Skip to main content
SpringerLink
Log in

The Möbius Geometry of Wintgen Ideal Submanifolds

  • Conference paper
  • First Online:
Real and Complex Submanifolds

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 106))

Abstract

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. They are Möbius invariant objects. The mean curvature sphere defines a conformal Gauss map into a Grassmann manifold. We show that any Wintgen ideal submanifold of dimension greater than or equal to 3 has a Riemannian submersion structure over a Riemann surface with the fibers being round spheres. Then the conformal Gauss map is shown to be a super-conformal and harmonic map from the underlying Riemann surface. Some of our previous results are surveyed in the final part.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    It was first noticed by Dajczer and Tojeiro in [8], based on an equivalent formulation of the DDVV inequality in [10].

  2. 2.

    The notion of the mean curvature sphere can be traced back to Blaschke [1] in 1920s.

  3. 3.

    This is an analog to the work of Bryant [2] and Ejiri [11] on Willmore surfaces in \(\mathbb{S}^{n}\).

References

  1. Blaschke, W.: Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. Springer Grundlehren XXIX, Berlin (1929)

    MATH  Google Scholar 

  2. Bryant, R.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20, 20–53(1984)

    Google Scholar 

  3. Bryant, R.: Some remarks on the geometry of austere manifolds. Bol. Soc. Bras. Mat. 21, 122–157 (1991)

    Article  Google Scholar 

  4. Chen, B. Y.: Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures. Ann. Glob. Anal. Geom. 38, 145–160 (2010)

    Article  MATH  Google Scholar 

  5. Choi, T., Lu, Z.: On the DDVV conjecture and the comass in calibrated geometry (I). Math. Z. 260, 409–429 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dajczer, M., Tojeiro, R.: A class of austere submanifolds. Illinois J. Math. 45(3), 735–755 (2001)

    MATH  MathSciNet  Google Scholar 

  7. Dajczer, M., Tojeiro, R.: All superconformal surfaces in R 4 in terms of minimal surfaces. Math. Z. 261(4), 869–890 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dajczer, M., Tojeiro, R.: Submanifolds of codimension two attaining equality in an extrinsic inequality. Math. Proc. Cambridge Philos. Soc. 146(2), 461–474 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. De Smet, P. J., Dillen, F., Verstraelen, L., Vrancken, L.: A pointwise inequality in submanifold theory. Arch. Math. 35, 115–128 (1999)

    MATH  Google Scholar 

  10. Dillen, F., Fastenakels, J., Van Der Veken, J.: Remarks on an inequality involving the normal scalar curvature. In: Proceedings of the International Congress on Pure and Applied Differential Geometry-PADGE Brussels, pp. 83–92. Shaker, Aachen (2007)

    Google Scholar 

  11. Ejiri, N.: Willmore surfaces with a duality in S N(1). Proc. Lond. Math. Soc. (3) 57, 383–416 (1988)

    Google Scholar 

  12. Ge, J., Tang, Z.: A proof of the DDVV conjecture and its equality case. Pac. J. Math., 237, 87–95 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Guadalupe, I., Rodríguez, L.: Normal curvature of surfaces in space forms. Pac. J. Math. 106, 95–103 (1983)

    Article  MATH  Google Scholar 

  14. Li, T., Ma, X., Wang, C.: Wintgen ideal submanifolds with a low-dimensional integrable distribution (I). http://arxiv.org/abs/1301.4742

  15. Li, T., Ma, X., Wang, C., Xie, Z.: Wintgen ideal submanifolds of codimension two, complex curves, and Moebius geometry. http://arxiv.org/abs/1402.3400

  16. Li, T., Ma, X., Wang, C., Xie, Z.: Classification of Moebius homogeneous Wintgen ideal submanifolds. http://arxiv.org/abs/1402.3430

  17. Lu, Z.: Normal Scalar Curvature Conjecture and its applications. J. Funct. Anal. 261, 1284–1308 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Petrovié-torgas̆ev, M., Verstraelen, L.: On Deszcz symmetries of Wintgen ideal Submanifolds. Arch. Math. 44, 57–67 (2008)

    Google Scholar 

  19. Wang, C. P.: Möbius geometry of submanifolds in S n. Manuscr. Math. 96, 517–534 (1998)

    Article  MATH  Google Scholar 

  20. Wintgen, P.: Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci. Paris 288, 993–995 (1979)

    MATH  MathSciNet  Google Scholar 

  21. Xie, Z., Li, T., Ma, X., Wang, C.: Möbius geometry of three dimensional Wintgen ideal submanifolds in \(\mathbb{S}^{5}\). Sci. China Math. doi:10.1007/s11425-013-4664-3. See also http://arxiv.org/abs/1402.3440

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Xiang Ma

  2. Department of Mathematics, China University of Mining and Technology (Beijing), Beijing, 100083, People’s Republic of China

    Zhenxiao Xie

Authors
  1. Xiang Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenxiao Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiang Ma .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Kyungpook National University, Daegu, Republic of Korea

    Young Jin Suh

  2. Department of Mathematics, King’s College London, London, UK

    Jürgen Berndt

  3. Department of Mathematics, Osaka City University, Osaka, Japan

    Yoshihiro Ohnita

  4. Department of Applied Mathematics, Kyung Hee University, Yongin, Republic of Korea

    Byung Hak Kim

  5. The Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang, Republic of Korea

    Hyunjin Lee

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Japan

About this paper

Cite this paper

Ma, X., Xie, Z. (2014). The Möbius Geometry of Wintgen Ideal Submanifolds. In: Suh, Y.J., Berndt, J., Ohnita, Y., Kim, B.H., Lee, H. (eds) Real and Complex Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 106. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55215-4_37

Download citation

Publish with us

Policies and ethics

Search

Navigation