Chen type of some classes of CR-submanifolds in CPm and CHm

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Abstract

We survey some of the fundamental classification results on low-type submanifolds of non-at model complex space forms (complex projective and hyperbolic spaces) via the standard embeddings by projection operators. These results include classification of submanifolds of type 1 in these spaces, of CMC and Hopf hypersurfaces of type 2, and investigation of the Chen type of totally real and Kähler submanifolds. Some examples of submanifolds of type 3 are presented. We also give some nonexistence results for certain families of CR-submanifolds of complex space forms of Chen type two. For example, there exist no holomorphic submanifolds of the complex hyperbolic space which are of type 2 via the standard embedding by projectors. This is contrasted with the situation in the complex projective space, where there exist some parallel Einstein Kähler submanifolds of type 2.

Original languageEnglish (US)
Pages (from-to)439-455
Number of pages17
JournalREVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES
Volume65
Issue number4
StatePublished - 2020

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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