Morse Thoery,Old And New

Morse theory gives a very direct way of analyzing the topology of a manifold by studying differentiable functions of that manifold.There are three main parts in this paper. First, we will see that Morse theory give a very direct way of analyzing the topology of a manifold by differentiable functions of that manifold, and finally, find out its CW structure. In the second part, we apply Morse theory to geodesics, and give an answer to the question that whether a compact connected C∞manifold always carry a nontrivial closed geodesic. Morse theory can be used to prove some strong results on the homology of manifold. The rank of the λ. th homology group Rλ (M) id less than or equal to the number of critical points of the index λ of a Morse function on M. In the third part, this fact is strengthened to obtain the Morse inequalities. And we will see Morse inequalities have a very surprising connection with supersymmetry theorem.