The Size of Critical and Tangency Sets

We emphasize several faces of the critical phenomena for maps between two manifolds Mm and Nn. In the nite dimensional setting, it basically concerns those points of the source manifold where the rank of the dierential is strictly less than its maximum possible value minfm; ng. Note that for n = 1 the critical points are those where the dierential vanishes and this fact is being considered as the denition of the critical points of real valued functions dened on innite dimensional Banach spaces. The set of images of such points is also part of the critical phenomena and the size of the critical sets and the sets of critical values is sometimes evaluated.