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Mathematische Annalen

, Volume 319, Issue 4, pp 707–714 | Cite as

A Moebius characterization of Veronese surfaces in $S^n$

  • Haizhong Li
  • Changping Wang
  • Faen Wu
Original article

Abstract.

Let \( M^m\) be an umbilic-free submanifold in \(S^n\) with I and II as the first and second fundamental forms. An important Moebius invariant for \(M^m\) in Moebius differential geometry is the so-called Moebius form \(\Phi\), defined by \(\Phi =-\rho^{-2}\sum_{i,\alpha}\{H^{\alpha}_{,i} +\sum_j(II^{\alpha}_{ij}-H^{\alpha}I_{ij})e_j(\log \rho)\{\omega_i\otimes e_{\alpha}\), where \(\{e_i\}\) is a local basis of the tangent bundle with dual basis \(\), \(\{e_{\alpha}\}\) is a local basis of the normal bundle, \(H=\sum_{\alpha}H^{\alpha}e_{\alpha}\) is the mean curvature vector and \(\rho =\sqrt{m\over{m-1}}\|II-HI\|\). In this paper we prove that if \(x: S^2\to S^n\) is an umbilics-free immersion of 2-sphere with vanishing Moebius form \(\Phi\), then there exists a Moebius transformation \(\tau: S^n\to S^n\) and a 2k-equator \(S^{2k}\subset S^n\) with \(2\le k\le [n/2]\) such that \(\tau\circ x:S^2\to S^{2k}\) is the Veronese surface.

Mathematics Subject Classification (2000): 53A30, 53C42, 53A10. 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Haizhong Li
    • 1
  • Changping Wang
    • 2
  • Faen Wu
    • 3
  1. 1.Department of Mathematical Sciences, Tsinghua Unviersity, Beijing 100084, People's Republic of China (e-mail: hli@math.tsinghua.edu.cn) CN
  2. 2.Department of Mathematics, Peking University, Beijing 1000871, People's Republic of China (e-mail: wangcp@pku.edu.cn) CN
  3. 3.Department of Mathematics, Northern Jiaotong University, Beijing, 100044, People's Republic of China (e-mail: fewu@center.njtu.edu.cn) CN

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