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Demonstratio Mathematica

Editor-in-Chief: Vetro, Calogero


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CiteScore 2017: 0.28
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2391-4661
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Volume 47, Issue 3

Issues

Examples of Non-Conformal Submersions from Spheres with Umbilical Fibers

Tomasz Zawadzki
Published Online: 2014-09-02 | DOI: https://doi.org/10.2478/dema-2014-0059

Abstract

We directly prove that the foliation of the sphere S2n+1, the leaves of which are intersections of all complex linear 2-dimensional subspaces of ℂn+1 translated by a constant vector p, defines a submersion that is horizontally conformal if and only if p=0. We generalise this result to the cases of S4n+3 and S15 with foliations constructed using quaternionic and octonionic structure (resp.) in an analogous way

Keywords: and phrases: foliation; conformal submersion; spheres

References

  • [1] F. E. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal Geometry of Surfaces in S4 and Quaternions, Springer, Berlin, 2002.Google Scholar

  • [2] D. B. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier 26(1) (1976), 265-282.CrossrefGoogle Scholar

  • [3] R. H. Escobales Jr, Riemannian submersions with totally geodesic fibers, J. Differential Geom. 10(2) (1975), 253-276.Google Scholar

  • [4] S. G. Heller, Conformal fibrations of S3 by circles, in Harmonic maps and differential geometry, 195-202, Contemp. Math. 542, AMS, Providence, R. I., 2011.Google Scholar

  • [5] B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469.Google Scholar

About the article

Received: 2013-02-18

Revised: 2013-04-16

Published Online: 2014-09-02

Published in Print: 2014-07-01


Citation Information: Demonstratio Mathematica, Volume 47, Issue 3, Pages 738–747, ISSN (Online) 2391-4661, ISSN (Print) 0420-1213, DOI: https://doi.org/10.2478/dema-2014-0059.

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© by Tomasz Zawadzki. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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