Laugwitz Differential and Riemannian Geometry

1. Auflage 2014
ISBN: 978-1-4832-6398-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 250 Seiten, Web PDF

Differential and Riemannian Geometry focuses on the methodologies, calculations, applications, and approaches involved in differential and Riemannian geometry. The book first offers information on local differential geometry of space curves and surfaces and tensor calculus and Riemannian geometry. Discussions focus on tensor algebra and analysis, concept of a differentiable manifold, geometry of a space with affine connection, intrinsic geometry of surfaces, curvature of surfaces, and surfaces and curves on surfaces. The manuscript then examines further development and applications of Riemannian geometry and selections from differential geometry in the large, including curves and surfaces in the large, spaces of constant curvature and non-Euclidean geometry, Riemannian spaces and analytical dynamics, and metric differential geometry and characterizations of Riemannian geometry. The publication elaborates on prerequisite theorems of analysis, as well as the existence and uniqueness theorem for ordinary first-order differential equations and systems of equations and integrability theory for systems of first-order partial differential equations. The book is a valuable reference for researchers interested in differential and Riemannian geometry.

Autoren/Hrsg.

Laugwitz, Detlef

Inhaltsverzeichnis

1;Front Cover;1
2;Differential and Riemannian Geometry;4
3;Copyright Page;5
4;Table of Contents;12
5;Preface to the German Edition;6
6;Translator's Preface;10
7;Author's Note;10
8;CHAPTER I.
Local Differential Geometry of Space Curves;14
8.1;§ 1. Differential-Geometric Properties of Curves;14
8.2;§ 2. The Complete System of Invariants for Space Curves;25
9;CHAPTER II.
Local Differential Geometry of Surfaces;32
9.1;§ 3. Surfaces, and Curves on Surfaces;32
9.2;§ 4. Intrinsic Geometry of Surfaces;50
9.3;§ 5. Curvature of Surfaces;65
9.4;§ 6. Special Topics in the Theory of Surfaces;79
10;CHAPTER III.
Tensor Calculus and Riemannian Geometry;92
10.1;§ 7. The Concept of a Differentiable Manifold;92
10.2;§ 8. Tensor Algebra;97
10.3;§ 9. Tensor Analysis;107
10.4;§ 10. The Geometry of a Space with Affine Connection;117
10.5;§ 11. Foundations of Riemannian Geometry;128
11;CHAPTER IV. Further Development and Applications
of Riemannian Geometry;147
11.1;§ 12. Spaces of Constant Curvature and Non-Euclidean Geometry;148
11.2;§ 13. Mappings;160
11.3;§ 14. Riemannian Spaces and Analytical Dynamics;180
11.4;§ 15. Metric Differential Geometry and Characterizations of Riemannian Geometry;189
12;CHAPTER V.
Selections from Differential Geometry in the Large;211
12.1;§16. Curves in the Large;211
12.2;§17. Surfaces in the Large;217
13;APPENDIX I:
From the History of Differential Geometry;234
14;APPENDIX II:
Some Prerequisite Theorems of Analysis;237
14.1;1. Existence and Uniqueness Theorem for Ordinary First-Order
Differential Equations and Systems of Equations;237
14.2;2. Integrability Theory for Systems of First-Order Partial Differential
Equations;238
14.3;3. From the Calculus of Variations;240
15;APPENDIX III:
Summary of Formulas;242
15.1;Theory of Curves;242
15.2;Theory of Surfaces;242
16;Index;246