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  • Contents

    1 Geotensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1 Linear Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Autovector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Oriented Autoparallel Segments on a Manifold . . . . . . . . . . . . . 30 1.4 Lie Group Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.5 Geotensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    2 Tangent Autoparallel Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1 Declinative (Autovector Spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.2 Declinative (Connection Manifolds) . . . . . . . . . . . . . . . . . . . . . . . . 87 2.3 Example: Mappings from Linear Spaces into Lie Groups . . . . . 92 2.4 Example: Mappings Between Lie Groups . . . . . . . . . . . . . . . . . . . 100 2.5 Covariant Declinative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

    3 Quantities and Measurable Qualities . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.1 One-dimensional Quality Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.2 Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.3 Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

    4 Intrinsic Physical Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.1 Intrinsic Laws in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.2 Example: Law of Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.3 Example: Ideal Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

    A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 A.1 Adjoint and Transpose of a Linear Operator . . . . . . . . . . . . . . . . . 155 A.2 Elementary Properties of Groups (in Additive Notation) . . . . . 159 A.3 Troupe Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 A.4 Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 A.5 Function of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 A.6 Logarithmic Image of SL(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 A.7 Logarithmic Image of SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 A.8 Central Matrix Subsets as Autovector Spaces . . . . . . . . . . . . . . . . 175 A.9 Geometric Sum on a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 A.10 Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

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  • XII Contents

    A.11 Total Riemann Versus Metric Curvature . . . . . . . . . . . . . . . . . . . . 184 A.12 Basic Geometry of GL(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 A.13 Lie Groups as Groups of Transformations . . . . . . . . . . . . . . . . . . . 204 A.14 SO(3) − 3D Euclidean Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.15 SO(3, 1) − Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . 218 A.16 Coordinates over SL(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 A.17 Autoparallel Interpolation Between Two Points . . . . . . . . . . . . . 224 A.18 Trajectory on a Lie Group Manifold . . . . . . . . . . . . . . . . . . . . . . . . 225 A.19 Geometry of the Concentration−Dilution Manifold . . . . . . . . . . 229 A.20 Dynamics of a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 A.21 Basic Notation for Deformation Theory . . . . . . . . . . . . . . . . . . . . . 234 A.22 Isotropic Four-indices Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 A.23 9D Representation of 3D Fourth Rank Tensors . . . . . . . . . . . . . . . 239 A.24 Some sketches of the configuration space (in 2D) . . . . . . . . . . . . 242 A.25 Saint-Venant Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 A.26 Electromagnetism versus Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 246

    Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

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  • XIV Contents

    List of Appendices

    A.1 Adjoint and Transpose of a Linear Operator . . . . . . . . . . . . . . . . 153 A.2 Elementary Properties of Groups (in Additive Notation) . . . . 157 A.3 Troupe Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.4 Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 A.5 Function of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 A.6 Logarithmic Image of SL(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.7 Logarithmic Image of SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 A.8 Central Matrix Subsets as Autovector Spaces . . . . . . . . . . . . . . . 173 A.9 Geometric Sum on a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 A.10 Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 A.11 Total Riemann Versus Metric Curvature . . . . . . . . . . . . . . . . . . . . 182 A.12 Basic Geometry of GL(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 A.13 Lie Groups as Groups of Transformations . . . . . . . . . . . . . . . . . . 203 A.14 SO(3) − 3D Euclidean Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 A.15 SO(3,1) − Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 217 A.16 Coordinates over SL(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 A.17 Autoparallel Interpolation Between Two Points . . . . . . . . . . . . 223 A.18 Trajectory on a Lie Group Manifold . . . . . . . . . . . . . . . . . . . . . . . . 224 A.19 Geometry of the Concentration−Dilution Manifold . . . . . . . . . 228 A.20 Dynamics of a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 A.21 Basic Notation for Deformation Theory . . . . . . . . . . . . . . . . . . . . 233 A.22 Isotropic Four-indices Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 A.23 9D Representation of Fourth Rank Symmetric Tensors . . . . . . 238 A.24 Rotation of Strain and Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 A.25 Macro-rotations, Micro-rotations, and Strain . . . . . . . . . . . . . . . 242 A.26 Elastic Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 A.27 Saint-Venant Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 A.28 Electromagnetism versus Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 249DR

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  • Overview

    One-dimensional Quality Spaces

    Consider a one-dimensional space, each point N of it representing a musical note. This line has to be imagined infinite in its two senses, with the infinitely acute tones at one “end” and the infinitely grave tones at the other “end”. Musicians can immediately give the distance between two points of the space, i.e., between two notes, using the octave as unit. To express this distance by a formula, we may choose to represent a note by its frequency, ν , or by its period, τ . The distance between two notes N1 and N2 is1

    Dmusic(N1,N2) = | log2 ν2 ν1 | = | log2

    τ2 τ1 | . (1)

    This distance is the only one that has the following properties:

    – its expression is identical when using the positive quantity ν = 1/τ or its inverse, the positive quantity τ = 1/ν ;

    – it is additive, i.e., for any set of three ordered points {N1,N2,N3} , the distance from point N1 to point N2 , plus the distance from point N2 to point N3 , equals the distance from point N1 to point N3 .

    This one-dimensional space (or, to be more precise, this one-dimensional manifold) is a simple example of a quality space. It is a metric manifold (the distance between points is defined). The quantities frequency ν and period τ are two of the coordinates that can be used on the quality space of the musical notes to characterize its points. Infinitely many more coordinates are, of course, possible, like the logarithmic frequency ν∗ = log(ν/ν0) , the cube of the frequency, η = ν3 , etc. Given the expression for the distance in some coordinate system, it is easy to obtain an expression for it using another coordinate system. For instance, it follows from equation (1) that the distance between two musical notes is, in terms of the logar

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