scholarly journals On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 150 ◽  
Author(s):  
Rifaqat Ali ◽  
Fatemah Mofarreh ◽  
Nadia Alluhaibi ◽  
Akram Ali ◽  
Iqbal Ahmad
Keyword(s):  
Type Inequality ◽  
Ambient Space ◽  
Holomorphic Sectional Curvature ◽  
First Eigenvalue ◽  
Space Forms ◽  
Standard Sphere ◽  
Sasakian Space Forms ◽  
Legendrian Submanifold ◽  
The Laplace Operator ◽  
Legendrian Submanifolds

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 ) .

scholarly journals Characterization of Lagrangian Submanifolds by Geometric Inequalities in Complex Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Lamia Saeed Alqahtani
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Lagrangian Submanifolds ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Complex Space Form ◽  
First Eigenvalue ◽  
Space Forms ◽  
Standard Sphere ◽  
The Laplace Operator

In this paper, we give an estimate of the first eigenvalue of the Laplace operator on a Lagrangian submanifold M n minimally immersed in a complex space form. We provide sufficient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere S n . We also obtain Simons-type inequality for same ambient space form.

scholarly journals Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1209-1215
Author(s):  
Aleksandar Sebekovic ◽  
Miroslava Petrovic-Torgasev ◽  
Anica Pantic
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Space Forms ◽  
Equality Case ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Sasakian Space Forms ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds ◽  
Weyl Conformal Curvature Tensor

For Legendrian submanifolds Mn in Sasakian space forms ?M2n+1(c), I. Mihai obtained an inequality relating the normalised scalar curvature (intrinsic invariant) and the squared mean curvature and the normalised scalar normal curvature of M in the ambient space ?M (extrinsic invariants) which is called the generalised Wintgen inequality, characterising also the corresponding equality case. And a Legendrian submanifold Mn in Sasakian space forms ?M2n+1(c) is said to be generalised Wintgen ideal Legendrian submanifold of ?M2n+1(c) when it realises at everyone of its points the equality in such inequality. Characterisations based on some basic intrinsic symmetries involving the Riemann-Cristoffel curvature tensor, the Ricci tensor and the Weyl conformal curvature tensor belonging to the class of pseudosymmetries in the sense of Deszcz of such generalised Wintgen ideal Legendrian submanifolds are given.

Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms

2008 ◽  
Vol 51 (3) ◽  
pp. 448-459 ◽  
Author(s):  
Toru Sasahara
Keyword(s):  
Critical Points ◽  
Harmonic Map ◽  
Biharmonic Maps ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
The Stability ◽  
Biharmonic Map ◽  
Legendrian Submanifolds

AbstractBiharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed.

Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms

2020 ◽  
Vol 155 ◽  
pp. 103768 ◽  
Author(s):  
Jae Won Lee ◽  
Chul Woo Lee ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Space Forms ◽  
Sasakian Space Forms ◽  
Legendrian Submanifolds

scholarly journals The First Fundamental Equation and Generalized Wintgen-Type Inequalities for Submanifolds in Generalized Space Forms

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1151 ◽  
Author(s):  
Mohd. Aquib ◽  
Michel Nguiffo Boyom ◽  
Mohammad Hasan Shahid ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Fundamental Equation ◽  
Type Inequality ◽  
Lagrangian Submanifold ◽  
Space Forms ◽  
Slant Submanifolds ◽  
Sasakian Space Forms ◽  
Generalized Complex ◽  
Complex Space Forms

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces.

scholarly journals Generalized Wintgen inequality for submanifolds in Kenmotsu space forms

2018 ◽  
Vol 50 (2) ◽  
pp. 155-164 ◽  
Author(s):  
Mohd Aquib ◽  
Mohammad Hasan Shahid
Keyword(s):  
Ambient Space ◽  
Slant Submanifold ◽  
Space Forms ◽  
Equality Case ◽  
Legendrian Submanifolds

In this paper, we obtain the generalized Wintgen inequality for Legendrian submanifolds in Kenmotsu space forms and discuss the equality case of the inequality. Further, we discuss the inequality for bi-slant submanifold in the same ambient space and derive its application in various slant cases.

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