Intrinsic invariants of cross caps

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 1.

    Bruce, J.W., West, J.M.: Functions on cross-caps. Math. Proc. Camb. Philos. Soc 123, 19–39 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Fujimori, S., Saji, K., Umehara, M., Yamada, K.: Singularities of maximal surfaces. Math. Z 259, 827–848 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Fukui, T., Hasegawa, M.: Height Functions on Whitney Umbrellas, to appear in RIMS Kôkyûroku Bessatsu 38 (2013)

  4. 4.

    Fukui, T., Nuño-Ballesteros, J.J.: Isolated roundings and flattenings of submanifolds in Euclidean spaces. Tôhoku Math. J 57, 469–503 (2005)

    Article  MATH  Google Scholar 

  5. 5.

    Fukui, T., Hasegawa, M.: Fronts of Whitney umbrella–a differential geometric approach via blowing up. J. Singul. 4, 35–67 (2012)

    MathSciNet  Google 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Gutierrez, C., Sotomayor, J.: Lines of principal curvature for mappings with Whitney umbrella singularities. Tôhoku Math. J. 38, 551–559 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Kuiper, N.H.: Stable surfaces in Euclidean three space. Math. Scand 36, 83–96 (1975)

    MATH  MathSciNet  Google 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)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Oliver, J.M.: On pairs of foliations of a parabolic cross-cap. Qual. Theory Dyn. Syst 10, 139–166 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Saji, K., Umehara, M., Yamada, K.: The geometry of fronts. Ann. Math 169, 491–529 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Tari, F.: Pairs of geometric foliations on a cross-cap. Tôhoku Math. J 59, 233–258 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    West, J.M.: The differential geometry of the cross-cap, Ph. D. thesis, The University of Liverpool, 1995

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Masaaki Umehara.

Additional information

The second and third authors were partly supported by the Grant-in-Aid for JSPS Fellows. The fourth and fifth authors were partially supported by Grant-in-Aid for Scientific Research (A) No. 22244006, and Scientific Research (B) No. 21340016, respectively, from the Japan Society for the Promotion of Science.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hasegawa, M., Honda, A., Naokawa, K. et al. Intrinsic invariants of cross caps. Sel. Math. New Ser. 20, 769–785 (2014). https://doi.org/10.1007/s00029-013-0134-6

Download citation

Keywords

  • Cross cap
  • Curvature
  • Isometric deformation

Mathematics Subject Classification (2010)

  • Primary 57R45
  • Secondary 53A05