Israel Journal of Mathematics

, Volume 100, Issue 1, pp 163–169 | Cite as

Einstein, conformally flat and semi-symmetric submanifolds satisfying chen’s equality

  • F. Dillen
  • M. Petrovic
  • L. Verstraelen


In a recent paper, B. Y. Chen proved a basic inequality between the intrinsic scalar invariants infK andτ ofM n , and the extrinsic scalar invariant |H|, being the length of the mean curvature vector fieldH ofM n in\(\mathbb{E}^m \). In the present paper we classify the submanifoldsM n of\(\mathbb{E}^m \) for which the basic inequality actually is an equality, under the additional assumption thatM n satisfies some of the most primitive Riemannian curvature conditions, such as to be Einstein, conformally flat or semi-symmetric.


Sectional Curvature Curvature Tensor Complex Space Form Basic Inequality Real Submanifolds 
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  1. [1]
    B.-Y. Chen,Some pinching and classification theorems for minimal submanifolds, Archiv för Mathematik60 (1993), 568–578.MATHCrossRefGoogle Scholar
  2. [2]
    B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken,Characterizing a class of totally real submanifolds of S 6 by their sectional curvatures, Tôhoku Mathematical Journal47 (1995), 185–198.MATHMathSciNetGoogle Scholar
  3. [3]
    B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken,Totally real minimal immersion of ℂP 3 satisfying a basic equality, Archiv för Mathematik63 (1994), 553–564.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Deprez,Semi-parallel immersions, inGeometry and Topology of Submanifolds, I, World Scientific, Singapore, 1989, pp. 73–88.Google Scholar
  5. [5]
    F. Dillen and S. Nölker,Semi-parallelity, multi-rotation surfaces and the helix-property, Journal für die reine und angewandte Mathematik435 (1993), 33–63.MATHGoogle Scholar
  6. [6]
    F. Dillen, M. Petrovic and L. Verstraelen,The conharmonic curvature tensor and 4-dimensional catenoids, Studia Babes-Bolyai Mathematica33 (1988), 16–23.MATHGoogle Scholar
  7. [7]
    Y. Ishii,On conharmonic transformations, Tensor7 (1957), 73–80.MATHMathSciNetGoogle Scholar
  8. [8]
    G. M. Lancaster,Canonical metrics for certain conformally Euclidean spaces of dimension three and codimension one, Duke Mathematical Journal40 (1973), 1–8.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. Petrovic, R. Rosca and L. Verstraelen,Exterior concurrent vector fields on Riemannian manifolds II. Examples, Soochow Journal of Mathematics19 (1993), 357–368.MATHMathSciNetGoogle Scholar
  10. [10]
    I. M. Singer and J. A. Thorpe,The curvature of 4-dimensional Einstein spaces, inGlobal Analysis: Papers in Honor of K. Kodaira, Princeton University Press, 1969, pp. 355–365.Google Scholar
  11. [11]
    Z. Szabó,Structure theorems on Riemannian spaces satisfying R(X, Y) · R = 0. I. The local version, Journal of Differential Geometry17 (1982), 531–582.MATHMathSciNetGoogle Scholar
  12. [12]
    L. Verstraelen and G. Zafindratafa,On the sectional curvature of conharmonically flat spaces, Rendiconti del Seminario Matematico di Messina, Seria 21 (1991), 247–254.MathSciNetGoogle Scholar
  13. [13]
    L. Verstraelen,Comments on pseudo-symmetry in the sense of Ryszard Deszcz, inGeometry and Topology of Submanifolds, VI, World Scientific, Singapore, 1994, pp. 199–209.Google Scholar

Copyright information

© Hebrew University 1997

Authors and Affiliations

  1. 1.Departement WiskundeKatholieke Universiteit LeuvenLeuvenBelgium
  2. 2.Department of MathematicsUniversity Svetozar Markovic Radoja DomanovicaKragujevacYugoslavia
  3. 3.Departement WiskundeKatholieke Universiteit LeuvenLeuvenBelgium
  4. 4.Group of Exact SciencesKatholieke Universiteit BrusselBrusselBelgium

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