Differential Geometry and Topology

Differential Geometry and Topology : With a View to Dynamical Systems

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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.show more

Product details

  • Hardback | 400 pages
  • 160 x 236.2 x 27.9mm | 680.4g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 132 black & white illustrations
  • 1584882530
  • 9781584882534

Table of contents

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 References Indexshow more