1st Edition
Differential Geometry and Topology With a View to Dynamical Systems
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.
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
Biography
Keith Burns, Marian Gidea
"The authors introduce important concepts by means of intuitive discussions and suggestive examples and follow them with significant applications, especially those related to dynamics. …The authors have succeeded in the integration of geometric theory, topological theory, and concrete applications to dynamical systems."
-Mathematical Reviews, Andrew Bucki
"The authors of this book treat a great many topics very concisely."
-MAA Reviews, William J. Satzer, 2005
"A noteworthy feature of the presentation is that dynamical systems, which are introduced in the second chapter, are used systematically to illustrate concepts and as a source of applications."
-CMS Notes, Vol. 38, No. 2, March, 2006". . . very well written, in a very pedagogical manner and it covers a lot of material in a very clear way. I think this is an ideal introduction to differential geometry and topology for beginning graduate students or advanced undergraduate students in mathematics, but it will be, also, useful to physicist or other scientists with an interest in differential geometry and dynamical systems."
– Paul Blaga, in Babes- Bolyai Mathematica, June 2007, Vol. 52, No. 2
"Numerous illustrations and exercises round off the picture of an original and very readable textbook."
– M. Kunzinger, in Monatshefte fur Math, 2007, Vol. 152, No. 1
