Lectures on Morse theory, old and new
Raoul Bott · Bulletin of the American Mathematical Society · 1982
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 00 manifold endowed with a fixed Riemannian structure.For instance you might think of the two-sphere S 2 with the Riemann structure inherited from an imbedding of S 2 in R 3 . 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 x -* 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 l to M: AM=Map(5 1 ,M), in the compact open topology.Also let A^Af denote the component of the constant maps of S l 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.