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.

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.

scholarly journals A fundamental theorem for submanifolds of multiproducts of real space forms

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Marie-Amélie Lawn ◽  
Julien Roth
Keyword(s):  
Minimal Surfaces ◽  
Fundamental Theorem ◽  
Arbitrary Number ◽  
Real Space ◽  
Isometric Immersions ◽  
Space Forms ◽  
Real Space Forms ◽  
Simply Connected ◽  
Bonnet Theorem

AbstractWe prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then we prove the existence of associate families of minimal surfaces in such products. Finally, in the case of 𝕊

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).

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