Differential and Riemannian Geometry
Published on 12. May 2014
250 pages
978-1-4832-6398-4 (ISBN)
Description
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.
More details
Content
Preface to the German EditionTranslator's PrefaceAuthor's NoteChapter I Local Differential Geometry of Space Curves 1. Differential-Geometric Properties of Curves 2. The Complete System of Invariants for Space CurvesChapter 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 SurfacesChapter 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 GeometryChapter 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 GeometryChapter V Selections from Differential Geometry in the Large 16. Curves in the Large 17. Surfaces in the LargeAppendix I From the History of Differential GeometryAppendix II Some Prerequisite Theorems of Analysis 1. Existence and Uniqueness Theorem for Ordinary First-Order Differential Equations and Systems of Equations 2. Integrability Theory for Systems of First-Order Partial Differential Equations 3. From the Calculus of VariationsAppendix III Summary of Formulas Theory of Curves Theory of SurfacesIndex
