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Journal of Zhejiang University SCIENCE A 2004 Vol.5 No.7 P.754~758


A general version of the Morse-Sard theorem

Author(s):  JIANG Hai-yi

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Corresponding email(s):   hyjiang@math.zju.edu.cn

Key Words:  Hausdorff measure, Rectifiable, Morse decomposition

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JIANG Hai-yi. A general version of the Morse-Sard theorem[J]. Journal of Zhejiang University Science A, 2004, 5(7): 754~758.

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Let k, m, n be positive integers, and k≤2, α∈(0,1], 0k,α(Rm, Rn), A=Cr(f)={x∈Rm|rank(Df(x))≤r}, then f(A) is d-null. Thus the statement posed by Arthur Sard in 1965 can be completely solved when k≥2.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1] Bates, S.M., 1993. Toward a precise smoothness hypothesis in Sard's theorem.Proc. Amer. Math. Soc.,117:279-283.

[2] Federer, H., 1969. Geometric Measure Theory. Grundlehren Math. Wiss., Vol.153. Springer, New York.

[3] Jiang, H.Y., Xi, L.F., 2000. On the Norton problem.Acta Math. Sinica,43:445-456 (in Chinese).

[4] Morse, A.P., 1939. The behavior of a function on its critical set.Ann. of Math.,40:62-70.

[5] Norton, A., 1994. The Zygmund Morse-Sard theorem.J. Geom. Analysis,4:403-424.

[6] Sard, A., 1942. The measure of the critical values of dif-ferentiable map.Bull. Amer. Math. Soc.,48:883-890.

[7] Sard, A., 1965. Hausdorff measures of critical images on Banach manifolds.Amer. J. Math.,87:158-174.

[8] Stein, E., 1970. Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton.

[9] Yomdin, Y., 1983. The geometry of critical and near critical values of differentiable mappings.Ann. of Math.,264:495-515.

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