scholarly journals THE FIRST EIGENVALUE OF THE TOTALLY REAL SUBMANIFOLDS WITH CONSTANT MEAN CURVATURE IN THE COMPLEX SPACE FORM

1993 ◽  
Vol 47 (2) ◽  
pp. 411-419
Author(s):  
Sun HUAFEI
Keyword(s):  
Mean Curvature ◽  
Complex Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Complex Space Form ◽  
First Eigenvalue ◽  
Real Submanifolds ◽  
The First Eigenvalue ◽  
Totally Real

scholarly journals Totally real submanifolds of a complex space form

1996 ◽  
Vol 19 (1) ◽  
pp. 39-44 ◽  
Author(s):  
U-Hang Ki ◽  
Young Ho Kim
Keyword(s):  
Mean Curvature ◽  
Complex Space ◽  
Space Form ◽  
Curvature Vector ◽  
Complex Space Form ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Real Submanifolds ◽  
Parallel Mean Curvature ◽  
Totally Real

Totally real submanifolds of a complex space form are studied. In particular, totally real submanifolds of a complex number space with parallel mean curvature vector are classified.

Totally real submanifolds in a generalized complex space form

2016 ◽  
Vol 10 (02) ◽  
pp. 1750035
Author(s):  
Majid Ali Choudhary
Keyword(s):  
Fundamental Form ◽  
Complex Space ◽  
Space Form ◽  
Second Fundamental Form ◽  
Text Structure ◽  
Complex Space Form ◽  
Real Submanifolds ◽  
Constant Length ◽  
Generalized Complex ◽  
Totally Real

In the present paper, we investigate totally real submanifolds in generalized complex space form. We study the [Formula: see text]-structure in the normal bundle of a totally real submanifold and derive some integral formulas computing the Laplacian of the square of the second fundamental form and using these formulas, we prove a pinching theorem. In fact, the purpose of this note is to generalize results proved in B. Y. Chen and K. Ogiue, On totally real manifolds, Trans. Amer. Math. Soc. 193 (1974) 257–266, S. S. Chern, M. Do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional Analysis and Related Fields (Springer-Verlag, 1970), pp. 57–75 to the case, when the ambient manifold is generalized complex space form.

scholarly journals A class of C-totally real submanifolds of Sasakian space forms

1998 ◽  
Vol 65 (1) ◽  
pp. 120-128
Author(s):  
Filip Defever ◽  
Ion Mihai ◽  
Leopold Verstraelen
Keyword(s):  
Riemannian Manifold ◽  
Mean Curvature ◽  
Space Form ◽  
Sasakian Space Form ◽  
Real Submanifolds ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Totally Real ◽  
Real Complex

AbstractRecently, Chen defined an invariant δM of a Riemannian manifold M. Sharp inequalities for this Riemannian invariant were obtained for submanifolds in real, complex and Sasakian space forms, in terms of their mean curvature. In the present paper, we investigate certain C-totally real submanifolds of a Sasakian space form M2m+1(C)satisfying Chen's equality.

scholarly journals Chen's problem on mixed foliate CR-submanifolds

1989 ◽  
Vol 40 (1) ◽  
pp. 157-160 ◽  
Author(s):  
Mohammed Ali Bashir
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Complex Submanifold ◽  
Hyperbolic Complex ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Simply Connected ◽  
Totally Real ◽  
Cr Submanifolds

We prove that the simply connected compact mixed foliate CR-submanifold in a hyperbolic complex space form is either a complex submanifold or a totally real submanifold. This is the problem posed by Chen.

scholarly journals THE FIRST EIGENVALUE OF THE DIRAC OPERATOR IN A CONFORMAL CLASS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 833-844 ◽  
Author(s):  
BERND AMMANN ◽  
EMMANUEL HUMBERT
Keyword(s):  
Dirac Operator ◽  
Variational Problem ◽  
Mean Curvature ◽  
Unit Volume ◽  
Constant Mean Curvature ◽  
Positive Eigenvalue ◽  
First Eigenvalue ◽  
Conformal Class ◽  
Open Problems ◽  
The First Eigenvalue

In this overview article, we study the first positive eigenvalue of the Dirac operator in a unit volume conformal class. In particular, we discuss the question whether the infimum is attained. In the first part, we explain the corresponding variational problem. In the following parts we discuss the relation to the spinorial mass endomorphism and an application to surfaces of constant mean curvature. The article also mentions some open problems and work in progress.

scholarly journals C-totally real submanifolds in (κ,μ)-contact space forms

2003 ◽  
Vol 67 (1) ◽  
pp. 51-65 ◽  
Author(s):  
Mukut Mani Tripathi ◽  
Jeong-Sik Kim
Keyword(s):  
Ricci Curvature ◽  
Mean Curvature ◽  
Space Form ◽  
Real Submanifolds ◽  
Space Forms ◽  
Left Hand ◽  
Real Submanifold ◽  
The Right ◽  
Totally Real ◽  
Contact Space

We obtain a basic B,-Y. Chen's inequality for a C-totally real submanifold in a (κ,μ)-contact space form involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Inequalities between the squared mean curvature and Ricci curvature and between the squared mean curvature and κ-Ricci curvature are also obtained. These results are applied to get corresponding results for C-totally real submanifolds in a Sasakian space form.

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