Jacobi's elliptic functions and Lagrangian immersions

1996 ◽  
Vol 126 (4) ◽  
pp. 687-704 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Complex Space ◽  
Lagrangian Submanifolds ◽  
Elliptic Functions ◽  
Complete Classification ◽  
Sharp Inequality ◽  
Space Forms ◽  
Nonflat Complex Space Forms ◽  
Basic Equality ◽  
Complex Space Forms

First, we establish a sharp inequality between the squared mean curvature and the scalar curvature for a Lagrangian submanifold in a nonflat complex-space-form. Then, by utilising the Jacobi's elliptic functions en and dn, we introduce three families of Lagrangian submanifolds and two exceptional Lagrangian submanifolds Fn, Ln in nonflat complex-space-forms which satisfy the equality case of the inequality. Finally, we obtain the complete classification of Lagrangian submanifolds in nonflat complex-space-forms which satisfy this basic equality.

scholarly journals Existence and uniqueness of inhomogeneous ruled hypersurfaces with shape operator of constant norm in the complex hyperbolic space

2021 ◽  
pp. 2150049
Author(s):  
Miguel Domínguez-Vázquez ◽  
Olga Pérez-Barral
Keyword(s):  
Hyperbolic Space ◽  
Complex Space ◽  
Shape Operator ◽  
Complex Hyperbolic Space ◽  
Real Hypersurfaces ◽  
Space Forms ◽  
Nonflat Complex Space Forms ◽  
Ruled Real Hypersurfaces ◽  
Complex Space Forms

We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.

scholarly journals Classification of δ(2,n− 2)-ideal Lagrangian submanifolds in n-dimensional complex space forms

2018 ◽  
Vol 458 (2) ◽  
pp. 1456-1485 ◽  
Author(s):  
Bang-Yen Chen ◽  
Franki Dillen ◽  
Joeri Van der Veken ◽  
Luc Vrancken
Keyword(s):  
Complex Space ◽  
Lagrangian Submanifolds ◽  
Space Forms ◽  
Complex Space Forms

COMPLETE CLASSIFICATION OF PARALLEL LORENTZIAN SURFACES IN LORENTZIAN COMPLEX SPACE FORMS

2010 ◽  
Vol 21 (05) ◽  
pp. 665-686 ◽  
Author(s):  
BANG-YEN CHEN ◽  
FRANKI DILLEN ◽  
JOERI VAN DER VEKEN
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Complex Space ◽  
Complete Classification ◽  
Second Fundamental Form ◽  
Space Forms ◽  
Complex Dimension ◽  
Point To Point ◽  
Complex Space Forms

A surface of a pseudo-Riemannian manifold is called parallel if its second fundamental form is parallel with respect to the Van der Waerden–Bortolotti connection. Such surfaces are fundamental since the extrinsic invariants of the surfaces do no change from point to point. In this article, we completely classify parallel Lorentzian surfaces in Lorentzian complex space forms of complex dimension two.

Classification of Casorati ideal Lagrangian submanifolds in complex space forms

2019 ◽  
Vol 63 ◽  
pp. 30-49 ◽  
Author(s):  
Mohd. Aquib ◽  
Jae Won Lee ◽  
Gabriel-Eduard Vîlcu ◽  
Dae Won Yoon
Keyword(s):  
Complex Space ◽  
Lagrangian Submanifolds ◽  
Space Forms ◽  
Complex Space Forms

Some new results on real hypersurfaces with generalized Tanaka-Webster connection

2019 ◽  
Vol 69 (3) ◽  
pp. 665-674
Author(s):  
Wenjie Wang ◽  
Ximin Liu
Keyword(s):  
Real Hypersurface ◽  
Complex Space ◽  
Shape Operator ◽  
Real Hypersurfaces ◽  
Space Forms ◽  
Complex Dimension ◽  
Nonflat Complex Space Forms ◽  
Complex Space Forms ◽  
Ruled Real Hypersurface

Abstract Let M be a real hypersurface in nonflat complex space forms of complex dimension two. In this paper, we prove that the shape operator of M is transversally Killing with respect to the generalized Tanaka-Webster connection if and only if M is locally congruent to a type (A) or (B) real hypersurface. We also prove that shape operator of M commutes with Cho operator on holomorphic distribution if and only if M is locally congruent to a ruled real hypersurface.

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