scholarly journals Delaunay surfaces in S3(ρ)

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1191-1200 ◽  
Author(s):  
J. Arroyo ◽  
O.J. Garay ◽  
A. Pámpano
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Energy Functional ◽  
Real Space ◽  
Space Forms ◽  
New Approach ◽  
Cmc Surfaces ◽  
Real Space Forms ◽  
Rotational Surfaces

Recently, invariant constant mean curvature (CMC) surfaces in real space forms have been characterized locally by using extremal curves of a Blaschke type energy functional [5]. Here, we use this characterization to offer a new approach to some global results for CMC rotational surfaces in the 3-sphere.

scholarly journals Three-dimensional CR-submanifolds in the nearly Kaehler six-sphere satisfying B.Y. Chen's basic equality

2000 ◽  
Vol 31 (4) ◽  
pp. 289-296
Author(s):  
Tooru Sasahara
Keyword(s):  
Mean Curvature ◽  
Three Dimensional ◽  
Real Space ◽  
Sharp Inequality ◽  
Space Forms ◽  
Equality Case ◽  
3 Dimensional ◽  
Real Space Forms ◽  
Basic Equality ◽  
Cr Submanifolds

B. Y. Chen introduced in [3] an important Riemannian invariant for a Riemannian manifold and obtained a sharp inequality between his invariant and the squared mean curvature for arbitrary submanifolds in real space forms. In this paper we investigate 3-dimensional CR-submanifolds in the nearly Kaehler 6-sphere which realize the equality case of the inequality.

Mean curvature of extrinsic spheres in submanifolds of real space forms

2004 ◽  
Vol 83 (4) ◽  
pp. 371-380
Author(s):  
Vicente Palmer ◽  
Mayte Pi�ero
Keyword(s):  
Mean Curvature ◽  
Real Space ◽  
Space Forms ◽  
Real Space Forms

Surfaces of Rotation with Constant Mean Curvature in the Direction of a Unitary Normal Vector Field in a Randers Space

2012 ◽  
Vol 64 (1) ◽  
pp. 44-80 ◽  
Author(s):  
T. M. M. Carvalho ◽  
H. N. Moreira ◽  
K. Tenenblat
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Euclidean Metric ◽  
Cmc Surfaces ◽  
Constant Solution ◽  
Rotational Surfaces

AbstractWe consider the Randers space (Vn, Fb) obtained by perturbing the Euclidean metric by a translation, Fb = α + β, where α is the Euclidean metric and β is a 1-form with norm b, 0 ≤ b < 1. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces (V3, Fb) of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when b = 0. It also reduces to the equation that characterizes the minimal rotational surfaces in (V3, Fb) when H = 0, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for . Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical systemand by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.

Chen Inequalities for Submanifolds of Real Space Forms with a Semi-Symmetric Non-Metric Connection

2012 ◽  
Vol 55 (3) ◽  
pp. 611-622 ◽  
Author(s):  
Cihan Özgür ◽  
Adela Mihai
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Real Space ◽  
Ambient Space ◽  
Space Forms ◽  
Real Space Forms ◽  
Metric Connection ◽  
Sectional Curvatures ◽  
The Mean

AbstractIn this paper we prove Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric non-metric connection, i.e., relations between the mean curvature associated with a semi-symmetric non-metric connection, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered.

scholarly journals A Riemannian invariant and its applications to Einstein manifolds

2004 ◽  
Vol 70 (1) ◽  
pp. 55-65 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Mean Curvature ◽  
Real Space ◽  
Einstein Manifolds ◽  
Space Forms ◽  
Real Space Forms ◽  
Optimal Inequalities

We introduce a Riemannian invariant and establish general optimal inequalities involving the invariants and the squared mean curvature for Einstein manifolds isometrically immersed in real space forms. We show that these inequalities do not hold for arbitrary submanifolds in real space forms in general. We also provide some immediate applications of the inequalities.

scholarly journals Mean curvature and shape operator of isometric immersions in real-space-forms

1996 ◽  
Vol 38 (1) ◽  
pp. 87-97 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Mean Curvature ◽  
Shape Operator ◽  
Real Space ◽  
Curvature Function ◽  
Isometric Immersions ◽  
Space Forms ◽  
Real Space Forms ◽  
The Mean

According to the well-known Nash's theorem, every Riemannian n-manifold admits an isometric immersion into the Euclidean space En(n+1)(3n+11)/2. In general, there exist enormously many isometric immersions from a Riemannian manifold into Euclidean spaces if no restriction on the codimension is made. For a submanifold of a Riemannian manifold there are associated several extrinsic invariants beside its intrinsic invariants. Among the extrinsic invariants, the mean curvature function and shape operator are the most fundamental ones.

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.

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