Isometric Immersions with Singularities Between Space Forms of the Same Positive Curvature

Abstract

In this paper, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front, which is a generalization of a theorem of O’Neill and Stiel: any isometric immersion of the n-sphere into the \((n+1)\)-sphere of the same sectional curvature is totally geodesic.

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Acknowledgements

The author would like to thank Professors Masaaki Umehara, Kotaro Yamada, Miyuki Koiso, Masatoshi Kokubu, Jun-ichi Inoguchi, Yu Kawakami, Masatomo Takahashi, and Kosuke Naokawa for their valuable comments and constant encouragements. He also expresses gratitude to Professor Udo Hertrich-Jeromin and Gudrun Szewieczek for careful reading of the first draft. This work is partially supported by Grant-in-Aid for Challenging Exploratory Research No. 26610016 of the Japan Society for the Promotion of Science.

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Correspondence to Atsufumi Honda.

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Honda, A. Isometric Immersions with Singularities Between Space Forms of the Same Positive Curvature. J Geom Anal 27, 2400–2417 (2017). https://doi.org/10.1007/s12220-017-9765-8

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Keywords

  • Isometric immersion
  • Wave front
  • (co-)orientability
  • Caustic
  • Dual

Mathematics Subject Classification

  • Primary 53C42
  • Secondary 57R45