A Comprehensive Introduction To Differential Geometry by Michael Spivak

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By Michael Spivak

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PROOF. There is a sequence of charts (Xi , Vi) with A c Ui Vi and each set xi (A n Vi) of measure 0. If (y, V ) is a chart on N, then J(A) n V = Ui J(A n Vi) n v. Each set y(J(A n Vi) n V ) = y 0 J o [l(x(A n Vi» has measure 0, by Lemma 6. Thus y(J(A) n V ) has measure 0. Since J (Ui Vi ) is contained in the Ullion of at most countably many components of N, it follows that J(A) has measure 0. (. Chapter 2 42 8. THEOREM (SARD 'S THEOREM). If J : M � N is a C1 map between n-manifolds, and M has at most countably many components, then the critical values of J form a set of measure 0 in N.

To deal with Coo functions effectively, one needs to know that there are lots of I hem. The existence of Coo functions on a manifold depends on the existence of Coo functions on �n which are 0 oUlside of a compact set. f. culus on Malfifolds, pg. 29). Differentiable Structures � 33 h: � � hex) { e-I /x' xx oft h(n>(O) j: �- � e I)( x ' . e (x+I>-' x E (-I,I) { J(x). x ¢ (-1,1) -I � � (I) The function 0 = o =0 = is Coo, and 0 for all � (2) The function = defined by n. defined by o i s Coo. Similarly, there is a Coo function k : elsewhere.

LEMMA. If J : �n � �n is C1 and A C �n has measure 0, then J(A) has measure o. PROOF. We can assume that A is contained in a compact set C (since �n is a countable union of compact sets). Lemma 5 implies that there is some K such that IJ(x) - J(y ) 1 ::; n 2 Klx - yl for all x, y E C. Thus J takes rectangles of diameter d into sets of diameter ::; n2 K d. This clearly implies that J(A) has measure ° if A does . •:. A subset A of a Coo n-manifold M has measure zero if there is a sequence of charts (XI, Vi), with A c Ui Vi, such that each set xi (A n Vi) c �" has measure 0.

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