Differential Geometry and Topology: With a View to Dynamical Systems

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