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Analysis and Algebra on Differentiable Manifolds : A Workbook for Students and Teachers / by P. M. Gadea, J. Muñoz Masqué
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerLink Bücher | Texts in the Mathematical Sciences ; 23Publisher: Dordrecht : Springer Science+Business Media B.V, 2009Description: Online-Ressource (XV, 438p. 49 illus, digital)ISBN:- 9789048135646
- Algebra
- Einführung
- Analysis
- Differenzierbare Mannigfaltigkeit
- Aufgabensammlung
- Differential geometry
- Applied mathematics
- Engineering mathematics
- Global analysis (Mathematics)
- Manifolds (Mathematics)
- Lie groups
- Geometry, Differential
- Global analysis
- Global differential geometry
- Mathematics
- Topological Groups
- 516.36
- QA641-670
- QA614.3.G34 2009eb
Contents:
Summary: Differentiable manifolds -- Tensor Fields and Differential Forms -- Integration on Manifolds -- Lie Groups -- Fibre Bundles -- Riemannian Geometry -- Some Definitions and Theorems -- Some Formulas and Tables -- Erratum to: Foreword.Summary: A famous Swiss professor gave a student’s course in Basel on Riemann surfaces. After a couple of lectures, a student asked him, “Professor, you have as yet not given an exact de nition of a Riemann surface.” The professor answered, “With Riemann surfaces, the main thing is to UNDERSTAND them, not to de ne them.” The student’s objection was reasonable. From a formal viewpoint, it is of course necessary to start as soon as possible with strict de nitions, but the professor’s - swer also has a substantial background. The pure de nition of a Riemann surface— as a complex 1-dimensional complex analytic manifold—contributes little to a true understanding. It takes a long time to really be familiar with what a Riemann s- face is. This example is typical for the objects of global analysis—manifolds with str- tures. There are complex concrete de nitions but these do not automatically explain what they really are, what we can do with them, which operations they really admit, how rigid they are. Hence, there arises the natural question—how to attain a deeper understanding? One well-known way to gain an understanding is through underpinning the d- nitions, theorems and constructions with hierarchies of examples, counterexamples and exercises. Their choice, construction and logical order is for any teacher in global analysis an interesting, important and fun creating task.PPN: PPN: 1648843123Package identifier: Produktsigel: ZDB-2-BAE | ZDB-2-SEB | ZDB-2-SMA | ZDB-2-SXMS | ZDB-2-SMA
""Foreword""; ""Preface""; ""Acknowledgements""; ""Contents""; ""Differentiable manifolds""; ""C manifolds""; ""Differentiable Structures Defined on Sets""; ""Differentiable Functions and Mappings""; ""Critical Points and Values""; ""Immersions, Submanifolds, Embeddings and Diffeomorphisms""; ""Constructing Manifolds by Inverse Image. Implicit Map Theorem""; ""Submersions. Quotient Manifolds""; ""The Tangent Bundle""; ""Vector Fields""; ""Working with Vector Fields""; ""Integral Curves""; ""Flows""; ""Transforming Vector Fields""; ""Tensor Fields and Differential Forms""; ""Vector Bundles""
""Tensor and Exterior Algebras. Tensor Fields""""Differential Forms. Exterior Product""; ""Lie Derivative. Interior Product""; ""Distributions and Integral Manifolds. Frobenius' Theorem. Differential Ideals""; ""Almost Symplectic Manifolds""; ""Integration on Manifolds""; ""Orientable manifolds. Orientation-preserving maps""; ""Integration on Chains. Stokes' Theorem I""; ""Integration on Oriented Manifolds. Stokes' Theorem II""; ""De Rham Cohomology""; ""Lie Groups""; ""Lie Groups and Lie Algebras""; ""Homomorphisms of Lie Groups and Lie Algebras""; ""Lie Subgroups and Lie Subalgebras""
""The Exponential Map""""The Adjoint Representation""; ""Lie Groups of Transformations""; ""Homogeneous Spaces""; ""Fibre Bundles""; ""Principal Bundles""; ""Connections in Bundles""; ""Characteristic Classes""; ""Linear Connections""; ""Torsion and Curvature""; ""Geodesics""; ""Almost Complex Manifolds""; ""Riemannian Geometry""; ""Riemannian Manifolds""; ""Riemannian Connections""; ""Geodesics""; ""The Exponential Map""; ""Curvature and Ricci Tensors""; ""Characteristic Classes""; ""Isometries""; ""Homogeneous Riemannian Manifolds and Riemannian Symmetric Spaces""
""Spaces of Constant Curvature""""Left-invariant Metrics on Lie Groups""; ""Gradient, Divergence, Codifferential, Curl, Laplacian, and Hodge Star Operator""; ""Affine, Killing, Conformal, Projective, Jacobi, and Harmonic Vector Fields""; ""Submanifolds. Second Fundamental Form""; ""Surfaces in I R3""; ""Pseudo-Riemannian Manifolds""; ""Some Definitions and Theorems""; ""Chapter 1. Differentiable Manifolds""; ""Chapter 2. Tensor Fields. Differential Forms""; ""Chapter 3. Integration on Manifolds""; ""Chapter 4. Lie Groups""; ""Chapter 5. Fibre Bundles""; ""Chapter 6. Riemannian Geometry""
""Some Formulas and Tables""""References""; ""List of Notations""; ""List of Figures""; ""Index""
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