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Transactions of the American Mathematical Society
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.


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Additional Information

Massimo Ferrarotti
Affiliation: Dipartimento di Matematica, Università di Pisa, 56127 Pisa, Italy
Email: ferraro@dm.unipi.it

Leslie C. Wilson
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
Email: les@math.hawaii.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01925-4
PII: S 0002-9947(98)01925-4
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