Curvature of Cr Manifolds

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

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(Pseudo) generalized Kaluza–Klein G-spaces and Einstein equations

International Journal of Mathematics ◽

10.1142/s0129167x1450116x ◽

2014 ◽

Vol 25 (12) ◽

pp. 1450116 ◽

Cited By ~ 3

Author(s):

Constantin M. Arcuş ◽

Esmaeil Peyghan

Keyword(s):

Vector Bundle ◽

Tangent Bundle ◽

Ricci Tensor ◽

Tensor Field ◽

General Framework ◽

Bianchi Type ◽

Einstein Equations ◽

Lie Algebroid ◽

Linear Connections ◽

Kaluza Klein

Introducing the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza–Klein vector bundle, we present the (g, h)-lift of a curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear (ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear (ρ, η)-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza–Klein G-spaces and we develop the Einstein equations in this general framework.

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A characterization of minimal ascreen null hypersurfaces of $(LCS)$-space forms

10.31730/osf.io/k3rej ◽

2019 ◽

Author(s):

Samuel Ssekajja

Keyword(s):

Constant Curvature ◽

Ricci Tensor ◽

Space Form ◽

Null Hypersurface ◽

Space Forms ◽

Screen Distribution ◽

Null Hypersurfaces ◽

Null Curve

In the present paper, we study nontotally geodesic minimal ascreen null hypersurface, $M$, of a Lorentzian concircular structure $(LCS)$-space form of constant curvature $0$ or $1$. We prove that; if the Ricci tensor of $M$ is parallel with respect to any leaf of its screen distribution, then $M$ is isometric to a product of a null curve and spheres.

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KALUZA–KLEIN THEORY WITH TORSION CONFINED TO THE EXTRA DIMENSION

Modern Physics Letters A ◽

10.1142/s0217732310033566 ◽

2010 ◽

Vol 25 (25) ◽

pp. 2121-2130 ◽

Cited By ~ 5

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.

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A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms

Mathematics ◽

10.3390/math8040642 ◽

2020 ◽

Vol 8 (4) ◽

pp. 642

Author(s):

George Kaimakamis ◽

Konstantina Panagiotidou ◽

Juan de Dios Pérez

Keyword(s):

Ricci Tensor ◽

Complex Space ◽

Space Form ◽

Flat Space ◽

Real Hypersurfaces ◽

Space Forms ◽

Ruled Real Hypersurfaces ◽

Levi Civita Connection ◽

Complex Space Forms

The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator F X ( k ) is defined and is related to both connections. If X belongs to the maximal holomorphic distribution D on M, the corresponding operator does not depend on k and is denoted by F X and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that F X S = S F X , where S denotes the Ricci tensor of M and a further condition is satisfied, are classified.

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A Characterization of Real Hypersurfaces in Complex Space Forms in Terms of the Ricci Tensor

Canadian Mathematical Bulletin ◽

10.4153/cmb-1997-031-5 ◽

1997 ◽

Vol 40 (3) ◽

pp. 257-265 ◽

Cited By ~ 2

Author(s):

Christos Baikoussis

Keyword(s):

Ricci Tensor ◽

Complex Space ◽

Space Form ◽

Complex Space Form ◽

Real Hypersurfaces ◽

Space Forms ◽

Complex Space Forms ◽

Orthogonal Distribution

AbstractWe study real hypersurfaces of a complex space form Mn(c), c ≠ 0 under certain conditions of the Ricci tensor on the orthogonal distribution T0.

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A LINEAR CONNECTION FOR BOTH SUB-RIEMANNIAN GEOMETRY AND NONHOLONOMIC MECHANICS (I)

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887811005361 ◽

2011 ◽

Vol 08 (04) ◽

pp. 725-752 ◽

Cited By ~ 1

Author(s):

AUREL BEJANCU

Keyword(s):

Riemannian Manifolds ◽

Riemannian Geometry ◽

Ricci Tensor ◽

Differential Operators ◽

Linear Connection ◽

Riemannian Metric ◽

Riemannian Connection ◽

Schur Theorem ◽

Horizontal Curvature ◽

Horizontal And Vertical Distribution

We study the geometry of a sub-Riemannian manifold (M, HM, VM, g), where HM and VM are the horizontal and vertical distribution respectively, and g is a Riemannian extension of the Riemannian metric on HM. First, without the assumption that HM and VM are orthogonal, we construct a sub- Riemannian connection ▽ on HM and prove some Bianchi identities for ▽. Then, we introduce the horizontal sectional curvature, prove a Schur theorem for sub-Riemannian geometry and find a class of sub-Riemannian manifolds of constant horizontal curvature. Finally, we define the horizontal Ricci tensor and scalar curvature, and some sub-Riemannian differential operators (gradient, divergence, Laplacian), extending some results from geometry to the sub-Riemannian setting.

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