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Geometric Inequalities for Warped Products in Riemannian Manifolds

Author

Listed:
  • Bang-Yen Chen

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

  • Adara M. Blaga

    (Department of Mathematics, West University of Timişoara, 300223 Timişoara, Romania
    These authors contributed equally to this work.)

Abstract

Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point of view during the last fifty years. In contrast, the study of warped products from extrinsic point of view was initiated around the beginning of this century by the first author in a series of his articles. In particular, he established an optimal inequality for an isometric immersion of a warped product N 1 × f N 2 into any Riemannian manifold R m ( c ) of constant sectional curvature c which involves the Laplacian of the warping function f and the squared mean curvature H 2 . Since then, the study of warped product submanifolds became an active research subject, and many papers have been published by various geometers. The purpose of this article is to provide a comprehensive survey on the study of warped product submanifolds which are closely related with this inequality, done during the last two decades.

Suggested Citation

  • Bang-Yen Chen & Adara M. Blaga, 2021. "Geometric Inequalities for Warped Products in Riemannian Manifolds," Mathematics, MDPI, vol. 9(9), pages 1-31, April.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:923-:d:540300
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    Citations

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    Cited by:

    1. Vladimir Rovenski, 2022. "Geometric Inequalities for a Submanifold Equipped with Distributions," Mathematics, MDPI, vol. 10(24), pages 1-11, December.
    2. 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.

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