Acta mathematica scientia,Series B ›› 2005, Vol. 25 ›› Issue (2): 223-232.

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SLANT IMMERSIONS OF COMPLEX SPACE FORMS AND CHEN’S INEQUALITY

 LI Guang-Han, TUN Chuan-Chi   

  1. 2School of Mathematical Science, Peking University, School of Mathematics and Computer Science, Hubei University,
  • Online:2005-04-20 Published:2005-04-20
  • Supported by:

    This project is supported by the NSFC(10271041); Tianyuan Youth Foundation of Mathematics.

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Abstract:

A submanifold in a complex space form is called slant if it has constant
Wirtinger angles. B. Y. Chen and Y. Tazawa proved that there do not exist minimal
proper slant surfaces in CP2 and CH2. So it seems that the slant immersion has some
interesting properties. The authors have great interest to consider slant immersions sat-
isfying some additional conditions, such as unfull first normal bundles or Chen’s equality
holding. They prove that there do not exist n-dimensional Kaehlerian slant immersions
in CPn and CHn with unfull first normal bundles. Next, it is seen that every Kaehlerian
slant submanifold satisfying an equality of Chen is minimal which is similar to that of
Lagrangian immersions. But in contrast, it is shown that a large class of slant immersions
do not exist thoroughly. Finally, they give an application of Chen’s inequality to general
slant immersions in a complex projective space, which generalizes a result of Chen.

Key words: Slant immersion；ideal, Chen’s inequality；complex space form

CLC Number: 

  • 53C40
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