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

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Abstract

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.

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Authors and Affiliations

  1. Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001, Leuven, Belgium

    F. Dillen

  2. Department of Mathematics, University Svetozar Markovic Radoja Domanovica, 34000, Kragujevac, Yugoslavia

    M. Petrovic

  3. Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001, Leuven, Belgium

    L. Verstraelen

  4. Group of Exact Sciences, Katholieke Universiteit Brussel, Vrijheidslaan 17, B-1080, Brussel, Belgium

    L. Verstraelen

Authors
  1. F. Dillen
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  2. M. Petrovic
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  3. L. Verstraelen
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Corresponding author

Correspondence to F. Dillen.

Additional information

The first author is Senior Research Assistant of the National Fund for Scientific Research (Belgium).

The second author was supported by the Research Council of the Katholieke Universiteit Leuven when this work was finished.

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Dillen, F., Petrovic, M. & Verstraelen, L. Einstein, conformally flat and semi-symmetric submanifolds satisfying chen’s equality. Isr. J. Math. 100, 163–169 (1997). https://doi.org/10.1007/BF02773638

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  • DOI: https://doi.org/10.1007/BF02773638

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