Differential geometry and topology : with a view to dynamical systems
著者
書誌事項
Differential geometry and topology : with a view to dynamical systems
（Studies in advanced mathematics）
Chapman & Hall/CRC, 2005
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
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.
目次
Manifolds. Vector Fields and Dynamical Systems. Riemannian Metrics. Riemannian Connections and Geodesics. Curvature. Tensors and Differential Forms. Fixed Points and Intersection Numbers. Morse Theory. Hyperbolic Systems. References. Index.
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