Smooth Mappings

Abstract

Sard’s Theorem is an important tool in differential topology and in Morse theory. We shall begin our discussion of Sard’s theorem by treating the most elementary form of the result, and then shall later develop generalizations and variants of the theorem. It is worth noting that the basic idea here was discovered by A. B. Brown in 1935 (see Brown [1]). Later, in 1939 and 1942, the result was rediscovered by A. P. Morse (see Morse [1]) and A. Sard (see Sard [1]). The method of proof of Sard’s theorem is quite robust and can be modified to yield a number of interesting results. We shall close the section with a discussion of one of these variants that is known as “hard Sard.”