A Möbius characterization of submanifolds in real space forms with parallel mean curvature and constant scalar curvature

2005 ◽  
Vol 117 (2) ◽  
pp. 135-152 ◽  
Author(s):  
Xingxiao Li ◽  
Fengyun Zhang
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Scalar Curvature ◽  
Real Space ◽  
Space Forms ◽  
Parallel Mean Curvature ◽  
Real Space Forms

scholarly journals Affine connections of non-integrable distributions

2020 ◽  
Vol 17 (08) ◽  
pp. 2050127
Author(s):  
Yong Wang
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Real Space ◽  
Space Forms ◽  
Affine Connections ◽  
Real Space Forms ◽  
Metric Connection ◽  
Statistical Connection

In this paper, we study non-integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a kind of semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non-integrable distributions with respect to the semi-symmetric metric connection, the semi-symmetric non-metric connection and the statistical connection. As applications, we obtain Chen’s inequalities for non-integrable distributions of real space forms endowed with a semi-symmetric metric connection and a kind of semi-symmetric non-metric connection. We give some examples of non-integrable distributions in a Riemannian manifold with affine connections. We find some new examples of Einstein distributions and distributions with constant scalar curvature.

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

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

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.

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.

scholarly journals Conformal Geometry of Hypersurfaces in Lorentz Space Forms

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Tongzhu Li ◽  
Changxiong Nie
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Lorentz Space ◽  
Constant Mean Curvature ◽  
Conformal Geometry ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Lagrange Equations ◽  
Conformal Metric ◽  
Space Forms

Let be a space-like hypersurface without umbilical points in the Lorentz space form . We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation of . We calculate the Euler-Lagrange equations of the volume functional of the hypersurface with respect to the conformal metric, whose critical point is called a Willmore hypersurface, and we give a conformal characteristic of the hypersurfaces with constant mean curvature and constant scalar curvature. Finally, we prove that if the hypersurface with constant mean curvature and constant scalar curvature is Willmore, then is a hypersurface in .

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