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Geometry of bi-warped product submanifolds of locally product Riemannian manifolds

  • Siraj Uddin
  • Adela Mihai
  • Ion MihaiEmail author
  • Awatif AL-Jedani
Original Paper
  • 81 Downloads

Abstract

In 2008, Chen and Dillen obtained a sharp estimation for the squared norm of the second fundamental form of multiply warped CR-submanifold \(M=M_1\times _{f_2}M_2\times \ldots \times _{f_k}M_k\) in an arbitrary Kähler manifold \({\tilde{M}}\) such that \(M_1\) is a holomorphic submanifold and \(M_\perp =_{f_2}M_2\times \cdots \times _{f_k}M_k\) is a totally real submanifold of \({\tilde{M}}\). In this paper, we study bi-warped product submanifolds of locally product Riemannian manifolds which are the generalizations of single warped products. We prove that the bi-warped products of the form \(M_T\times _{f_1}M_\perp \times _{f_2}M_\theta \) and \(M_\perp \times _{f_1}M_T\times _{f_2}M_\theta \) in an arbitrary locally product Riemannian manifold \({\tilde{M}}\), where \(M_T\) is an invariant submanifold, \(M_\perp \) an anti-invariant submanifold and \(M_\theta \) a slant submanifold of \({\tilde{M}}\), are either Riemannian products or single warped products. Then, we investigate the geometry of bi-warped product submanifolds \(M_\theta \times _{f_1}M_T\times _{f_2}M_\perp \) in a locally product Riemannian manifold \({\tilde{M}}\). We provide non-trivial examples of such submanifolds and a sharp estimation for the squared norm of the second fundamental form is obtained in terms of the warping functions \(f_1\) and \(f_2\). The equality case is also considered. Further, we give some applications of our main result.

Keywords

Warped products Bi-warped products Multiply warped products Invariant submanifold Slant Semi-slant Pseudo-slant Locally product Riemannian manifolds 

Mathematics Subject Classification

53C15 53C40 53C42 53B25 

Notes

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Department of Mathematics and Computer ScienceTechnical University of Civil Engineering BucharestBucharestRomania
  3. 3.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

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