scholarly journals Maslovian Lagrangian immersions of real space forms into complex space forms

2004 ◽  
Vol 30 (2) ◽  
pp. 227-281 ◽  
Author(s):  
Bang-Yen CHEN ◽  
Oscar J. GARAY
Keyword(s):  
Complex Space ◽  
Real Space ◽  
Space Forms ◽  
Real Space Forms ◽  
Complex Space Forms

Isotropic Immersions with Low Codimension of Complex Space Forms into Real Space Forms

2004 ◽  
Vol 47 (4) ◽  
pp. 492-503
Author(s):  
Nobutaka Boumuki
Keyword(s):  
Complex Space ◽  
Real Space ◽  
Space Forms ◽  
Real Space Forms ◽  
Complex Space Forms

AbstractThe main purpose of this paper is to determine isotropic immersions of complex space forms into real space forms with low codimension. This is an improvement of a result of S. Maeda.

Indefinite rigidity of complex submanifolds and maximal surfaces

1989 ◽  
Vol 106 (3) ◽  
pp. 481-494 ◽  
Author(s):  
Kinetsu Abe ◽  
Martin A. Magid
Keyword(s):  
Minimal Surfaces ◽  
Complex Space ◽  
Real Space ◽  
Rigidity Theorem ◽  
Space Forms ◽  
Maximal Surfaces ◽  
Real Space Forms ◽  
Complex Submanifolds ◽  
Complex Space Forms

In 1953, Calabi proved a rigidity theorem for Kählerian submanifolds in complex space forms [3]. The Calabi rigidity theorem, since then, has been successfully applied to various areas in geometry. Among them is the study of minimal surfaces in real space forms; see [4] for example.

Horizontal Newton operators and high-order Minkowski formula

2020 ◽  
pp. 2050004
Author(s):  
Chiara Guidi ◽  
Vittorio Martino
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Real Space ◽  
Second Fundamental Form ◽  
Space Forms ◽  
Nonlinear Operators ◽  
The Second Fundamental Form ◽  
Real Space Forms ◽  
Newton Transformations ◽  
Minkowski Formula

In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of nonlinearity and we will show how a Minkowski-type formula can be obtained in this case.

scholarly journals Parallel submanifolds of complex space forms I

1983 ◽  
Vol 90 ◽  
pp. 85-117 ◽  
Author(s):  
Hiroo Naitoh
Keyword(s):  
Symmetric Space ◽  
Sectional Curvature ◽  
Complex Space ◽  
Space Form ◽  
Real Space ◽  
Hermitian Symmetric Space ◽  
Holomorphic Sectional Curvature ◽  
Space Forms ◽  
Simply Connected ◽  
Complex Space Forms

Complete parallel submanifolds of a real space form of constant sectional curvature k have been completely classified by Ferus [3] when k ≧ 0, and by Takeuchi [19] when k < 0. A complex space form is by definition a 2n-dimensional simply connected Hermitian symmetric space of constant holomorphic sectional curvature c and will be denoted by (c).

Lagrangian isometric immersions of a real-space-form Mn(c) into a complex-space-form M˜n(4c)

1998 ◽  
Vol 124 (1) ◽  
pp. 107-125 ◽  
Author(s):  
B.-Y. CHEN ◽  
F. DILLEN ◽  
L. VERSTRAELEN ◽  
L. VRANCKEN
Keyword(s):  
Sectional Curvature ◽  
Complex Space ◽  
Space Form ◽  
Real Space ◽  
Constant Sectional Curvature ◽  
Complex Space Form ◽  
Product Decomposition ◽  
Twisted Product ◽  
Space Forms ◽  
Real Space Forms

It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form M˜n(4c) of constant holomorphic sectional curvature 4c are real-space-forms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form M˜n(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1I1×… ×fkIk×1Nn−k(c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-space-form M˜n(4c). Conversely, if L: Mn(c)→ M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.

Sign in / Sign up

Export Citation Format

Share Document