scholarly journals Second order parallel tensor in real and complex space forms

1989 ◽  
Vol 12 (4) ◽  
pp. 787-790 ◽  
Author(s):  
Ramesh Sharma
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Killing Vector ◽  
Real Space ◽  
Second Order ◽  
Complex Space Form ◽  
Metric Tensor ◽  
Space Forms ◽  
Lévy’S Theorem ◽  
Order Parallel

Levy's theorem ‘A second order parallel symmetric non-singular tensor in a real space form is proportional to the metric tensor’ has been generalized by showing that it holds even if one assumes the second order tensor to be parallel (not necessarily symmetric and non-singular) in a real space form of dimension greater than two. Analogous result has been established for a complex space form.It has been shown that an affine Killing vector field in a non-flat complex space form is Killing and analytic.

Explicit construction of Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c)

2002 ◽  
Vol 132 (3) ◽  
pp. 481-508 ◽  
Author(s):  
YUN MYUNG OH
Keyword(s):  
Sectional Curvature ◽  
Isometric Immersion ◽  
Complex Space ◽  
Space Form ◽  
Real Space ◽  
Constant Sectional Curvature ◽  
Complex Space Form ◽  
Product Decomposition ◽  
Twisted Product ◽  
Space Forms

In [4], it is proved that there exists a ‘unique’ adapted Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c) of constant sectional curvature 4c associated with each twisted product decomposition of a real-space-form if its twistor form is twisted closed. Conversely, if L: Mn(c) → M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(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 immersion L is determined by the corresponding adapted Lagrangian isometric immersion of the twisted product decomposition. It is natural to ask the explicit expressions of adapted Lagrangian isometric immersions of twisted product decompositions of real-space-forms Mn(c) into complex-space-forms M˜n(4c) for each case: c = 0, c > 0 and c < 0.

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 Quasi-Einstein Hypersurfaces of Complex Space Forms

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Xiaomin Chen ◽  
Xuehui Cui
Keyword(s):  
Real Hypersurface ◽  
Complex Space ◽  
Space Form ◽  
Ricci Soliton ◽  
Complex Space Form ◽  
Einstein Metric ◽  
Space Forms ◽  
The Real ◽  
Complex Euclidean Space ◽  
Complex Space Forms

Based on a well-known fact that there are no Einstein hypersurfaces in a nonflat complex space form, in this article, we study the quasi-Einstein condition, which is a generalization of an Einstein metric, on the real hypersurface of a nonflat complex space form. For the real hypersurface with quasi-Einstein metric of a complex Euclidean space, we also give a classification. Since a gradient Ricci soliton is a special quasi-Einstein metric, our results improve some conclusions of Cho and Kimura.

Remarks on η-parallel real hypersurfaces in ℂP2 and ℂH2

2020 ◽  
Vol 17 (05) ◽  
pp. 2050073
Author(s):  
Yaning Wang
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Three Dimensional ◽  
Shape Operator ◽  
Complex Space Form ◽  
Real Hypersurfaces ◽  
Geodesic Sphere ◽  
Principal Curvatures ◽  
Space Forms ◽  
Complex Dimension

Let [Formula: see text] be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that [Formula: see text] is [Formula: see text]-parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in [Formula: see text] or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in [Formula: see text]. Moreover, [Formula: see text]-parallel real hypersurfaces in [Formula: see text] and [Formula: see text] under some other conditions are classified and these results extend Suh’s in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo’s in [On characterizations of real hypersurfaces in a complex space form with [Formula: see text]-parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126].

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 On some inequalities for submanifolds of Bochner-Kaehler manifolds

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5511-5523
Author(s):  
Mehraj Lone ◽  
Mohammed Jamali ◽  
Mohammad Shahid
Keyword(s):  
Mean Curvature ◽  
Complex Space ◽  
Space Form ◽  
Real Space ◽  
Complex Space Form ◽  
Warping Function ◽  
Kaehler Manifold ◽  
Kaehler Manifolds

Chen established sharp inequalities between certain Riemannian invariants and the squared norm of mean curvature for submanifolds in real space form as well as in complex space form. In this paper we generalize Chen inequalities for submanifolds of Bochner-Kaehler manifolds. Moreover, we study CRwarped product submanifolds of Bochner-Kaehler manifold and establish an inequality for the Laplacian of the warping function, from which we conclude some obstructions to the existence of such immersions.

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).

An exotic totally real minimal immersion of S3 in ℂP3 and its characterisation

1996 ◽  
Vol 126 (1) ◽  
pp. 153-165 ◽  
Author(s):  
B.-Y. Chen ◽  
F. Dillen ◽  
L. Verstraelen ◽  
L. Vrancken
Keyword(s):  
Riemannian Manifold ◽  
Complex Space ◽  
Space Form ◽  
Three Dimensional ◽  
Minimal Immersion ◽  
Real Space ◽  
Sharp Inequality ◽  
Complex Space Form ◽  
Curvature Function ◽  
Totally Real

In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.

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