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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Generalized Hestenes' Lemma
and extension of functions

Authors: Massimo Ferrarotti and Leslie C. Wilson
Journal: Trans. Amer. Math. Soc. 350 (1998), 1957-1975
MSC (1991): Primary 58C20; Secondary 53C40
MathSciNet review: 1422896
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Abstract: Suppose we have an $m$-jet field on $V\subset \mathbf{R}^{n}$ which is a Whitney field on the nonsingular part $M$ of $V$. We show that, under certain hypotheses about the relationship between geodesic and euclidean distance on $V$, if the field is flat enough at the singular part $S$, then it is a Whitney field on $V$ (the order of flatness required depends on the coefficients in the hypotheses). These hypotheses are satisfied when $V$ is subanalytic. In Section II, we show that a $C^{2}$ function $f$ on $M$ can be extended to one on $V$ if the differential $df$ goes to $0$ faster than the order of divergence of the principal curvatures of $M$ and if the first covariant derivative of $df$ is sufficiently flat. For the general case of $C^{m}$ functions with $m >2$, we give a similar result for $\operatorname{codim} M=1$ in Section III.

References [Enhancements On Off] (What's this?)

  • [B] K. Bekka, Thèse: Sur les propriétés topologiques et métriques des espaces stratifiés, Univ. de Paris-Sud, 1988.
  • [BT] Karim Bekka and David Trotman, Propriétiés métriques de familles Φ-radiales de sous-variétés différentiables, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 9, 389–392 (French, with English summary). MR 910377
  • [Bi] Edward Bierstone, Extension of Whitney fields from subanalytic sets, Invent. Math. 46 (1978), no. 3, 277–300. MR 0481081
  • [BiMP] E. Bierstone, P. Milman, W. Pawlucki, Composite differentiable functions, Duke Math. Jour. 83 (1996), 607-620. CMP 96:13
  • [H] M. R. Hestenes, Extension of the range of a differentiable function, Duke Math. J. 8 (1941), 183–192. MR 0003434
  • [Hi] Heisuke Hironaka, Subanalytic sets, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453–493. MR 0377101
  • [KO] K. Kurdyka, P. Orro, Distance géodésique sur un sous-analytique, Proceedings of the RAAG Conference Segovia 1995, Revista matematica de la Universidad Complutense de Madrid 10, num. suplementaire (1997).
  • [KP] K. Kurdyka, W. Pawlucki, Subanalytic version of Whitney's extension theorem, Studia Math. 124 (1997), 269-280. CMP 97:14
  • [M] B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
  • [Mi] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331
  • [S] R. T. Seeley, Extension of 𝐶^{∞} functions defined in a half space, Proc. Amer. Math. Soc. 15 (1964), 625–626. MR 0165392, 10.1090/S0002-9939-1964-0165392-8
  • [St] Jacek Stasica, The Whitney condition for subanalytic sets, Zeszyty Nauk. Uniw. Jagielloń. Prace Mat. 23 (1982), 211–221. MR 670588
  • [T] Jean-Claude Tougeron, Idéaux de fonctions différentiables, Springer-Verlag, Berlin-New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71. MR 0440598
  • [W] C. T. C. Wall, On finite 𝐶^{𝑘}-left determinacy, Invent. Math. 70 (1982/83), no. 3, 399–405. MR 683690, 10.1007/BF01391798
  • [Wh] H. Whitney, Analytic extension of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89.
  • [Wi] Leslie C. Wilson, Infinitely determined map germs, Canad. J. Math. 33 (1981), no. 3, 671–684. MR 627650, 10.4153/CJM-1981-053-3

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Additional Information

Massimo Ferrarotti
Affiliation: Dipartimento di Matematica, Università di Pisa, 56127 Pisa, Italy

Leslie C. Wilson
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822

Keywords: Whitney fields, singularities
Received by editor(s): January 24, 1996
Received by editor(s) in revised form: August 12, 1996
Additional Notes: The first author was partially supported by GNSAGA (CNR), MURST. This work was partially supported by Eurocontract CHRX-CT94-0506.
Article copyright: © Copyright 1998 American Mathematical Society