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A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms
Article

On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms

1
Department of Mathematics, College of Sciences and Arts, Muhayil, King Khalid University, Abha 9004, Saudi Arabia
2
Mathematical Science Department, Faculty of Science, Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi Arabia
3
Department of Mathematics, Science and Arts College, Rabigh Campus, King Abdulaziz University, Jeddah 21911, Saudi Arabia
4
Department of Mathematics, College of Science, King Khalid University, Abha 9004, Saudi Arabia
5
College of Engineering, Qassim University, Buraidah 51452, Al-Qassim, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(2), 150; https://doi.org/10.3390/math8020150
Received: 10 December 2019 / Revised: 14 January 2020 / Accepted: 15 January 2020 / Published: 21 January 2020
(This article belongs to the Special Issue Inequalities in Geometry and Applications)
In this paper, we give an estimate of the first eigenvalue of the Laplace operator on minimally immersed Legendrian submanifold N n in Sasakian space forms N ˜ 2 n + 1 ( ϵ ) . We prove that a minimal Legendrian submanifolds in a Sasakian space form is isometric to a standard sphere S n if the Ricci curvature satisfies an extrinsic condition which includes a gradient of a function, the constant holomorphic sectional curvature of the ambient space and a dimension of N n . We also obtain a Simons-type inequality for the same ambient space forms N ˜ 2 n + 1 ( ϵ ) . View Full-Text
Keywords: legendrian submanifolds; sasakian space forms; obata differential equation; isometric immersion legendrian submanifolds; sasakian space forms; obata differential equation; isometric immersion
MDPI and ACS Style

Ali, R.; Mofarreh, F.; Alluhaibi, N.; Ali, A.; Ahmad, I. On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms. Mathematics 2020, 8, 150. https://doi.org/10.3390/math8020150

AMA Style

Ali R, Mofarreh F, Alluhaibi N, Ali A, Ahmad I. On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms. Mathematics. 2020; 8(2):150. https://doi.org/10.3390/math8020150

Chicago/Turabian Style

Ali, Rifaqat, Fatemah Mofarreh, Nadia Alluhaibi, Akram Ali, and Iqbal Ahmad. 2020. "On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms" Mathematics 8, no. 2: 150. https://doi.org/10.3390/math8020150

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