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Chen–Ricci Inequality for CR-Warped Products and Related Open Problems

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Abstract

Chen–Ricci inequality is derived for CR-warped products in complex space forms, Theorem 4.1, involving an intrinsic invariant (Ricci curvature) controlled by extrinsic one (the mean curvature vector), which provides an answer for Problem 1. As a geometric application, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in a complex Euclidean space, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. Moreover, various applications are given. In addition, a rich geometry of CR-warped products appeared when the equality cases are discussed. Also, we extend this inequality to generalized complex space forms. In further research directions, we address a couple of open problems, namely Problems 3 and 4.

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Notes

  1. Throughout this work, \(M^n=N_1\times _fN_2\) denotes for the isometrically immersed warped product submanifold in \(\tilde{M}^m\). The numbers \(m,~n,~n_1,\) and \(n_2\) are the dimensions of \(\tilde{M}^m\), \(M^n\), \(N_1\) and \(N_2\), respectively.

  2. Throughout this work, we use the following convention on the range of indices unless otherwise stated, the indices i, j run from 1 to n, the lowercase letters a, b from 1 to \(n_1\), the uppercase letters A, B from \(n_1\) to n and r from n to m.

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Acknowledgements

The authors are grateful to the anonymous referees and the handling editor for their constructive comments. Also, the first author (Abdulqader Mustafa) would like to thank the Palestine Technical University Kadoori, PTUK, for its supports to accomplish this work.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Science, Palestine Technical University, Kadoorei, Tulkarm, Palestine

    Abdulqader Mustafa

  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589, Jeddah, Saudi Arabia

    Siraj Uddin

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  1. Abdulqader Mustafa
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  2. Siraj Uddin
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Correspondence to Siraj Uddin.

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Mustafa, A., Uddin, S. Chen–Ricci Inequality for CR-Warped Products and Related Open Problems. Mediterr. J. Math. 18, 67 (2021). https://doi.org/10.1007/s00009-021-01722-8

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  • DOI: https://doi.org/10.1007/s00009-021-01722-8

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