Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-02T15:13:07.724Z Has data issue: false hasContentIssue false
  • English
  • Français

Conformal Geometry and the Cyclides of Dupin

Published online by Cambridge University Press:  20 November 2018

Thomas E. Cecil  and
Patrick J. Ryan
Thomas E. Cecil
Affiliation:
College of the Holy Cross, Worcester, Massachusetts
Patrick J. Ryan
Affiliation:
Indiana University at South Bend, South Bend, Indiana
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.

A Riemannian manifold (M, g) is said to be conformally flat if every point has a neighborhood conformai to an open set in Euclidean space. Over the past thirty years, many papers have appeared attacking, with varying degrees of success, the problem of classifying the conformally flat spaces which occur as hypersurfaces in Euclidean space. Most of these start from the following pointwise result of Schouten.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 4 , April 1980 , pp. 767 - 782
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Banchoff, T., The spherical two-piece property and tight surfaces in spheres, J. Differential Geometry 4 (1970), 193–205.Google Scholar
2. Carter, S. and West, A., Tight and taut immersions, Proc. London Math. Soc. 25 (1972), 701–720.Google Scholar
3. Cecil, T., Taut immersions of non-compact surfaces into a Euclidean 3-space, J. Differential Geometry 11 (1976), 451–459.Google Scholar
4. Cecil, T. and Ryan, P., Focal sets, taut embeddings and the cyclides of Dupin, Math. Ann. 236 (1978), 177–190.Google Scholar
5. Cecil, T. and Ryan, P., pocai sets of submanifolds, Pacific J. Math. 78 (1978), 27–39.Google Scholar
6. Chen, B.-Y. and Yano, K., Special conformally flat spaces and canal hyper surf aces, Tôhoku Math. J. 25 (1973), 177–184.Google Scholar
7. Chen, B.-Y., Geometry of submanifolds (Marcel Dekker, New York, 1973).Google Scholar
8. Eisenhart, L., A treatise on the differential geometry of curves and surfaces (Ginn, Boston, 1909).Google Scholar
9. Hartman, P. and Nirenberg, L., On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901–920.Google Scholar
10. Hilbert, D. and Cohn-Vossen, S., Geometry and the imagination (Chelsea, New York, 1952).Google Scholar
11. Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Vol. II (John Wiley, New York, 1969).Google Scholar
12. Kulkarni, R. S., Conform ally flat manifolds, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2675–2676.Google Scholar
13. Lancaster, G. M., A characterization of certain conformally Euclidean spaces of class one, Proc. Amer. Math. Soc. 21 (1969), 623–628.Google Scholar
14. Lancaster, G. M., Canonical metrics for certain conformally Euclidean spaces of dimension three and codimension one, Duke Math. j. 40 (1973), 1–8.Google Scholar
15. Lilienthal, R., Besondere Flächen, in Encyklopadie der Math. Wissenschaften, Vol. III, 3, 269–354, Leipzig: B. G. Teubner, 1902–1927.Google Scholar
16. Moore, J. D., Conformally flat submanifolds of Euclidean space, Math. Ann. 225 (1977), 89–97.Google Scholar
17. Nishikawa, S., Conformally flat hypersurfaces in a Euclidean space, Tôhoku Math. J. 26 (1974), 563–572.Google Scholar
18. Nishikawa, S. and Maeda, Y., Conformally flat hypersurfaces in a conformally flat Riemannian manifold, Tôhoku Math. J. 26 (1974), 159–168.Google Scholar
19. Nomizu, K., On hypersurfaces satisfying a certain condition on the curvature tensor, Tôhoku Math. J. 20 (1968), 46–59.Google Scholar
20. Nomizu, K. and Rodriguez, L., Umbilical submanifolds and Morse functions, Nagoya Math. J. 48 (1972), 197–201.Google Scholar
21. Palais, R., A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. 22 (1957).Google Scholar
22. Ryan, P., Homogeneity and some curvature conditions for hypersurfaces, Tôhoku Math. J. 21 (1969), 363–388.Google Scholar
23. Samelson, H., Orientability of hypersurfaces in Rn, Proc. Amer. Math. Soc. 22 (1969), 301–302.Google Scholar
24. Schouten, J. A., Uber die konforme Abbildung n-dimensionaler Mannigfaltigkeiter nut quadratischer Maβ bestimmung auf eine Mannigfaltigkeit mit euklidischer Maβ bestimmung, Math. Z. 11 (1921), 58–88.Google Scholar