Critical Point Theory in Global Analysis and Differential Topology (eBook)

Front Cover 1
Critical Point Theory in Global Analysis and Differential Topology: An Introduction 4
Copyright Page 5
Contents 12
Preface 8
Introduction 14
Part I: Analysis of Nondegenerate Functions 18
Chapter 1. Differentiable Mappings 20
Chapter 2. Laws of the Mean 23
Chapter 3. Real, Symmetric, Quadratic Forms 28
Chapter 4. The Reduction Theorem for ƒ 33
Chapter 5. Regular Manifolds in Euclidean Spaces 42
Chapter 6. First Theorems on Nondegeneracy 52
Chapter 7. Nondegeneracy: Theorem 7.1 63
Chapter 8. S-Normal f-Coordinates 71
Chapter 9. Nondegenerate Functions under Boundary Conditions A 77
Chapter 10. Nondegenerate Functions under Boundary Conditions B 82
Chapter 11. Proof of Theorem 10.1 under Boundary Conditions B 87
Chapter 12. ƒ-level Boundaries 96
Part II : Abstract Differentiable Manifolds 102
Chapter 13. The Manifolds Defined 104
Chapter 14. Families of Differentiable Functions on Mn 111
Chapter 15. Focal Points of Regular Manifolds 119
Chapter 16. Differentiable Mappings of Manifolds into Manifolds 124
Chapter 17. Dynamical Systems on a Manifold Mn 140
Chapter 18. Tangent and Cotangent Vectors of Mn 149
Chapter 19. Mn As Riemannian Manifold 156
Chapter 20. ƒƒ-Presentations in DMnƒ 171
Chapter 21, ƒ-Trajectories on Mnf 179
Chapter 22. ƒ-Preferred Riemannian Structures S´ 185
Chapter 23. The Basic Homotopy Theorem 197
Part III : Singular Homology Theory 216
Chapter 24. Introduction 218
Chapter 25. Model Polyhedral Complexes P 228
Chapter 26. Singular Complexes S(x). Eilenberg 237
Chapter 27. Chain-Homotopies on S(x) 247
Chapter 28. Relative Homologies 257
Chapter 29. Comparison of the Homology Groups on fa and on fa — pa 268
Chapter 30. Type Numbers and Connectivities 280
Part IV: Other Applications of Critical Point Theory 292
Chapter 31. Normals from a Point to a Manifold 294
Chapter 32. Equilibrium Points of an Electrostatic Potential 300
Chapter 33. Symmetric Squares of Manifolds and Critical Chords 308
Chapter 34. The Symmetric Square of an n-Sphere 325
Chapter 35. The Complex Projective n-Space CPn 331
Chapter 36. Caps and Saddles 337
Chapter 37. The Real Projective n-Space Pn 348
Chapter 38. Stein Manifolds. A Theorem of Lefschetz 361
Chapter 39. Supplementary Concepts and Theorems 368
Appendix I. Preliminary Definitions 376
Appendix II. On Elevating Manifold Differentiability 379
Appendix III. Singular Homology Theory on Mn over Z 381
Fundamental Symbols 387
Bibliography 393
Index of Terms 400
Pure and Applied Mathematics 403