scholarly journals Almost Kahler 4-manifolds with J-invariant ricci tensor and special weyl tensor

2000 ◽  
Vol 51 (3) ◽  
pp. 275-294 ◽  
Author(s):  
V. Apostolov
Keyword(s):  
Weyl Tensor ◽  
Ricci Tensor ◽  
Almost Kähler

scholarly journals WEYL COLLINEATIONS THAT ARE NOT CURVATURE COLLINEATIONS

2005 ◽  
Vol 14 (08) ◽  
pp. 1431-1437 ◽  
Author(s):  
IBRAR HUSSAIN ◽  
ASGHAR QADIR ◽  
K. SAIFULLAH
Keyword(s):  
Linear Combination ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Weyl Tensor ◽  
Ricci Tensor ◽  
Lie Symmetries ◽  
Ricci Scalar ◽  
Finite Dimensional ◽  
Fundamental Reason ◽  
Curvature Collineations

Though the Weyl tensor is a linear combination of the curvature tensor, Ricci tensor and Ricci scalar, it does not have all and only the Lie symmetries of these tensors since it is possible, in principle, that "asymmetries cancel." Here we investigate if, when and how the symmetries can be different. It is found that we can obtain a metric with a finite dimensional Lie algebra of Weyl symmetries that properly contains the Lie algebra of curvature symmetries. There is no example found for the converse requirement. It is speculated that there may be a fundamental reason for this lack of "duality."

scholarly journals A SPLITTING THEOREM FOR KÄHLER MANIFOLDS WHOSE RICCI TENSORS HAVE CONSTANT EIGENVALUES

2001 ◽  
Vol 12 (07) ◽  
pp. 769-789 ◽  
Author(s):  
VESTISLAV APOSTOLOV ◽  
TEDI DRĂGHICI ◽  
ANDREI MOROIANU
Keyword(s):  
Ricci Tensor ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Complex Dimension ◽  
Negative Eigenvalues ◽  
Kähler Surfaces ◽  
Almost Kähler ◽  
Compact Kähler Manifold

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler–Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kähler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kähler metrics of any even real dimension greater than 4.

scholarly journals General properties of f(R) gravity vacuum solutions

2020 ◽  
Vol 29 (13) ◽  
pp. 2050089
Author(s):  
Salvatore Capozziello ◽  
Carlo Alberto Mantica ◽  
Luca Guido Molinari
Keyword(s):  
Weyl Tensor ◽  
Ricci Tensor ◽  
Curvature Scalar ◽  
General Properties ◽  
Walker Metric ◽  
Vacuum Solutions

General properties of vacuum solutions of [Formula: see text] gravity are obtained by the condition that the divergence of the Weyl tensor is zero and [Formula: see text]. Specifically, a theorem states that the gradient of the curvature scalar, [Formula: see text], is an eigenvector of the Ricci tensor and, if it is timelike, the spacetime is a Generalized Friedman–Robertson–Walker metric; in dimension four, it is Friedman–Robertson–Walker.

scholarly journals Weyl compatible tensors

2014 ◽  
Vol 11 (08) ◽  
pp. 1450070 ◽  
Author(s):  
Carlo Alberto Mantica ◽  
Luca Guido Molinari
Keyword(s):  
General Relativity ◽  
Constant Curvature ◽  
Weyl Tensor ◽  
Ricci Tensor ◽  
Algebraic Property ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Symmetric Tensors ◽  
Magnetic Part ◽  
Necessary And Sufficient

We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular, it is shown that the existence of a Weyl compatible vector implies that the Weyl tensor is algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdziński and Shen [11], Hall [15]) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as in McIntosh et al. [16, 17]). A simple example of Weyl compatible tensor is the Ricci tensor of an hypersurface in a manifold with constant curvature.

scholarly journals On conformally recurrent manifolds of dimension greater than 4

2016 ◽  
Vol 13 (05) ◽  
pp. 1650053
Author(s):  
Carlo Alberto Mantica ◽  
Luca Guido Molinari
Keyword(s):  
Explicit Expression ◽  
Riemannian Manifolds ◽  
Weyl Tensor ◽  
Ricci Tensor ◽  
Symmetric Tensor ◽  
Scalar Function ◽  
Dimension Formula ◽  
Null Recurrence ◽  
Conformally Recurrent

Conformally recurrent pseudo-Riemannian manifolds of dimension [Formula: see text] are investigated. The Weyl tensor is represented as a Kulkarni–Nomizu product. If the square of the Weyl tensor is non-zero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak’s theorem, the explicit expression of the traceless part of the Ricci tensor is obtained, up to a scalar function. The Ricci tensor has at most two distinct eigenvalues, and the recurrence vector is an eigenvector. Lorentzian conformally recurrent manifolds are then considered. If the square of the Weyl tensor is non-zero, the manifold is decomposable. A null recurrence vector makes the Weyl tensor of algebraic type IId or higher in the Bel–Debever–Ortaggio classification, while a time-like recurrence vector makes the Weyl tensor purely electric.

scholarly journals The Adjunction Inequality for Weyl-Harmonic Maps

2020 ◽  
Vol 7 (1) ◽  
pp. 129-140
Author(s):  
Robert Ream
Keyword(s):  
Minimal Surfaces ◽  
Twistor Space ◽  
Harmonic Maps ◽  
Holomorphic Curves ◽  
Weyl Connection ◽  
Almost Kähler ◽  
Conformal Manifolds

AbstractIn this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality\chi \left( {{T_f}\sum } \right) + \chi \left( {{N_f}\sum } \right) \le \pm {c_1}\left( {f*{T^{\left( {1,0} \right)}}M} \right).The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.

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