Surjective mappings whose differential is nowhere surjective

Author:
Y. Yomdin

Journal:
Proc. Amer. Math. Soc. **111** (1991), 267-270

MSC:
Primary 58C25

DOI:
https://doi.org/10.1090/S0002-9939-1991-1039267-2

MathSciNet review:
1039267

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Examples of -mappings , are given, with rank at any , but , for any . Thus a weak form of Sard's theorem (if all the points in the source are critical, the image has measure zero) does not hold for mappings of low smoothness.

**[1]**M. Briskin and Y. Yomdin,*Critical and near-critical values in nonlinear control problems*, preprint.**[2]**W. H. Fleming,*Future directions in control theory: a mathematical perspective*, Report of the Panel on Future Directions in Control Theory, SIAM Reports, Philadelphia, 1988.**[3]**M. Morse,*The calculus of variations in the large*, Amer. Math. Soc. Publ. 18, 1934. MR**1451874 (98f:58070)****[4]**A. Sard,*The measure of the critical values of differentiable maps*, Bull. Amer. Math. Soc.**48**(1942), 883-890. MR**0007523 (4:153c)****[5]**H. Whitney,*A function not constant on a connected set of critical point*, Duke Math. J.**1**(1935), 514-517. MR**1545896****[6]**-,*Analytic extensions of differentiable functions defined in closed sets*, Trans. Amer. Math. Soc.**36**(1934), 63-89. MR**1501735****[7]**Y. Yomdin,*The geometry of critical and near-critical values of differentiable mappings*, Math. Ann.**264**(1983), 495-515. MR**716263 (86f:58017)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58C25

Retrieve articles in all journals with MSC: 58C25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1039267-2

Article copyright:
© Copyright 1991
American Mathematical Society