Sard’s Theorem

Abstract

This chapter introduces a powerful tool in smooth manifold theory, Sard’s theorem, which says that the set of critical values of a smooth function has measure zero. After proving the theorem, we use it to prove three important results about smooth manifolds. The first result is the Whitney embedding theorem, which says that every smooth manifold can be smoothly embedded in some Euclidean space. (This justifies our habit of visualizing manifolds as subsets of ℝⁿ.) The second result is the Whitney approximation theorem, which comes in two versions: every continuous real-valued or vector-valued function can be uniformly approximated by smooth ones, and every continuous map between smooth manifolds is homotopic to a smooth map. The third result is the transversality homotopy theorem, which says, among other things, that embedded submanifolds can always be deformed slightly so that they intersect “nicely” in a certain sense that we will make precise.