformation theory of space Affine and Related Invariants: 10 Tensors; 11 Invariants; 12 Parallel displacement of a vector around an infinitesimal closed circuit; 13 Covariant differentiation; 14 Alternative methods of covariant differentiation. Extension; 15 Differential parameters Projective Invariants: 16 Affine representation of projective spaces; 17 Some geometrical interpretations; 18 Projective tensors and invariants; 19 Transformations of the group $\star\mathfrak G$ Conformal Invariants: 20 Fundamental conformal-affine tensor; 21 Affine representation of conformal spaces; 22 Conformal tensors and invariants; 23 Completion of the incomplete covariant derivative. General case; 24 An extension of the preceding method; 25 Systems algebraically equivalent to the system of equations of transformation of the components of a conformal tensor; 26 Exceptional case $K=0$; 27 Exceptional case $L=0$; 28 The complete conformal curvature tensor and its successive covariant derivatives Normal Coordinates: 29 Affine normal coordinates; 30 Absolute normal coordinates; 31 Projective normal coordinates; 32 General theory of extension; 33 Some formulae of extension; 34 Scalar differentiation in a space of distant parallelism; 35 Differential invariants defined by means of normal coordinates. Normal tensors; 36 A generalization of the affine normal tensors; 37 Formulae of repeated extension; 38 A theorem on the affine connection; 39 Replacement theorems Spatial Identities: 40 Complete sets of identities; 41 Identities in the components of the normal tensors; 42 Identities of the space of distant parallelism; 43 Determination of the components of the normal tensors in terms of the components of their extensions; 44 Generalization of the preceding identities; 45 Space determination by tensor invariants; 46 Relations between the components of the extensions of the normal tensors; 47 Convergence proofs; 48 Relations between the components of certain invariants of the space of distant parallelism; 49 Determination of the components of the affine normal tensors in terms of the components of the curvature tensor and its covariant derivatives; 50 Curvature. Theorem of Schur; 51 Identities in the components of the projective curvature tensor; 52 Certain divergence identities; 53 A general method for obtaining divergence identities; 54 Numbers of algebraically independent components of certain spatial invariants Absolute Scalar Differential Invariants and Parameters: 55 Abstract groups; 56 Finite continuous groups; 57 Essential parameters; 58 The parameter groups; 59 Fundamental differential equations of an $r$-parameter group; 60 Transformation theory connected with the fundamental differential equations; 61 Equivalent $r$-parameter groups; 62 Constants of composition; 63 Group space and its structure; 64 Infinitesimal transformations; 65 Transitive and intransitive groups. Invariant subspaces; 66 Invariant functions; 67 Groups defined by the equations of transformation of the components of tensors; 68 Infinitesimal transformations of the affine and metric groups; 69 Differential equations of absolute affine and metric scalar differential invariants; 70 Absolute metric differential invariants of order zero; 71 General theorems on the independence of the differential equations; 72 Number of independent differential equations. Affine case; 73 Number of independent differential equations. Metric case; 74 Exceptional case of two dimensions; 75 Fundamental sets of absolute scalar differential invariants; 76 Rational invariants; 77 Absolute scalar differential parameters; 78 Independence of the differential equations of the differential parameters; 79 Fundamental sets of differential parameters; 80 Extension to relative tensor differential invariants The Equivalence Problem: 81 Equivalence of generalized spaces; 82 Normal coordinates and the equivalence problem; 83 Complete sets of invariants; 84 A theorem on mixed systems of partial differential equations; 85 Finite equivalence theorem for affinely connected spaces; 86 Finite equivalence theorem for metric spaces; 87 Finite; equivalence theorem for spaces of distant parallelism; 88 Finite equivalence theorem for projective spaces; 89 Equivalence of two dimensional conformal spaces; 90 Finite equivalence theorem for conformal spaces of three or more dimensions; 91 Spatial arithmetic invariants Reducibility of Spaces: 92 Differential conditions of reducibility; 93 Flat spaces; 94 Reducibility of the general affinely connected space to a space of distant parallelism; 95 Algebraic conditions for the reducibility of the affine space of paths to a metric space; 96 Algebraic conditions for the reducibility of the affine space of paths to a Weyl space Functional Arbitrariness of Spatial Invariants: 97 Regular systems of partial differential equations; 98 Extension to tensor differential equations; 99 General existence theorem for regular systems; 100 Groups of independent components; 101 Special case of two dimensions; 102 General case of $n(\geqq 3)$ dimensions; 103 The existence theorems in normal coordinates; 104 Convergence of the $A$ series; 105 Convergence of the $g$ series Index.