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Classification of Quadratic Harmonic Maps of \(S^{7}\) into \(S^{7}\)

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Abstract

A complete classification of full quadratic harmonic maps of the seven-sphere \(S^7\) into itself is obtained. It is shown that any full quadratic harmonic map of the seven-sphere \(S^7\) into itself is equivalent to either the gradient map of the cubic isoparametric polynomial of \(\acute{\hbox {E}}.\) Cartan or a map depending only on one parameter. The components of the map in the latter case are explicitly given.

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Acknowledgments

The original version of this paper was completed after his stay at the Polytechnic Institute of NYU of the first author as a visiting scholar. He wishes to thank the Department of Mathematics of Polytech for its hospitality. The authors thank the referee’s careful reading and providing references [12, 28] and informing the authors that the result of Lemma 3 is available in [12]. The first author is supported by NSFC number 11171016.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Faen Wu

  2. Academy of Mathematics and System Science, Chinese Academy of Science, Beijing, 100190, People’s Republic of China

    Yueshan Xiong

  3. GuangXi University of Technology, Lushan College, Liuzhou, 545610, People’s Republic of China

    Xinnuan Zhao

Authors
  1. Faen Wu
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  2. Yueshan Xiong
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  3. Xinnuan Zhao
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Correspondence to Faen Wu.

Additional information

Communicated by Jiri Dadok.

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Wu, F., Xiong, Y. & Zhao, X. Classification of Quadratic Harmonic Maps of \(S^{7}\) into \(S^{7}\) . J Geom Anal 25, 1992–2010 (2015). https://doi.org/10.1007/s12220-014-9501-6

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  • DOI: https://doi.org/10.1007/s12220-014-9501-6

Keywords

Mathematics Subject Classification (2010)

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