Differential Geometry and Topology
With a View to Dynamical Systems
Keith Burns, Marian Gidea - Collection Studies in Advanced Mathematics
Résumé
Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow.
Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models.
The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow.
The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field.
L'auteur - Keith Burns
Keith Burns Northwestern University, Evanston, Illinois, USA
L'auteur - Marian Gidea
Marian Gidea Northeastern Illinois University, Chicago, USA
Sommaire
- Manifolds
- Introduction
- Review of topological concepts
- Smooth manifolds
- Smooth maps
- Tangent vectors and the tangent bundle
- Tangent vectors as derivations
- The derivative of a smooth map
- Orientation
- Immersions, embeddings and submersions
- Regular and critical points and values
- Manifolds with boundary
- Sard's theorem
- Transversality
- Stability
- Exercises
- Vector Fields and Dynamical Systems
- Introduction
- Vector fields
- Smooth dynamical systems
- Lie derivative, Lie bracket
- Discrete dynamical systems
- Hyperbolic fixed points and periodic orbits
- Exercises
- Riemannian Metrics
- Introduction
- Riemannian metrics
- Standard geometries on surfaces
- Exercises
- Riemannian Connections and Geodesics
- Introduction
- Affine connections
- Riemannian connections
- Geodesics
- The exponential map
- Minimizing properties of geodesics
- The Riemannian distance
- Exercises
- Curvature
- Introduction
- The curvature tensor
- The second fundamental form
- Sectional and Ricci curvatures
- Jacobi fields
- Manifolds of constant curvature
- Conjugate points
- Horizontal and vertical sub-bundles
- The geodesic flow
- Exercises
- Tensors and Differential Forms
- Introduction
- Vector bundles
- The tubular neighborhood theorem
- Tensor bundles
- Differential forms
- Integration of differential forms
- Stokes' theorem
- De Rham cohomology
- Singular homology
- The de Rham theorem
- Exercises
- Fixed Points and Intersection Numbers
- Introduction
- The Brouwer degree
- The oriented intersection number
- The fixed point index
- The Lefschetz number
- The Euler characteristic
- The Gauss-Bonnet theorem
- Exercises
- Morse Theory
- Introduction
- Nondegenerate critical points
- The gradient flow
- The topology of level sets
- Manifolds represented as CW complexes
- Morse inequalities
- Exercises
- Hyperbolic Systems
- Introduction
- Hyperbolic sets
- Hyperbolicity criteria
- Geodesic flows
- Exercises
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Chapman and Hall / CRC |
| Auteur(s) | Keith Burns, Marian Gidea |
| Collection | Studies in Advanced Mathematics |
| Parution | 04/07/2005 |
| Nb. de pages | 400 |
| Format | 16 x 24 |
| Couverture | Relié |
| Poids | 700g |
| Intérieur | Noir et Blanc |
| EAN13 | 9781584882534 |
| ISBN13 | 978-1-58488-253-4 |
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