Sard's theorem

The set of Sard, as Lemma Sard or set of Morse - Sard known, is a basis of differential topology, and there the Morse theory and the Transversalitätstheorie to classification of germs of differentiable maps in the singularity theory or the thomschen catastrophe theory.

This sentence makes a statement about the measure of the set of critical values ​​of a differentiable map between two differentiable manifolds. This is called a value exactly critical if it is image of a critical point.

The set of Sard states that the critical values ​​of a mapping between two manifolds have the Lebesgue measure zero, if the picture is from, so - times continuously differentiable, for one.

Special cases of these are:

• If a differentiable function, so does the set of critical values ​​of degree 0
• A submanifold of smaller dimension has always measure 0, for example, the graph of a differentiable function as a subset of.
• A differentiable map between two manifolds can not be surjective for.

For pictures from the the sentence was proved in 1942 by Arthur Sard, which he was able to generalize the special case three years earlier shown by Marston Morse.