Differential Geometry and Topology: With a View to Dynamical Systems

by Burns; Keith

• ISBN13: 9781584882534
• ISBN10: 1584882530
• Edition: 1st
• Format: Hardcover
• Copyright: 2005-05-27
• Publisher: Chapman & Hall/

Summary

Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. The authors' intuitive approach forms a treatment that is comprehensible to relative beginners, yet rigorous enough for professional mathematicians. Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, integration on manifolds, intersection theory provide the foundation for many applications in dynamical systems and mechanics. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow.

Table of Contents

Manifolds	1	(70)
Introduction	1	(3)
Review of topological concepts	4	(5)
Smooth manifolds	9	(7)
Smooth maps	16	(3)
Tangent vectors and the tangent bundle	19	(8)
Tangent vectors as derivations	27	(3)
The derivative of a smooth map	30	(3)
Orientation	33	(3)
Immersions, embeddings and submersions	36	(9)
Regular and critical points and values	45	(3)
Manifolds with boundary	48	(5)
Sard's theorem	53	(6)
Transversality	59	(3)
Stability	62	(4)
Exercises	66	(5)
Vector Fields and Dynamical Systems	71	(38)
Introduction	71	(3)
Vector fields	74	(6)
Smooth dynamical systems	80	(6)
Lie derivative, Lie bracket	86	(8)
Discrete dynamical systems	94	(3)
Hyperbolic fixed points and periodic orbits	97	(9)
Exercises	106	(3)
Riemannian Metrics	109	(18)
Introduction	109	(3)
Riemannian metrics	112	(9)
Standard geometries on surfaces	121	(4)
Exercises	125	(2)
Riemannian Connections and Geodesics	127	(44)
Introduction	127	(4)
Affine connections	131	(5)
Riemannian connections	136	(6)
Geodesics	142	(7)
The exponential map	149	(6)
Minimizing properties of geodesics	155	(7)
The Riemannian distance	162	(5)
Exercises	167	(4)
Curvature	171	(54)
Introduction	171	(5)
The curvature tensor	176	(8)
The second fundamental form	184	(11)
Sectional and Ricci curvatures	195	(6)
Jacobi fields	201	(7)
Manifolds of constant curvature	208	(2)
Conjugate points	210	(3)
Horizontal and vertical sub-bundles	213	(4)
The geodesic flow	217	(5)
Exercises	222	(3)
Tensors and Differential Forms	225	(54)
Introduction	225	(2)
Vector bundles	227	(4)
The tubular neighborhood theorem	231	(2)
Tensor bundles	233	(5)
Differential forms	238	(9)
Integration of differential forms	247	(4)
Stokes' theorem	251	(6)
De Rham cohomology	257	(6)
Singular homology	263	(8)
The de Rham theorem	271	(5)
Exercises	276	(3)
Fixed Points and Intersection Numbers	279	(48)
Introduction	279	(3)
The Brouwer degree	282	(9)
The oriented intersection number	291	(2)
The fixed point index	293	(10)
The Lefschetz number	303	(3)
The Euler characteristic	306	(7)
The Gauss-Bonnet theorem	313	(11)
Exercises	324	(3)
Morse Theory	327	(30)
Introduction	327	(2)
Nondegenerate critical points	329	(8)
The gradient flow	337	(3)
The topology of level sets	340	(8)
Manifolds represented as CW complexes	348	(3)
Morse inequalities	351	(5)
Exercises	356	(1)
Hyperbolic Systems	357	(22)
Introduction	357	(2)
Hyperbolic sets	359	(9)
Hyperbolicity criteria	368	(5)
Geodesic flows	373	(3)
Exercises	376	(3)
References	379	(6)
Index	385