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


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.


Cross cap Curvature Isometric deformation 

Mathematics Subject Classification (2010)

Primary 57R45 Secondary 53A05 



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.


  1. 1.
    Bruce, J.W., West, J.M.: Functions on cross-caps. Math. Proc. Camb. Philos. Soc 123, 19–39 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Fujimori, S., Saji, K., Umehara, M., Yamada, K.: Singularities of maximal surfaces. Math. Z 259, 827–848 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Fukui, T., Hasegawa, M.: Height Functions on Whitney Umbrellas, to appear in RIMS Kôkyûroku Bessatsu 38 (2013)Google Scholar
  4. 4.
    Fukui, T., Nuño-Ballesteros, J.J.: Isolated roundings and flattenings of submanifolds in Euclidean spaces. Tôhoku Math. J 57, 469–503 (2005)CrossRefzbMATHGoogle Scholar
  5. 5.
    Fukui, T., Hasegawa, M.: Fronts of Whitney umbrella–a differential geometric approach via blowing up. J. Singul. 4, 35–67 (2012)MathSciNetGoogle Scholar
  6. 6.
    Garcia, R., Gutierrez, C., Sotomayor, J.: Lines of principal curvature around umbilics and Whitney umbrellas. Tôhoku Math. J 52, 163–172 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gutierrez, C., Sotomayor, J.: Lines of principal curvature for mappings with Whitney umbrella singularities. Tôhoku Math. J. 38, 551–559 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kuiper, N.H.: Stable surfaces in Euclidean three space. Math. Scand 36, 83–96 (1975)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Nuño-Ballesteros, J.J., Tari, F.: Surfaces in \({\bf R}^4\) and their projections to 3-spaces. Proc. Roy. Soc. Edinburgh Sect. A 137, 1313–1328 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Oliver, J.M.: On pairs of foliations of a parabolic cross-cap. Qual. Theory Dyn. Syst 10, 139–166 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Saji, K., Umehara, M., Yamada, K.: The geometry of fronts. Ann. Math 169, 491–529 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Tari, F.: Pairs of geometric foliations on a cross-cap. Tôhoku Math. J 59, 233–258 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    West, J.M.: The differential geometry of the cross-cap, Ph. D. thesis, The University of Liverpool, 1995Google Scholar

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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