On weakly conformally symmetric pseudo-Riemannian manifolds

2017 ◽  
Vol 29 (03) ◽  
pp. 1750007
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh
Keyword(s):  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Topological Properties ◽  
Lorentzian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Integral Curves ◽  
Pontryagin Form ◽  
Lorentzian Case

In this paper, we study the properties of weakly conformally symmetric pseudo- Riemannian manifolds focusing particularly on the [Formula: see text]-dimensional Lorentzian case. First, we provide a new proof of an important result found in literature; then several new others are stated. We provide a decomposition for the conformal curvature tensor in [Formula: see text]. Moreover, some important identities involving two particular covectors are stated; for example, it is proven that under certain conditions the Ricci tensor and other tensors are Weyl compatible. Topological properties involving the vanishing of the first Pontryagin form are then stated. Further, we study weakly conformally symmetric [Formula: see text]-dimensional Lorentzian manifolds (space-times): it is proven that one of the previously defined co-vectors is null and unique up to a scaling. Moreover, it is shown that under certain conditions, the same vector is an eigenvector of the Ricci tensor and its integral curves are geodesics. Finally, it is stated that such space-time is of Petrov type N with respect to the same vector.

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.

scholarly journals Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1209-1215
Author(s):  
Aleksandar Sebekovic ◽  
Miroslava Petrovic-Torgasev ◽  
Anica Pantic
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Space Forms ◽  
Equality Case ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Sasakian Space Forms ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds ◽  
Weyl Conformal Curvature Tensor

For Legendrian submanifolds Mn in Sasakian space forms ?M2n+1(c), I. Mihai obtained an inequality relating the normalised scalar curvature (intrinsic invariant) and the squared mean curvature and the normalised scalar normal curvature of M in the ambient space ?M (extrinsic invariants) which is called the generalised Wintgen inequality, characterising also the corresponding equality case. And a Legendrian submanifold Mn in Sasakian space forms ?M2n+1(c) is said to be generalised Wintgen ideal Legendrian submanifold of ?M2n+1(c) when it realises at everyone of its points the equality in such inequality. Characterisations based on some basic intrinsic symmetries involving the Riemann-Cristoffel curvature tensor, the Ricci tensor and the Weyl conformal curvature tensor belonging to the class of pseudosymmetries in the sense of Deszcz of such generalised Wintgen ideal Legendrian submanifolds are given.

scholarly journals QUASI-CONFORMAL CURVATURE TENSOR OF GENERALIZED SASAKIAN-SPACE-FORMS

2020 ◽  
Vol 35 (1) ◽  
pp. 089
Author(s):  
Braj B. Chaturvedi ◽  
Brijesh K. Gupta
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Space Form ◽  
Riemannian Curvature ◽  
Space Forms ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Sasakian Space Forms ◽  
Riemannian Curvature Tensor

The present paper deals the study of generalised Sasakian-space-forms with the conditions Cq(ξ,X).S = 0, Cq(ξ,X).R = 0 and Cq(ξ,X).Cq = 0, where R, S and Cq denote Riemannian curvature tensor, Ricci tensor and quasi-conformal curvature tensor of the space-form, respectively and at last, we have given some examples to improve our results.

Some characterizations of Lorentzian manifolds

2019 ◽  
Vol 16 (01) ◽  
pp. 1950016
Author(s):  
Uday Chand De ◽  
Young Jin Suh
Keyword(s):  
Curvature Tensor ◽  
Lorentzian Manifold ◽  
Lorentzian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor

Generalized Robertson–Walker (GRW) spacetime is the generalization of the Robertson–Walker (RW) spacetime and a further generalization of GRW spacetime is the twisted spacetime. In this paper, we generalize the results of the paper [C. A. Mantica, Y. J. Suh and U. C. De, A note on generalized Robertson–Walker spacetimes, Int. J. Geom. Methods Mod. Phys. 13 (2016), Article Id: 1650079, 9 pp., doi: 101142/s0219887816500791 ]. We prove that a Ricci simple Lorentzian manifold with vanishing quasi-conformal curvature tensor is a RW spacetime. Further, we prove that a Ricci simple Lorentzian manifold with harmonic quasi-conformal curvature tensor is a GRW spacetime. As a consequence, we obtain several corollaries. Finally, we have cited some examples of the obtained results.

scholarly journals On some classes of generalized quasi Einstein manifolds

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 443-456 ◽  
Author(s):  
Sinem Güler ◽  
Sezgin Demirbağ
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Einstein Manifold ◽  
Sufficient Conditions ◽  
Riemannian Curvature ◽  
Einstein Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Riemannian Curvature Tensor ◽  
Concircular Curvature Tensor

In the present paper, we investigate generalized quasi Einstein manifolds satisfying some special curvature conditions R?S = 0,R?S = LSQ(g,S), C?S = 0,?C?S = 0,?W?S = 0 and W2?S = 0 where R, S, C,?C,?W and W2 respectively denote the Riemannian curvature tensor, Ricci tensor, conformal curvature tensor, concircular curvature tensor, quasi conformal curvature tensor and W2-curvature tensor. Later, we find some sufficient conditions for a generalized quasi Einstein manifold to be a quasi Einstein manifold and we show the existence of a nearly quasi Einstein manifolds, by constructing a non trivial example.

scholarly journals Deszcz Pseudo Symmetry Type LP-Sasakian Manifolds

2016 ◽  
Vol 54 (1) ◽  
pp. 35-53
Author(s):  
Kanak Kanti Baishya ◽  
Partha Roy Chowdhury
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Symmetry Type ◽  
Sasakian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor ◽  
Concircular Curvature Tensor ◽  
Curvature Tensors

Abstract Recently the present authors introduced the notion of generalized quasi-conformal curvature tensor which bridges Conformal curvature tensor, Concircular curvature tensor, Projective curvature tensor and Conharmonic curvature tensor. This paper attempts to charectrize LP-Sasakian manifolds with ω(X, Y) · 𝒲 = L{(X ∧ɡ Y) · 𝒲}. On the basis of this curvature conditions and by taking into account, the permutation of different curvature tensors we obtained and tabled the nature of the Ricci tensor for the respective pseudo symmetry type LP-Sasakian manifolds.

scholarly journals On the quasi-conformal curvature tensor of an almost Kenmotsu manifold with nullity distributions

2018 ◽  
Vol 33 (2) ◽  
pp. 255
Author(s):  
Dibakar Dey ◽  
Pradip Majhi
Keyword(s):  
Vector Field ◽  
Curvature Tensor ◽  
Conformally Flat ◽  
Kenmotsu Manifold ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Kenmotsu Manifolds ◽  
Almost Kenmotsu Manifolds ◽  
Nullity Distribution ◽  
Characteristic Vector Field

The object of the present paper is to characterize quasi-conformally flat and $\xi$-quasi-conformally flat almost Kenmotsu manifolds with  $(k,\mu)$-nullity and $(k,\mu)'$-nullity distributions respectively. Also we characterize almost Kenmotsu manifolds with vanishing extended quasi-conformal curvature tensor and extended $\xi$-quasi-conformally flat almost Kenmotsu manifolds such that the characteristic vector field $\xi$ belongs to the $(k,\mu)$-nullity distribution.

A note on generalized Robertson–Walker space-times

2016 ◽  
Vol 13 (06) ◽  
pp. 1650079 ◽  
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh ◽  
Uday Chand De
Keyword(s):  
Vector Field ◽  
Scalar Field ◽  
Perfect Fluid ◽  
Curvature Tensor ◽  
Weyl Tensor ◽  
Space Time ◽  
Stiff Matter ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Mass Less Scalar Field

A generalized Robertson–Walker (GRW) space-time is the generalization of the classical Robertson–Walker space-time. In the present paper, we show that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a GRW space-time. Further, we show that a stiff matter perfect fluid space-time or a mass-less scalar field with time-like gradient and with divergence-free Weyl tensor are GRW space-times.

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.

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