scholarly journals A NEW RESULT ABOUT ALMOST UMBILICAL HYPERSURFACES OF REAL SPACE FORMS

2014 ◽  
Vol 91 (1) ◽  
pp. 145-154 ◽  
Author(s):  
JULIEN ROTH
Keyword(s):  
Space Form ◽  
Short Note ◽  
Real Space ◽  
Round Sphere ◽  
Space Forms ◽  
Compact Hypersurface ◽  
Geodesic Spheres ◽  
Real Space Forms

AbstractIn this short note, we prove that an almost umbilical compact hypersurface of a real space form with almost Codazzi umbilicity tensor is embedded, diffeomorphic and quasi-isometric to a round sphere. Then, we derive a new characterisation of geodesic spheres in space forms.

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.

GOLDEN- AND PRODUCT-SHAPED HYPERSURFACES IN REAL SPACE FORMS

2013 ◽  
Vol 10 (04) ◽  
pp. 1320006 ◽  
Author(s):  
MIRCEA CRASMAREANU ◽  
CRISTINA-ELENA HREŢCANU ◽  
MARIAN-IOAN MUNTEANU
Keyword(s):  
Hyperbolic Space ◽  
Space Form ◽  
Shape Operator ◽  
Real Space ◽  
Clifford Torus ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Real Space Forms ◽  
Unit Hypersphere

We define two classes of hypersurfaces in real space forms, golden- and product-shaped, respectively, by imposing the shape operator to be of golden or product type. We obtain the whole families of above hypersurfaces, based on the classification of isoparametric hypersurfaces, as follows: in the golden case all are hyperspheres, a hyperbolic space and a generalized Clifford torus, while for the product case we obtain the unit hypersphere, the hyperplane, a hypersphere and its associated Clifford torus, respectively, according to the type of the ambient space form namely parabolic, hyperbolic or elliptic, respectively.

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.

scholarly journals A characterisation of Helices and Cornu spirals in real space forms

1997 ◽  
Vol 56 (1) ◽  
pp. 37-49 ◽  
Author(s):  
J. Arroyo ◽  
M. Barros ◽  
O.J. Garay
Keyword(s):  
Mean Curvature ◽  
Space Form ◽  
Arbitrary Dimension ◽  
Curvature Vector ◽  
Real Space ◽  
Curvature Function ◽  
Mean Curvature Vector ◽  
Space Forms ◽  
Real Space Forms ◽  
Unit Speed

We classify unit speed curves contained in a real space form of arbitrary dimension Nm(c), whose mean curvature vector is proper for the Laplacian. Then we use these results to classify Hopf cylinders of S3 and semi-Riemannian Hopf cylinders of with proper mean curvature function.

scholarly journals A Class of Special Hypersurfaces in Real Space Forms

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Yan Zhao ◽  
Ximin Liu
Keyword(s):  
Real Space ◽  
Principal Curvatures ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Real Space Forms ◽  
Constant Principal Curvatures

We define the generalized golden- and product-shaped hypersurfaces in real space forms. A hypersurfaceMin real space formsRn+1,Sn+1, andHn+1is isoparametric if it has constant principal curvatures. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the generalized golden- and product-shaped hypersurfaces in real space forms.

scholarly journals A new characterization of complete linear Weingarten hypersurfaces in real space forms

2013 ◽  
Vol 261 (1) ◽  
pp. 33-43 ◽  
Author(s):  
Cícero Aquino ◽  
Henrique de Lima ◽  
Marco Velásquez
Keyword(s):  
Real Space ◽  
Space Forms ◽  
Linear Weingarten Hypersurfaces ◽  
Real Space Forms
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