Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T20:51:23.608Z Has data issue: false hasContentIssue false
  • English
  • Français

A Topological Characterization of Conjugate Nets

Published online by Cambridge University Press:  20 November 2018

Paul A. Vincent
Paul A. Vincent*
Affiliation:
Université de Moncton, Moncton, Nouveau Brunswick
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

One aspect of topological analysis that authors, such as G. T. Whyburn and Marston Morse, have pointed to ([16; 6] for instance) as being fundamental in the development of function theory is the topological study of the level sets of analytic and harmonic functions or of their topological analogues, light open maps and pseudo-harmonic functions. The first step in this direction seems to have been made by H. Whitney [14] when he studied families of curves, given abstractly using a condition of regularity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 4 , 01 August 1977 , pp. 707 - 721
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Anderson, R. D., On monotone interior mappings of the plane, Trans. Amer. Math. Soc. 73 (1952), 211–222.Google Scholar
2. Boothby, W. M., The topology of regular curve families with multiple saddle points, Amer. J. Math. 73 (1951), 405–438.Google Scholar
3. Boothby, W. M. The topology of level curves of harmonic functions with critical points, Amer. J. Math. 73 (1951), 512–538.Google Scholar
4. Fox, W. C., The critical points of peano-interior functions defined on 2-manifolds, Trans. Amer. Math. Soc. 83 (1956), 338–370.Google Scholar
5. Hocking, J. G. and Young, G. S., Topology (Addison-Wesley, Reading, Mass., 1961).Google Scholar
6. Jenkins, J. and Morse, M., The existence of pseudo-conjugates on Riemann surfaces, Fund. Math 39 (1952), 269–287.Google Scholar
7. Jenkins, J. and Morse, M. Topological methods on Riemann surfaces, Annals of Math. Studies, 30 111–139 (Princeton, 1953).Google Scholar
8. Jenkins, J. and Morse, M. Conjugate nets on an open Riemann surface, in Lectures on Functions of a Complex Variable, ed. W. Kaplan et al., 123–185 (The University of Michigan Press, 1955).Google Scholar
9. Kaplan, W., Regular curve families filling the plane, Duke Math. Jour. 7 (1940) 154-185; 8 (1941), 11–45.Google Scholar
10. Moore, R. L., Foundations of point set theory, Amer. Math. Soc. Colloquium Publications 13 (1932).Google Scholar
11. Roberts, J. H., Concerning collections of continua not all bounded, Amer. J. Math. 52 (1930), 551–562.Google Scholar
12. Rutt, N. E., On certain types of plane continua, Trans. Amer. Math. Soc. 33 (1931), 806–816.Google Scholar
13. Vincent, P., A metrization theorem for 2-manifolds, Can. Math. Bull. 19 (1976), 95–104.Google Scholar
14. Whitney, H., Regular families of curves, Ann. of Math. 34 (1933), 244–270.Google Scholar
15. Whyburn, G. T., Analytic topology, Amer. Math. Soc. Colloquium Publications, 28, New York, 1942.Google Scholar
16. Whyburn, G. T. Continuous decompositions, Amer. Jour. Math. 71 (1949), 218–226.Google Scholar