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Introductory Differential Geometry for Physicists - A. Visconti

Introductory Differential Geometry for Physicists


Published: 1992
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This book develops the mathematics of differential geometry in a way more intelligible to physicists and other scientists interested in this field. This book is basically divided into 3 levels; level 0, the nearest to intuition and geometrical experience, is a short summary of the theory of curves and surfaces; level 1 repeats, comments and develops upon the traditional methods of tensor algebra analysis and level 2 is an introduction to the language of modern differential geometry. A final chapter (chapter IV) is devoted to fibre bundles and their applications to physics. Exercises are provided to amplify the text material.

"... the book is very suitable for advanced undergraduates or researchers in other areas interested in an introduction to the subject ... A very interesting feature of this book is that all the examples and exercises are motivated by important physical theories. Therefore the book is also of interest to mathematicians." Jair Koiller Mathematical Reviews "The present book certainly provides the basic tools in an easily digestible form and with plenty of concrete mathematical detail. As such I am sure it will be found useful by a research student intending to take up research in this area, or someone needing to brush up on a particular topic ... it should prove useful for a postgraduate lecture course." G W Gibbons Contemporary Physics "The many complements and exercises and their connections to physics are the strong point of this book; they are due to the author's philosophy that an abstract introduction of mathematical concepts should be motivated to students in physics by examples and applications while a purely intuitive and heuristic differential geometric education of physicists is not sufficient." U Simon Mathematics Abstracts

Level 0: The Intuitive Approach - Theory of Surfaces (19th Century)p. 1
Curvesp. 2
Surfaces - Topological Invariantsp. 5
Geometry on a Surface or Riemannian Geometryp. 15
Geodesicsp. 23
Generalization of the Concept of Tangent and of Tangent Plane to a Surfacep. 28
Level 1: The Taxonomic Approachp. 67
Manifolds - Tensor Fields - Covariant Differentiationp. 68
Some Remindersp. 68
Tangent Vector Spaces and Contravariant Vector Fieldsp. 69
Jacobian, Jacobian Matrix and their Applicationsp. 75
Tensor Fieldsp. 79
Covariant Derivativesp. 81
Parallel Displacement and Self-Parallel Curvesp. 86
Curvature Tensor Field or Riemannian Tensor or Curvature Tensorp. 89
Riemannian Geometryp. 92
Riemannian Metricp. 92
Riemannian Connectionp. 95
Riemannian Tensorp. 97
Miscellaneous Questionsp. 98
Deformation of Manifolds - Lie Derivativesp. 98
Isometriesp. 102
Geodesic and Self-Parallel Curvesp. 104
Level 2: The Intrinsic Approachp. 185
Differentiable Manifoldsp. 186
Differentiable Manifoldsp. 186
Functions and Curves Defined on a Manifoldp. 189
Tangent Vector Spacep. 191
Tangent Vector Spacep. 191
Local Expression of a Tangent Vector v[subscript M]p. 193
Change of Chartp. 198
Map of a Manifold M[subscript N] into a Manifold M'[subscript N']p. 199
Active Transformation and Lie Derivativep. 202
Cotangent Vector Spacep. 208
Some Remindersp. 208
Cotangent Vector Space. Covectors and Covariant Tensors. Exterior Derivationp. 212
Change of Chart Map of a Manifold M[subscript N] into a Manifold M'[subscript N'] Lie Derivativep. 219
Total Differential, Closed and Exact Differential Formsp. 223
Integration of Differential Formsp. 225
Some Classical Remindersp. 226
Integration of a Differential Formp. 227
p-Rectangles, Chains and Boundariesp. 228
Stokes' Theoremp. 230
Stokes' and Green's Formulae Revisitedp. 232
Cohomology. Betti's Numbersp. 234
Theory of Linear Connectionsp. 235
Linear Connections, Covariant (or Absolute) Derivativep. 235
Curvature Tensorp. 241
Torsion Tensorp. 244
Curvature and Torsion 2-Formsp. 245
Riemannian Geometryp. 246
The ds[superscript 2] of a Differentiable Manifold M[subscript N]p. 247
Connection on a Riemannian Manifoldp. 248
Geodesics and Autoparallel Curvesp. 251
Vector Fields and Lie Groupsp. 253
Vector Fields and One-Parameter Pseudo Groupsp. 253
Lie Groups and Lie Groups of Transformationsp. 257
Fibre Spacesp. 372
Fibre Bundlesp. 372
The Bundle of Linear Framesp. 372
Examples of Bundlesp. 374
Axiomatic Definition of a Principal Bundlep. 377
More Examples of Principal Bundlesp. 380
Vectors and Tensors Associated Vector Bundlesp. 381
Non-Vectorial Associated Bundlesp. 384
Connectionsp. 386
Some Introductory Commentsp. 386
Connection on Principal Bundlesp. 386
The Curvature 2-Form and Torsion 2-Formp. 391
Connection Form and Curvature Form in Associated Vector Bundlesp. 393
Gauge Group - Characteristic Classesp. 394
Physical Outlookp. 396
Bibliographyp. 400
Indexp. 403
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9789971501860
ISBN-10: 9971501864
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 424
Published: 1992
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 22.86 x 16.51  x 2.54
Weight (kg): 0.79