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Sheets of Real Analytic Varieties

Published online by Cambridge University Press:  20 November 2018

Andrew H. Wallace
Andrew H. Wallace*
Affiliation:
University of Toronto
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In a previous paper (4) the author worked out some results on the analytic connectivity properties of real algebraic varieties, that is to say, properties associated with the joining of points of the variety by analytic arcs lying on the variety. It is natural to ask whether these properties can be carried over to analytic varieties, since the proofs in the algebraic case depend mainly on local properties. But although this generalization can be carried out to a large extent, there are, nevertheless, difficulties in the analytic case, owing mainly to the fact (cf. 2, § 11) that a real analytic variety may not be definable by means of a set of global equations. Thus, although the general idea of the treatment given here is the same as in (4), some variation in the details of the method has proved to be necessary, and some of the final results are slightly weaker in form.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 51 - 67
Copyright
Copyright © Canadian Mathematical Society 1960

References

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3. van der Waerden, B. L., Moderne algebra (Berlin, 1931).Google Scholar
4. Wallace, A.H., Algebraic approximation of curves, Can. J. Math. 10 (1958), 242–278.Google Scholar
5. Whitney, H., Analytic coordinate systems and arcs in a manifold, Ann. Math., 38 (1937), 809–818.Google Scholar
6. Whitney, H. and Bruhat, F., Quelques propriétés fondamentales des ensembles analytiques-réels, Comm. Math. Helvet., 33 (1959), 132–160.Google Scholar