AMS Chelsea Publishing 1934; 240 pp; hardcover Reprint/Revision History: reprinted 1991 ISBN10: 0828403368 ISBN13: 9780828403368 List Price: US$34 Member Price: US$30.60 Order Code: CHEL/336
 This work is intended to give the student a connected account of the subject of the differential invariants of generalized spaces, including the interesting and important discoveries in the field by LeviCivita, Weyl, and the author himself, and theories of Schouten, Veblen, Eisenhart and others. Table of Contents  \(N\)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 conformalaffine 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
