scholarly journals On generalized Quasi Einstein manifolds

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 811-820 ◽  
Author(s):  
Avik De ◽  
Ahmet Yildiz ◽  
Uday De
Keyword(s):  
Einstein Manifold ◽  
Natural Generalization ◽  
Einstein Manifolds ◽  
Geometric Properties

Quasi Einstein manifold is a simple and natural generalization of an Einstein manifold. The object of the present paper is to study some geometric properties of generalized quasi Einstein manifolds. Two non-trivial examples have been constructed to prove the existence of a generalized quasi Einstein manifold.

scholarly journals On generalized quasi-Einstein manifolds admitting certain vector fields

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 599-609 ◽  
Author(s):  
Uday De ◽  
Sahanous Mallick
Keyword(s):  
Vector Fields ◽  
Einstein Manifold ◽  
Einstein Manifolds ◽  
Geometric Properties

The object of the present paper is to study some geometric properties of a generalized quasi-Einstein manifold. The existence of such a manifold have been proved by several non-trivial examples.

scholarly journals Some Results on Generalized Quasi-Einstein Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
D. G. Prakasha ◽  
H. Venkatesha
Keyword(s):  
Curvature Tensor ◽  
Einstein Manifold ◽  
Einstein Manifolds ◽  
Geometric Properties ◽  
Conharmonic Curvature Tensor

This paper deals with generalized quasi-Einstein manifold satisfying certain conditions on conharmonic curvature tensor. Here we study some geometric properties of generalized quasi-Einstein manifold and obtain results which reveal the nature of its associated 1-forms.

scholarly journals Stability of Riemannian manifolds with Killing spinors

2017 ◽  
Vol 28 (01) ◽  
pp. 1750005 ◽  
Author(s):  
Changliang Wang
Keyword(s):  
Riemannian Manifolds ◽  
Einstein Manifold ◽  
Stability Result ◽  
Warped Products ◽  
Einstein Manifolds ◽  
Killing Spinors ◽  
Parallel Spinors ◽  
Unstable Manifolds ◽  
The Stability ◽  
Complete Riemannian Manifolds

Riemannian manifolds with nonzero Killing spinors are Einstein manifolds. Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in [Stable and unstable Einstein warped products, preprint (2015), arXiv:1507.01782v1 ]. In this paper, we obtain a new proof for this stability result by using a Bochner-type formula in [X. Dai, X. Wang and G. Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math. 161(1) (2005) 151–176; M. Wang, Preserving parallel spinors under metric deformations, Indiana Univ. Math. J. 40 (1991) 815–844]. Moreover, existence of real Killing spinors is closely related to the Sasaki–Einstein structure. A regular Sasaki–Einstein manifold is essentially the total space of a certain principal [Formula: see text]-bundle over a Kähler–Einstein manifold. We prove that if the base space is a product of two Kähler–Einstein manifolds then the regular Sasaki–Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors.

scholarly journals ON PSEUDO CYCLIC RICCI SYMMETRIC MANIFOLDS

2009 ◽  
Vol 02 (02) ◽  
pp. 227-237
Author(s):  
Absos Ali Shaikh ◽  
Shyamal Kumar Hui
Keyword(s):  
Riemannian Manifold ◽  
Einstein Manifold ◽  
Conformally Flat ◽  
Geometric Properties ◽  
Symmetric Manifold ◽  
Euclidean Manifold ◽  
Flat Riemannian Manifold

The object of the present paper is to introduce a type of non-flat Riemannian manifold called pseudo cyclic Ricci symmetric manifold and study its geometric properties. Among others it is shown that a pseudo cyclic Ricci symmetric manifold is a special type of quasi-Einstein manifold. In this paper we also study conformally flat pseudo cyclic Ricci symmetric manifolds and prove that such a manifold can be isometrically immersed in a Euclidean manifold as a hypersurface.

scholarly journals On some classes of mixed-super quasi-Einstein manifolds

2016 ◽  
Vol 8 (1) ◽  
pp. 32-52
Author(s):  
Santu Dey ◽  
Buddhadev Pal ◽  
Arindam Bhattacharyya
Keyword(s):  
Warped Product ◽  
Einstein Manifold ◽  
Warped Products ◽  
Einstein Manifolds

Abstract Quasi-Einstein manifold and generalized quasi-Einstein manifold are the generalizations of Einstein manifold. The purpose of this paper is to study the mixed super quasi-Einstein manifold which is also the generalizations of Einstein manifold satisfying some curvature conditions. We define both Riemannian and Lorentzian doubly warped product on this manifold. Finally, we study the completeness properties of doubly warped products on MS(QE)4 for both the Riemannian and Lorentzian cases.

scholarly journals Generalized η-Einstein 3-Dimensional Trans-Sasakian Manifold

2015 ◽  
Vol 2 ◽  
pp. 28-32 ◽  
Author(s):  
Barnali Laha
Keyword(s):  
Present Note ◽  
Einstein Manifold ◽  
Sasakian Manifold ◽  
Geometric Properties ◽  
3 Dimensional

In the present note we have introduced a new concept called generalized η-Einstein manifold in a 3-dimensional trans-Sasakian manifold and have given some preliminary ideas about the same. Inthe next section some geometric properties about this manifold have been deduced. Finally we havejustified the existence of such a manifold by citing an example.

scholarly journals Bochner-type Formulas for the Weyl Tensor on Four-dimensional Einstein Manifolds

2018 ◽  
Vol 2020 (12) ◽  
pp. 3794-3823
Author(s):  
Giovanni Catino ◽  
Paolo Mastrolia
Keyword(s):  
Covariant Derivative ◽  
Weyl Tensor ◽  
Einstein Manifold ◽  
Higher Order ◽  
Einstein Metric ◽  
Einstein Manifolds ◽  
Rigidity Result ◽  
Type Formula ◽  
Definition Of ◽  
Integral Identities

Abstract The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner-type formulas for the Weyl tensor on a four-dimensional Einstein manifold. In particular, we prove a 2nd Bochner-type formula that, formally, extends to the covariant derivative level the classical one for the Weyl tensor obtained by Derdziński in 1983. As a consequence, we deduce new integral identities involving the Weyl tensor and its derivatives on a compact four-dimensional Einstein manifold and we derive a new rigidity result.

On geometry of sub-Riemannian η-Einstein manifolds

2018 ◽  
pp. 68-81
Author(s):  
S. Galaev
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Einstein Manifold ◽  
Riemannian Structure ◽  
Einstein Manifolds ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
Curvature Tensors ◽  
Sasaki Manifold

On a sub-Riemannian manifold of contact type a connection  with torsion is considered, called in the work a Ψ-connection. A Ψ- connection is a particular case of an N-connection. On a sub-Riemannian manifold, a Ψ-connection is defined up to an endomorphism  :DD of a distribution D, this endomorphism is called in the work the structure endomorphism. The endomorphism ψ is uniquely defined by the following relations:  0,   (x, y)  g( x, y), x, yD. If the distribution of a sub-Riemannian manifold is integrable, then the Ψ-connection is of the class of the quarter-symmetric connections. It is proved that the Ψ- connection is a metric connection if and only if the structure vector field of the sub-Riemannian structure is integrable. A formula expressing the Ψ-connections in terms of the Levi-Civita connection of the sub- Riemannian manifold is obtained. The components of the curvature tensors and the Ricci-tensors of the Ψ-connection and of the Levi-Civita connection are computed. It is proved that if a sub-Riemannian manifold is an η-Einstein manifold, then it is also an η-Einstein manifold with respect to the Ψ-connection. The converse holds true only under the condition that the trace of the structure endomorphism Ψ is a constant not depending on a point of the manifold. The paper is completed by the theorem stating that a Sasaki manifold is an η-Einstein manifold if and only if M is an η-Einstein manifold with respect to the Ψ-connection.

scholarly journals WARPED EMBEDDINGS BETWEEN EINSTEIN MANIFOLDS

2010 ◽  
Vol 25 (18) ◽  
pp. 1521-1530 ◽  
Author(s):  
HUAN-XIONG YANG ◽  
LIU ZHAO
Keyword(s):  
Einstein Manifold ◽  
Explicit Solutions ◽  
Einstein Manifolds ◽  
Higher Dimensional ◽  
Potential Applications ◽  
Lower Dimensional

Warped embeddings from a lower dimensional Einstein manifold into a higher dimensional one are analyzed. Explicit solutions for the embedding metrics are obtained for all cases of codimension 1 embeddings and some of the codimension n>1 cases. Some of the interesting features of the embedding metrics are pointed out and potential applications of the embeddings are discussed.

A note on the circle actions on Einstein manifolds

2001 ◽  
Vol 63 (1) ◽  
pp. 83-91
Author(s):  
Seungsu Hwang
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Scalar Curvature ◽  
Einstein Manifold ◽  
Orbit Space ◽  
Circle Action ◽  
Fundamental Result ◽  
Einstein Manifolds ◽  
Circle Actions ◽  
Zero Scalar Curvature

A fundamental result in the theory of black holes due to Hawking asserts that the event horizon of a black hole in the stationary space-time is a 2-sphere topologically. In this article we prove the Riemannian analogue of Hawking's result. In other words, we prove that each bolt of a 4-dimensional complete noncompact Einstein manifold of zero scalar curvature admitting a semifree isometric circle action is a 2-sphere topologically. We also study the structure of the orbit space of an Einstein manifold admitting a free isometric circle action.

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