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Recent Developments on the First Chen Inequality in Differential Geometry

Author

Listed:
  • Bang-Yen Chen

    (Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA
    These authors contributed equally to this work.)

  • Gabriel-Eduard Vîlcu

    (Department of Mathematics and Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania
    “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie 13, 050711 Bucharest, Romania
    These authors contributed equally to this work.)

Abstract

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. In this respect, the first author established, in 1993, a basic inequality involving the first δ -invariant, δ ( 2 ) , and the squared mean curvature of submanifolds in real space forms, known today as the first Chen inequality or Chen’s first inequality. Since then, there have been many papers dealing with this inequality. The purpose of this article is, thus, to present a comprehensive survey on recent developments on this inequality performed by many geometers during the last three decades.

Suggested Citation

  • Bang-Yen Chen & Gabriel-Eduard Vîlcu, 2023. "Recent Developments on the First Chen Inequality in Differential Geometry," Mathematics, MDPI, vol. 11(19), pages 1-50, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4186-:d:1254639
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    References listed on IDEAS

    as
    1. Dragoş Cioroboiu, 2003. "B.-Y. Chen inequalities for semislant submanifolds in Sasakian space forms," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-8, January.
    2. Jeong-Sik Kim & Jaedong Choi, 2003. "A basic inequality for submanifolds in a cosymplectic space form," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-9, January.
    3. Bang-Yen Chen & Adara M. Blaga & Gabriel-Eduard Vîlcu, 2022. "Differential Geometry of Submanifolds in Complex Space Forms Involving δ -Invariants," Mathematics, MDPI, vol. 10(4), pages 1-38, February.
    Full references (including those not matched with items on IDEAS)

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