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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 C manifold endowed with a fixed Riemannian structure. For instance you might think of the two-sphere S with the Riemann structure inherited from an imbedding of S in R. 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:S -* 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 S to M: A M = M a p ( 5 1 , M ) ,

Lectures on Morse theory, old and new