Low-type submanifolds of real space forms via the immersions by projectors

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Abstract

We undertake a comprehensive study of submanifolds of low Chen-type (1, 2, or 3) in non-flat real space forms, immersed into a suitable (pseudo) Euclidean space of symmetric matrices by projection operators. Some previous results for submanifolds of the unit sphere (obtained in [A. Ros, Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z. 187 (1984) 393-404; M. Barros, B.Y. Chen, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, Nagoya Math. J. 108 (1987) 77-91; I. Dimitrić, Spherical hypersurfaces with low type quadric representation, Tokyo J. Math. 13 (1990) 469-492; J.T. Lu, Hypersurfaces of a sphere with 3-type quadric representation, Kodai Math. J. 17 (1994) 290-298]) are generalized and extended to real projective and hyperbolic spaces as well as to the sphere. In particular, we give a characterization of 2-type submanifolds of these space forms with parallel mean curvature vector. We classify 2-type hypersurfaces in these spaces and give two sets of necessary conditions for a minimal hypersurface to be of 3-type and for a hypersurface with constant mean curvature to be mass-symmetric and of 3-type. These conditions are then used to classify such hypersurfaces of dimension n ≤ 5. For example, the complete minimal hypersurfaces of the unit sphere Sn + 1 which are of 3-type via the immersion by projectors are exactly the 3-dimensional Cartan minimal hypersurface SO (3) / (Z2 × Z2) and the Clifford minimal hypersurfaces Mk, n - k for n ≠ 2 k. An interesting characterization of horospheres in R Hn + 1 is also obtained.

Original languageEnglish (US)
Pages (from-to)507-526
Number of pages20
JournalDifferential Geometry and its Application
Volume27
Issue number4
DOIs
StatePublished - Aug 1 2009

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology
  • Computational Theory and Mathematics

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