There is a (tiny) insight that I find missing in this introduction, the fact that the complexity can lie either on the function or on the manifold. The two ext

There is a (tiny) insight that I find missing in this introduction, the fact that the complexity can lie either on the function or on the manifold. The two extreme--and most common--cases of Morse theory are: a very complicated manifold with a simple function on it (e.g. one coordinate), or a very complicated function defined on a flat manifold. The theory applies the same to both cases but the insights are very different. May I add two beautiful texts of historical interest but amazingly readable: * J.C. Maxwell, "On Hills and Dales", 1870. https://www.maths.ed.ac.uk/~v1ranick/papers/hilldale.pdf https://www.maths.ed.ac.uk/~v1ranick/papers/hilldale.pdf * A. Cayley, "On contour and slope lines", 1859. http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf "http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf