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Volume 8
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Article

Geometric Inequalities of Warped Product Submanifolds and Their Applications

by
Nadia Alluhaibi
1,
Fatemah Mofarreh
2,
Akram Ali
3,* and
Wan Ainun Mior Othman
4
1
Department of Mathematics, Science and Arts College, Rabigh Campus, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2
Mathematical Science Department, Faculty of Science, Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi Arabia
3
Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia
4
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(5), 759; https://doi.org/10.3390/math8050759
Submission received: 17 April 2020 / Revised: 4 May 2020 / Accepted: 7 May 2020 / Published: 11 May 2020
(This article belongs to the Special Issue Inequalities in Geometry and Applications)
Versions Notes

Abstract

In the present paper, we prove that if Laplacian for the warping function of complete warped product submanifold M m = B p × h F q in a unit sphere S m + k satisfies some extrinsic inequalities depending on the dimensions of the base B p and fiber F q such that the base B p is minimal, then M m must be diffeomorphic to a unit sphere S m . Moreover, we give some geometrical classification in terms of Euler–Lagrange equation and Hamiltonian of the warped function. We also discuss some related results.
Keywords: warped product; sphere theorem; Laplacian; inequalities; diffeomorphic warped product; sphere theorem; Laplacian; inequalities; diffeomorphic

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MDPI and ACS Style

Alluhaibi, N.; Mofarreh, F.; Ali, A.; Mior Othman, W.A. Geometric Inequalities of Warped Product Submanifolds and Their Applications. Mathematics 2020, 8, 759. https://doi.org/10.3390/math8050759

AMA Style

Alluhaibi N, Mofarreh F, Ali A, Mior Othman WA. Geometric Inequalities of Warped Product Submanifolds and Their Applications. Mathematics. 2020; 8(5):759. https://doi.org/10.3390/math8050759

Chicago/Turabian Style

Alluhaibi, Nadia, Fatemah Mofarreh, Akram Ali, and Wan Ainun Mior Othman. 2020. "Geometric Inequalities of Warped Product Submanifolds and Their Applications" Mathematics 8, no. 5: 759. https://doi.org/10.3390/math8050759

APA Style

Alluhaibi, N., Mofarreh, F., Ali, A., & Mior Othman, W. A. (2020). Geometric Inequalities of Warped Product Submanifolds and Their Applications. Mathematics, 8(5), 759. https://doi.org/10.3390/math8050759

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