scholarly journals ON CONFORMAL AND QUASI-CONFORMAL CURVATURE TENSORS OF AN N(κ)-QUASI EINSTEIN MANIFOLD

2012 ◽  
Vol 27 (2) ◽  
pp. 317-326 ◽  
Author(s):  
Aliakbar Hosseinzadeh ◽  
Abolfazl Taleshian
Keyword(s):  
Einstein Manifold ◽  
Conformal Curvature ◽  
Curvature Tensors

Conformai Curvature Tensors

2011 ◽  
Author(s):  
Charles Fefferman ◽  
C. Robin Graham
Keyword(s):  
Normal Form ◽  
Curvature Tensor ◽  
Riemannian Metric ◽  
Conformal Group ◽  
Conformal Change ◽  
Conformal Curvature ◽  
Covariant Derivatives ◽  
Curvature Tensors ◽  
Derivatives Of

This chapter studies conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on tensors. It is assumed throughout this chapter that n ≥ 3.

scholarly journals Conformal curvature tensors in a generalized Riemannian space in Eisenhart sense

2020 ◽  
pp. 34-34
Author(s):  
Ana Velimirovic
Keyword(s):  
Curvature Tensor ◽  
Riemannian Space ◽  
Conformal Transformation ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Generalized Riemannian Space ◽  
Curvature Tensors ◽  
Special Case

In the present paper generalizations of conformal curvature tensor from Riemannian space are given for five independent curvature tensors in generalized Riemannian space (GRN ), i.e. when the basic tensor is non-symmetric. In earlier works of S. Mincic and M. Zlatanovic et al a special case has been investigated, that is the case when in the conformal transformation the torsion remains invariant (equitorsion transformation). In the present paper this condition is not supposed and for that reason the results are more general and new.

scholarly journals Some curvature tensors in N(k)-contact metric manifold

BIBECHANA ◽  
2018 ◽  
Vol 16 ◽  
pp. 55-63
Author(s):  
Riddhi Jung Shah
Keyword(s):  
Ricci Tensor ◽  
Einstein Manifold ◽  
Contact Metric Manifold ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Curvature Tensors

The purpose of this paper is to study W7and W9-curvature tensors on N(k)-contact metric manifolds. We prove that a N(k)-contact metric manifold satisfying the condition W7( xi,X).W9=0 is eta-Einstein manifold. We also obtain the Ricci tensor S of type (0, 2) for phi-W9flat and divW9=0 conditions on N(k)-contact metric manifolds. Finally, we give an example of 3-dimensional N(k)-contact metric manifold.BIBECHANA 16 (2019) 55-63

The Ambient Metric (AM-178)

2011 ◽  
Author(s):  
Charles Fefferman ◽  
C. Robin Graham
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Conformal Geometry ◽  
Four Dimensions ◽  
Conformal Curvature ◽  
Special Cases ◽  
Einstein Spaces ◽  
Explicit Analysis ◽  
Curvature Tensors ◽  
Formal Power

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric in n+1 dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics. The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.

PSEUDO Z SYMMETRIC RIEMANNIAN MANIFOLDS WITH HARMONIC CURVATURE TENSORS

2012 ◽  
Vol 09 (01) ◽  
pp. 1250004 ◽  
Author(s):  
CARLO ALBERTO MANTICA ◽  
YOUNG JIN SUH
Keyword(s):  
Symmetric Spaces ◽  
General Notion ◽  
Dimensional Manifold ◽  
Tensor Fields ◽  
Harmonic Curvature ◽  
Symmetric Manifold ◽  
Conformal Curvature ◽  
New Class ◽  
Conformal Curvature Tensor ◽  
Curvature Tensors

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)4 spacetime manifolds.

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.

Solitons of Kählerian space-time manifolds

2020 ◽  
pp. 2150021
Author(s):  
M. M. Praveena ◽  
C. S. Bagewadi ◽  
M. R. Krishnamurthy
Keyword(s):  
Energy Density ◽  
Cosmological Constant ◽  
Gravitational Constant ◽  
Space Time ◽  
Constant Energy ◽  
Conformal Curvature ◽  
Isotropic Pressure ◽  
Constant Energy Density ◽  
Curvature Tensors

We study solitons of almost pseudo symmetric Kählerian space-time manifold. It is considered that different curvature tensors like projective, conharmonic and conformal curvature tensors in almost pseudo symmetric Kählerian space-time manifolds are flat. It is shown that solitons are steady, expanding or shrinking under different relations of isotropic pressure, the cosmological constant, energy density and gravitational constant..

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