Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor

2010 ◽  
Vol 62 (5) ◽  
pp. 1037-1057 ◽  
Author(s):  
E. Calviño-Louzao ◽  
E. García-Río ◽  
R. Vázquez-Lorenzo
Keyword(s):  
Ricci Tensor ◽  
Curvature Operator ◽  
Locally Homogeneous ◽  
Torsion Free

AbstractCorrespondence between torsion-free connections with nilpotent skew-symmetric curvature operator and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces.

scholarly journals Real hypersurfaces of a complex projective space

1986 ◽  
Vol 33 (3) ◽  
pp. 383-387 ◽  
Author(s):  
M. Kimura
Keyword(s):  
Projective Space ◽  
Ricci Tensor ◽  
Complex Projective Space ◽  
Real Hypersurfaces ◽  
Principal Curvatures ◽  
Locally Homogeneous ◽  
Complex Projective

We study real hypersurfaces M of a complex projective space and show that a condition on the derivative of the Ricci Tensor of M implies M is locally homogeneous with two or three principal curvatures.

scholarly journals KALUZA–KLEIN THEORY WITH TORSION CONFINED TO THE EXTRA DIMENSION

2010 ◽  
Vol 25 (25) ◽  
pp. 2121-2130 ◽  
Author(s):  
KARTHIK H. SHANKAR ◽  
KAMESHWAR C. WALI
Keyword(s):  
Ricci Tensor ◽  
Vector Fields ◽  
Dynamical Variable ◽  
Einstein Equations ◽  
Cosmic Acceleration ◽  
Scalar And Vector Fields ◽  
Klein Theory ◽  
Cartan Theory ◽  
Torsion Free ◽  
Kaluza Klein

Here we consider a variant of the five-dimensional Kaluza–Klein (KK) theory within the framework of Einstein–Cartan formalism that includes torsion. By imposing a set of constraints on torsion and Ricci rotation coefficients, we show that the torsion components are completely expressed in terms of the metric. Moreover, the Ricci tensor in 5D corresponds exactly to what one would obtain from torsion-free general relativity on a 4D hypersurface. The contributions of the scalar and vector fields of the standard KK theory to the Ricci tensor and the affine connections are completely nullified by the contributions from torsion. As a consequence, geodesic motions do not distinguish the torsion free 4D spacetime from a hypersurface of 5D spacetime with torsion satisfying the constraints. Since torsion is not an independent dynamical variable in this formalism, the modified Einstein equations are different from those in the general Einstein–Cartan theory. This leads to important cosmological consequences such as the emergence of cosmic acceleration.

scholarly journals Some Geometric Properties of a Non-Strict Eight Dimensional Walker Manifold

2021 ◽  
pp. 107-119
Author(s):  
Silas Longwap ◽  
Gukat G. Bitrus ◽  
Chibuisi Chigozie
Keyword(s):  
Coordinate System ◽  
Ricci Tensor ◽  
Curvature Operator ◽  
Geometric Properties ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
The One ◽  
Walker Manifold

An 8 dimensional Walker manifold (M; g) is a strict walker manifold if we can choose a coordinate system fx1; x2; x3; x4; x5; x6; x7; x8g on (M,g) such that any function f on the manfold (M,g), f(x1; x2; x3; x4; x5; x6; x7; x8) = f(x5; x6; x7; x8): In this work, we dene a Non-strict eight dimensional walker manifold as the one that we can choose the coordinate system such that for any f in (M; g); f(x1; x2; x3; x4; x5; x6; x7; x8) = f(x1; x2; x3; x4): We derive cononical form of the Levi-Civita connection, curvature operator, (0; 4)-curvature tansor, the Ricci tensor, Weyl tensorand study some of the properties associated with the class of Non-strict 8 dimensionalWalker manifold. We investigate the Einstein property and establish a theorem for the metric to be locally conformally at.

Curvature of Cr Manifolds

2013 ◽  
Vol 59 (1) ◽  
pp. 43-72
Author(s):  
Aurel Bejancu ◽  
Hani Reda Farran
Keyword(s):  
Existence And Uniqueness ◽  
Ricci Tensor ◽  
Tensor Field ◽  
Differential Operators ◽  
Linear Connection ◽  
Horizontal Distribution ◽  
Einstein Equations ◽  
Space Forms ◽  
Torsion Free

Abstract We prove the existence and uniqueness of a torsion-free and h-metric linear connection ▽(CR connection) on the horizontal distribution of a CR manifold M. Then we define the CR sectional curvature of M and obtain a characterization of the CR space forms. Also, by using the CR Ricci tensor and the CR scalar curvature we define the CR Einstein gravitational tensor field on M. Thus, we can write down Einstein equations on the horizontal distribution of the 5-dimensional CR manifold involved in the Penrose correspondence. Finally, some CR differential operators are defined on M and two examples are given to illustrate the theory developed in the paper. Most of the results are obtained for CR manifolds that do not satisfy the integrability conditions

scholarly journals QUASIHOMOGENEOUS ANALYTIC AFFINE CONNECTIONS ON SURFACES

2013 ◽  
Vol 05 (04) ◽  
pp. 491-532 ◽  
Author(s):  
SORIN DUMITRESCU ◽  
ADOLFO GUILLOT
Keyword(s):  
Affine Connections ◽  
Open Set ◽  
Real Analytic ◽  
Locally Homogeneous ◽  
Torsion Free ◽  
Local Result

We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies on a local result that classifies germs of torsion-free real-analytic affine connections on a neighborhood of the origin in the plane which are quasihomogeneous, in the sense that they are locally homogeneous on an open set containing the origin in its closure, but not locally homogeneous in the neighborhood of the origin.

scholarly journals The structure of the Ricci tensor on locally homogeneous Lorentzian gradient Ricci solitons

2018 ◽  
Vol 148 (3) ◽  
pp. 461-482 ◽  
Author(s):  
M. Brozos-Vázquez ◽  
E. García-Río ◽  
S. Gavino-Fernández ◽  
P. Gilkey
Keyword(s):  
Ricci Tensor ◽  
Ricci Soliton ◽  
Complete Classification ◽  
Ricci Solitons ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Locally Homogeneous

We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show that the soliton is rigid in dimensions 3 and 4. In the steady case we give a complete classification in dimension 3.

How many torsionless affine connections exist in general dimension?

2016 ◽  
Vol 16 (1) ◽  
Author(s):  
Zdenek Dušek ◽  
Oldrich Kowalski
Keyword(s):  
Ricci Tensor ◽  
Smooth Manifold ◽  
Analytic Torsion ◽  
Affine Connections ◽  
General Dimension ◽  
Real Analytic ◽  
Torsion Free

AbstractWe study the question how many real analytic torsion-free affine connections exist locally on a smooth manifold M of dimension n. The families of torsion-free connections with skew-symmetric Ricci tensor and those with symmetric Ricci tensor are determined in terms of the number of arbitrary functions of n variables.

Jordan k-Homomorphisms onto Completely Prime ΓN Rings

1970 ◽  
Vol 30 ◽  
pp. 32-40
Author(s):  
Sujoy Charaborty ◽  
Akhil Chandra Paul
Keyword(s):  
Torsion Free

By introducing the notions of k-homomorphism, anti-k-homomorphism and Jordan khomomorphism of Nobusawa Γ -rings, we establish some significant results related to these concepts. If M1 is a Nobusawa Γ1 -ring and M2 is a 2-torsion free completely prime Nobusawa Γ2 -ring, then we prove that every Jordan k-homomorphism θ of M1 onto M2 such that k(Γ1 ) = Γ2 is either a k-homomorphism or an anti-k-homomorphism. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 32-40 DOI: http://dx.doi.org/10.3329/ganit.v30i0.8500  

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