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 Conformally Recurrent Semi-Riemannian Manifolds

2005 ◽  
Vol 35 (1) ◽  
pp. 285-307 ◽  
Author(s):  
Young Jin Suh ◽  
Jung-Hwan Kwon
Keyword(s):  
Riemannian Manifolds ◽  
Conformally Recurrent

scholarly journals Weighted Hardy inequality on Riemannian manifolds

2016 ◽  
Vol 18 (06) ◽  
pp. 1550072 ◽  
Author(s):  
El Hadji Abdoulaye Thiam
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Hardy Inequality ◽  
Compact Riemannian Manifold ◽  
Sufficient Conditions ◽  
Extremal Functions ◽  
Dimension Formula ◽  
Existence Of Minimizers ◽  
Necessary And Sufficient ◽  
Funct Anal

Let [Formula: see text] be a smooth compact Riemannian manifold of dimension [Formula: see text] and let [Formula: see text] to be a closed submanifold of dimension [Formula: see text]. In this paper, we study existence and non-existence of minimizers of Hardy inequality with weight function singular on [Formula: see text] within the framework of Brezis–Marcus–Shafrir [Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000) 177–191]. In particular, we provide necessary and sufficient conditions for existence of minimizers.

Pseudo B-symmetric manifolds

2017 ◽  
Vol 14 (09) ◽  
pp. 1750119
Author(s):  
Young Jin Suh ◽  
Carlo Alberto Mantica ◽  
Uday Chand De ◽  
Prajjwal Pal
Keyword(s):  
Riemannian Manifold ◽  
Explicit Form ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Sufficient Condition ◽  
Conformally Flat ◽  
Harmonic Curvature ◽  
Codazzi Type ◽  
Curvature Tensors

In this paper, we introduce a new tensor named [Formula: see text]-tensor which generalizes the [Formula: see text]-tensor introduced by Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-[Formula: see text]-symmetric manifolds [Formula: see text] which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a [Formula: see text]. Next, we prove that a pseudo-Riemannian manifold is [Formula: see text]-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a [Formula: see text] to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a [Formula: see text] if the [Formula: see text]-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-[Formula: see text]-symmetric manifolds and prove that a [Formula: see text] spacetime is a [Formula: see text]-wave under certain conditions.

Recurrent conformal 2-forms on pseudo-Riemannian manifolds

2014 ◽  
Vol 11 (06) ◽  
pp. 1450056 ◽  
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Higher Dimensions ◽  
Topological Properties ◽  
Lorentzian Manifolds ◽  
Differential Structure ◽  
Arbitrary Signature

In this paper, we introduce the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature. Some theorems already proved for the same differential structure on a Riemannian manifold are proven to hold in this more general contest. Moreover other interesting results are pointed out; it is proven that if the associated covector is closed, then the Ricci tensor is Riemann compatible or equivalently, Weyl compatible: these notions were recently introduced and investigated by one of the present authors. Further some new results about the vanishing of some Weyl scalars on a pseudo-Riemannian manifold are given: it turns out that they are consequence of the generalized Derdziński–Shen theorem. Topological properties involving the vanishing of Pontryagin forms and recurrent conformal 2-forms are then stated. Finally, we study the properties of recurrent conformal 2-forms on Lorentzian manifolds (space-times). Previous theorems stated on a pseudo-Riemannian manifold of arbitrary signature are then interpreted in the light of the classification of space-times in four or in higher dimensions.

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