Sard's theorem

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

This theorem makes a statement about the measure of the set of critical values of a differentiable mapping between two differentiable manifolds . A value is called critical if and only if it is an image of a critical point . For differentiable manifolds there is generally no meaningful generalization of the Lebesgue measure, the term Lebesgue - null sets can still be transferred meaningfully: Let a -dimensional differentiable manifold and , then a Lebesgue null set is called if for every card with the set is a Lebesgue null set in . ${\ displaystyle M}$ ${\ displaystyle n}$ ${\ displaystyle C \ subset M}$ ${\ displaystyle C}$ ${\ displaystyle h \ colon U \ rightarrow V}$ ${\ displaystyle V \ subset \ mathbb {R} ^ {n}}$ ${\ displaystyle h \ left (C \ cap U \ right)}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

Sard's theorem says that the critical values of a mapping between two differentiable manifolds are Lebesgue null sets if the mapping is off, i.e. is continuously differentiable-times, for a . ${\ displaystyle f \ colon M \ to N}$ ${\ displaystyle C ^ {r} (M, N)}$ ${\ displaystyle r}$ ${\ displaystyle r> \ max (0, \ dim (M) - \ dim (N))}$

Special cases of this are:

If there is a differentiable function, then the set of critical values has a measure .

{\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}

{\ displaystyle \ {f (x) \ mid f '(x) = 0 \}}

{\ displaystyle 0}

A submanifold of smaller dimensions always has measure 0, for example the graph of a differentiable function as a subset of . ${\ displaystyle \ mathbb {R} \ to \ mathbb {R}}$ ${\ displaystyle \ mathbb {R} ^ {2}}$

A differentiable mapping between two manifolds can be for non- surjective . ${\ displaystyle f \ colon M \ to N}$ ${\ displaystyle \ dim (M) <\ dim (N)}$

For illustrations from in the the theorem was proved by Arthur Sard in 1942 , whereby he was able to generalize the special case shown three years earlier by Anthony Morse . ${\ displaystyle \ mathbb {R} ^ {m}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle n = 1}$

literature

A. Sard: The measure of the critical values of differentiable maps. Bull. Amer. Math. Soc. 48, (1942). 883-890.
M. Golubitsky, V. Guillemin : Stable Mappings and Their Singularities (= Graduate Texts in Mathematics 14). Springer-Verlag, New York NY et al. 1973, ISBN 0-387-90073-X .
Victor Guillemin, Alan Pollack: Differential Topology. Prentice Hall, Englewood Cliffs NJ 1974, ISBN 0-13-212605-2 .
Morris W. Hirsch : Differential Topology (= Graduate Texts in Mathematics 33). Springer-Verlag, New York NY et al. 1976, ISBN 0-387-90148-5 .
Michel Demazure : Catastrophes et Bifurcations. Editions Marketing, Paris 1989, ISBN 2-7298-8946-9 (French), (English: Bifurcations and Catastrophes. Geometry of Solutions to Nonlinear Problems. Springer, Berlin et al. 2000, ISBN 3-540-52118-6 ).

Individual evidence

↑ Theodor Bröcker , Klaus Jänich : Introduction to the differential topology (= Heidelberg pocket books . Volume 143 ). Springer Verlag, Berlin / Heidelberg a. a. 1990, ISBN 3-540-06461-3 , § 6. Sard's theorem, definition 6.3, p. 58–59 (Corrected reprint. “Differentiable” is always meant here .). ${\ displaystyle C ^ {\ infty}}$

[broecker_jaenich-1] Theodor Bröcker , Klaus Jänich : Introduction to the differential topology (= Heidelberg pocket books . Volume 143 ). Springer Verlag, Berlin / Heidelberg a. a. 1990, ISBN 3-540-06461-3 , § 6. Sard's theorem, definition 6.3, p. 58–59 (Corrected reprint. “Differentiable” is always meant here .). ${\ displaystyle C ^ {\ infty}}$