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The Differential Invariants of Generalized Spaces
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AMS Chelsea Publishing
1934; 240 pp; hardcover
Reprint/Revision History:
reprinted 1991
ISBN-10: 0-8284-0336-8
ISBN-13: 978-0-8284-0336-8
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 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
• 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
• 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 sub-spaces; 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
• 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