Differential and Riemannian Geometry

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.

Contenu

Preface to the German Edition
Translator's Preface
Author's Note
Chapter I Local Differential Geometry of Space Curves

1. Differential-Geometric Properties of Curves
2. The Complete System of Invariants for Space Curves
   Chapter II Local Differential Geometry of Surfaces
3. Surfaces, and Curves on Surfaces
4. Intrinsic Geometry of Surfaces
5. Curvature of Surfaces
6. Special Topics in the Theory of Surfaces
   Chapter III Tensor Calculus and Riemannian Geometry
7. The Concept of a Differentiable Manifold
8. Tensor Algebra
9. Tensor Analysis
10. The Geometry of a Space with Affine Connection
11. Foundations of Riemannian Geometry
    Chapter IV Further Development and Applications of Riemannian Geometry
12. Spaces of Constant Curvature and Non-Euclidean Geometry
13. Mappings
14. Riemannian Spaces and Analytical Dynamics
15. Metric Differential Geometry and Characterization of Riemannian Geometry
    Chapter V Selections from Differential Geometry in the Large
16. Curves in the Large
17. Surfaces in the Large
    Appendix I From the History of Differential Geometry
    Appendix II Some Prerequisite Theorems of Analysis
18. Existence and Uniqueness Theorem for Ordinary First-Order Differential Equations and Systems of Equations
19. Integrability Theory for Systems of First-Order Partial Differential Equations
20. From the Calculus of Variations
    Appendix III Summary of Formulas
    Theory of Curves
    Theory of Surfaces
    Index