A Method Using Differential Equations for Characterizing Warped Product Lagrangian Submanifolds in Complex Space Forms
- 1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
- 2Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia
Abstract
In this paper, we establish new inequalities for the length of the second fundamental form of an n-dimensional compact warped product Lagrangian submanifold Qn in a complex space form Qcn(4c). These estimates are expressed in terms of the Laplacian and the gradient of the warping function, together with a positive lower bound on the Ricci curvature. To the best of our knowledge, this provides the first nontrivial result of this type obtained without imposing higher-order curvature assumptions. The characterization results are derived using Obata-type and Tashiro-type differential equations on suitable Riemannian manifolds. By combining the Bochner formula with second-order differential equation techniques, we further characterize the warped product Qn via the first nonzero eigenvalue associated with the warping function. In particular, under appropriate conditions on the second fundamental form, we show that Qn is isometric to either the Euclidean space Rn or the standard sphere Sn.
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BibTeX
@article{alshehri2026method,
title = {{A Method Using Differential Equations for Characterizing Warped Product Lagrangian Submanifolds in Complex Space Forms}},
author = {Norah Alshehri and Ali H. Alkhaldi and Akram Ali},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193568},
url = {https://www.mdpi.com/2227-7390/14/19/3568},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Method Using Differential Equations for Characterizing Warped Product Lagrangian Submanifolds in Complex Space Forms
AU - Norah Alshehri
AU - Ali H. Alkhaldi
AU - Akram Ali
PY - 2026
DA - 2026/10/01
JO - Mathematics
DO - 10.3390/math14193568
UR - https://www.mdpi.com/2227-7390/14/19/3568
AB - In this paper, we establish new inequalities for the length of the second fundamental form of an n-dimensional compact warped product Lagrangian submanifold Qn in a complex space form Qcn(4c). These estimates are expressed in terms of the Laplacian and the gradient of the warping function, together with a positive lower bound on the Ricci curvature. To the best of our knowledge, this provides the first nontrivial result of this type obtained without imposing higher-order curvature assumptions. The characterization results are derived using Obata-type and Tashiro-type differential equations on suitable Riemannian manifolds. By combining the Bochner formula with second-order differential equation techniques, we further characterize the warped product Qn via the first nonzero eigenvalue associated with the warping function. In particular, under appropriate conditions on the second fundamental form, we show that Qn is isometric to either the Euclidean space Rn or the standard sphere Sn.
SN - 2227-7390
ER -