Proceedings of Symposia in Pure Mathematics Volume 39 (1983), Part 2 Lectures on Morse Theory, Old and New RAOUL BOTT1 Morse Theory is a beautiful and natural extension of the minimum principle for a continuous function on a compact space. In these lectures I would like to discuss it in the context of two problems in analysis which have self-evident geometric interest as well as physical origins. The first question is simply this. Let M be a compact connected C00 manifold endowed with a fixed Riemannian structure. For instance you might think of the two-sphere S2 with the Riemann structure inherited from an imbedding of S2 in R3. Question. Does such an M always carry a nontrivial closed geodesic? Recall here first of all that on a compact manifold any two points P and Q can be joined by a geodesic which minimizes the length of all piecewise smooth curves joining P to Q in M. In one way or another this is then an application of the minimum principle, and conceptually you should think of pulling a string confined to M and joining P and Q as tight as possible. When the string has assumed a position in which it cannot be tightened any more, then it describes a geodesic joining P to Q. If it cannot be tightened further even after a "jiggling", then it describes the minimal geodesic in question. This "pulling tight" principle works also for finding closed geodesies, provided only that we have some constraint to pull against. Thus if a is a piecewise smooth map of the circle a: Sl - M which cannot be deformed to a point in M, then shortening a in its homotopy class will indeed produce a closed geodesic. Put differently, let AM, denote the space of continuous maps from Sl to M: A M = M a p ( S 1 , M ) , in the compact open topology. Also let A^M denote the component of the constant maps of Sl to M. Then a classical theorem going back to Hadamard, Cartan, etc., asserts that THEOREM. Every component of AM other than A^M contains a bona fide closed geodesic. Reprinted from Bulletin Amer. Math. Soc. (N.S.) 7 (1982), 331-358. 1980 Mathematics Subject Classification. Primary 58E05. 'The following is the text of a series of four lectures which I delivered in Peking during the summer of 1980. However the content is an expanded and somewhat altered version of my lecture at the Poincare Symposium. Rather than giving a survey I tried in both instances to produce as self-contained an account as possible of two specific problems in the subject. © 1982 American Mathematical Society 0082-0717/83 $1.00 + $.25 per page 3
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