One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory. Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, "The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master." Milnor has published five books with Princeton University Press.
Author(s): John Milnor
Series: Annals of Mathematics Studies 51
Edition: 1
Publisher: Princeton University Press
Year: 1963
Language: English
Pages: 163
Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Contents......Page 8
1 Introduction......Page 10
2. Definitions and Lemmas......Page 13
3. Homotopy Type in Terms of Critical Values......Page 21
4. Examples.......Page 34
5. The Morse Inequalities......Page 37
6. Manifolds in Euclidean Space: The Existence of Non-degenerate Functions......Page 41
7. The Lefschetz Theorem on Hyperplane Sections.......Page 48
8. Covariant Differentiation......Page 52
9. The Curvature Tensor......Page 60
10. Geodesics and Completeness......Page 64
11. The Path Space of a Smooth Manifold......Page 76
12. The Energy of a Path......Page 79
13. The Hessian of the Energy Function at a Critical Path......Page 83
14. Jacobi Fields: The Null-space of E......Page 86
15. The Index Theorem......Page 91
16. A Finite Dimensional Approximation to \Omega^c......Page 97
17. The Topology of the Full Path Space......Page 102
18. Existence of Non-conjugate Points......Page 107
19. Some Relations Between Topology and Curvature......Page 109
20. Symmetric Spaces......Page 118
21. Lie Groups as Symmetric Spaces......Page 121
22. Whole Manifolds of Minimal Geodesics......Page 127
23. The Bott Periodicity Theorem for the Unitary Group......Page 133
24. The Periodicity Theorem for the Orthogonal Group.......Page 142
APPENDIX. THE HOMOTOPY TYPE OF A MONOTONE UNION......Page 158
Back Cover......Page 163
