Differential Geometry and Topology: With a View to Dynamical Systems

ISBN-13: 9781584882534 / Angielski / Twarda / 2005 / 400 str.

Marian Gidea; Keith Howard Burns; Burns Burns

Opis

Provides a clear introduction to three related subjects whose interplay enhances intuitive understanding Focuses on geodesics as a tool for better understanding Riemannian geometry and as a source of examples in dynamical systems Offers a unique presentation of the very modern theory of the Conley index as a generalization of Morse theory and as a means for understanding topological dynamical systems Takes an example-based approach to the more cryptic notions of differential geometry Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. The author's intuitive approach forms a treatment that is comprehensible to relative beginners, yet rigorous enough for professional mathematicians. Integration on manifolds plays an important role in the text and provides the foundation for many concrete applications. Special attention is given to chaotic phenomena, including the persistence of chaos under small perturbations. The author also provides a unique presentation of the Conley index theory, which then forms the basis of an original approach to hyperbolic dynamics.

Spis treści

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
Index

O autorze

Keith Burns, Marian Gidea