Differential Geometry (eBook)

Heinrich W. Guggenheimer (Autor)

eBook Download: EPUB

2012
400 Seiten
Dover Publications (Verlag)
978-0-486-15720-7 (ISBN)

Infos

This text contains an elementary introduction to continuous groups and differential invariants; an extensive treatment of groups of motions in euclidean, affine, and riemannian geometry; more. Includes exercises and 62 figures.

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This is a text of local differential geometry considered as an application of advanced calculus and linear algebra. The discussion is designed for advanced undergraduate or beginning graduate study, and presumes of readers only a fair knowledge of matrix algebra and of advanced calculus of functions of several real variables. The author, who is a Professor of Mathematics at the Polytechnic Institute of New York, begins with a discussion of plane geometry and then treats the local theory of Lie groups and transformation groups, solid differential geometry, and Riemannian geometry, leading to a general theory of connections. The author presents a full development of the Erlangen Program in the foundations of geometry as used by Elie Cartan as a basis of modern differential geometry; the book can serve as an introduction to the methods of E. Cartan. The theory is applied to give a complete development of affine differential geometry in two and three dimensions. Although the text deals only with local problems (except for global problems that can be treated by methods of advanced calculus), the definitions have been formulated so as to be applicable to modern global differential geometry. The algebraic development of tensors is equally accessible to physicists and to pure mathematicians. The wealth of specific resutls and the replacement of most tensor calculations by linear algebra makes the book attractive to users of mathematics in other disciplines.

Inhalt

PrefaceChapter 1. Elementary Differential Geometry 1-1 Curves 1-2 Vector and Matrix Functions 1-3 Some FormulasChapter 2. Curvature 2-1 Arc Length 2-2 The Moving Frame 2-3 The Circle of CurvatureChapter 3. Evolutes and Involutes 3-1 The Riemann-Stieltjès Integral 3-2 Involutes and Evolutes 3-3 Spiral Arcs 3-4 Congruence and Homothety 3-5 The Moving PlaneChapter 4. Calculus of Variations 4-1 Euler Equations 4-2 The Isoperimetric ProblemChapter 5. Introduction to Transformation Groups 5-1 Translations and Rotations 5-2 Affine TransformationsChapter 6. Lie Group Germs 6-1 Lie Group Germs and Lie Algebras 6-2 The Adjoint Representation 6-3 One-parameter SubgroupsChapter 7. Transformation Groups 7-1 Transformation Groups 7-2 Invariants 7-3 Affine Differential GeometryChapter 8. Space Curves 8-1 Space Curves in Euclidean Geometry 8-2 Ruled Surfaces 8-3 Space Curves in Affine GeometryChapter 9. Tensors 9-1 Dual Spaces 9-2 The Tensor Product 9-3 Exterior Calculus 9-4 Manifolds and Tensor FieldsChapter 10. Surfaces 10-1 Curvatures 10-2 Examples 10-3 Integration Theory 10-4 Mappings and Deformations 10-5 Closed Surfaces 10-6 Line CongruencesChapter 11. Inner Geometry of Surfaces 11-1 Geodesics 11-2 Clifford-Klein Surfaces 11-3 The Bonnet FormulaChapter 12. Affine Geometry of Surfaces 12-1 Frenet Formulas 12-2 Special Surfaces 12-3 Curves on a SurfaceChapter 13. Riemannian Geometry 13-1 Parallelism and Curvature 13-2 Geodesics 13-3 Subspaces 13-4 Groups of Motions 13-5 Integral TheoremsChapter 14. ConnectionsAnswers to Selected ExercisesIndex

Produkt-Details

• Erscheint lt. Verlag: 27.4.2012
• Reihe/Serie: Dover Books on Mathematics
• Sprache: englisch
• Maße: 140 x 140 mm
• Gewicht: 431 g
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• ISBN-10: 0-486-15720-2 / 0486157202
• ISBN-13: 978-0-486-15720-7 / 9780486157207