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Selecta Mathematica

, Volume 20, Issue 3, pp 769–785 | Cite as

Intrinsic invariants of cross caps

  • Masaru Hasegawa
  • Atsufumi Honda
  • Kosuke Naokawa
  • Masaaki UmeharaEmail author
  • Kotaro Yamada
Article

Abstract

It is classically known that generic smooth maps of \(\varvec{R}^2\) into \(\varvec{R}^3\) admit only isolated cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap \(f_{\mathrm{std }}(u,v)=(u,uv,v^2)\) has non-trivial isometric deformations with infinite-dimensional freedom. Since there are several geometric invariants for cross caps, the existence of isometric deformations suggests that one can ask which invariants of cross caps are intrinsic. In this paper, we show that there are three fundamental intrinsic invariants for cross caps. The existence of extrinsic invariants is also shown.

Keywords

Cross cap Curvature Isometric deformation 

Mathematics Subject Classification (2010)

Primary 57R45 Secondary 53A05 

Notes

Acknowledgments

The authors thank Shyuichi Izumiya, Toshizumi Fukui and Wayne Rossman for valuable comments. The fourth author thanks Huili Liu for fruitful discussions on this subject at 8th Geometry Conference for Friendship of China and Japan at Chengdu.

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Masaru Hasegawa
    • 1
  • Atsufumi Honda
    • 2
  • Kosuke Naokawa
    • 3
  • Masaaki Umehara
    • 4
    Email author
  • Kotaro Yamada
    • 3
  1. 1.Department of Mathematics, Faculty of ScienceSaitama UniversitySakura-ku, SaitamaJapan
  2. 2.Miyakonojo National College of TechnologyMiyakonojo, MiyazakiJapan
  3. 3.Department of MathematicsTokyo Institute of TechnologyTokyoJapan
  4. 4.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan

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