# On generalized projective P-curvature tensor

TL;DR: In this paper, the P -curvature tensor is introduced and investigated, which generalizes projective, conharmonic, M -projective and the set of W i curvature tensors introduced by Pokhariyal and Mishra.

About: This article is published in Journal of Geometry and Physics.The article was published on 2021-01-01. It has received 5 citations till now. The article focuses on the topics: Riemann curvature tensor & Constant curvature.

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TL;DR: In this paper, it was shown that pseudo-Ricci symmetric spacetimes are Ricci flat in trivial,, and subspaces, whereas perfect fluid in subspace,, and, has zero scalar curvature.

Abstract: In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that spacetimes are Ricci flat in trivial, , and subspaces, whereas perfect fluid in subspaces , , and , and have zero scalar curvature in subspace . Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.

3 citations

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TL;DR: In this article , a generalized Ricci recurrent spacetimes (GR)n are investigated in Gray's seven subspaces, and it is shown that the Ricci tensor of GR n is Riemann compatible if Al is closed.

2 citations

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TL;DR: In this article , sufficient conditions on a pseudoprojective symmetric spacetime were given for the Ricci tensor to be either a perfect fluid or an Einstein spacetime.

Abstract: Sufficient conditions on a pseudoprojective symmetric spacetime
$\text{}{\left(\text{PPS}\right)}_{n}$
whose Ricci tensor is of Codazzi type to be either a perfect fluid or Einstein spacetime are given. Also, it is shown that a
$\text{}{\left(\text{PPS}\right)}_{n}$
is Einstein if its Ricci tensor is cyclic parallel. Next, we illustrate that a conformally flat
$\text{}{\left(\text{PPS}\right)}_{n}$
spacetime is of constant curvature. Finally, we investigate conformally flat
$\text{}{\left(\text{PPS}\right)}_{4}$
spacetimes and conformally flat
$\text{}{\left(\text{PPS}\right)}_{4}$
perfect fluids in
$f\left(R,\mathcal{G}\right)$
theory of gravity, and amongst many results, it is proved that the isotropic pressure and the energy density of conformally flat perfect fluid
$\text{}{\left(\text{PPS}\right)}_{4}$
spacetimes are constants and such perfect fluid behaves like a cosmological constant. Further, in this setting, we consider some energy conditions.

1 citations

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TL;DR: In this article , an almost pseudo symmetric spacetime with Codazzi type tensor is shown to be a perfect fluid whose velocity vector field is parallel and the energy density is constant.

Abstract: We prove that an almost pseudo [Formula: see text] symmetric spacetime with Codazzi type of [Formula: see text] tensor is perfect fluid whose velocity vector field [Formula: see text] is parallel. The energy density [Formula: see text] of such perfect fluid spacetime is constant and the state equation is obtained. Also, this spacetime is shown to be a static spacetime. Next, it is shown that such spacetime is a GRW spacetime with Einstein fiber. This kind of spacetime is investigated in f (R) theory of gravity; in this case, we find the forms of the isotropic pressure p and the energy density [Formula: see text] which are constant. Further, some energy conditions are studied in this spacetime. Finally, a concrete example of almost pseudo [Formula: see text] symmetric spacetimes is considered.

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261 citations

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TL;DR: In this article, the authors derived the conditions on curvature that make the Laplacian Azp? 0 under certain conditions and concluded the non-existence of harmonic tensors and consequently the vanishing of the Betti numbers.

Abstract: S. Bochner [1, 2, 3] has recently developed a beautiful theory on curvature and Betti numbers of an orientable compact Riemannian space V. with positive definite metric. He starts from the lemma: In V, , if the Laplacian Azp= gt'j;j; j _ 0 everywhere for a certain scalar (p, then we have Asp = 0 everywhere. In the above statement, the semi-colon denotes the covariant differentiation. He then takes a harmonic tensor tiji2...ip and put (p = ilit2 ...ilt2. ..ip . He calculates the Azp and seeks for the conditions on curvature that make the Laplacian Azp ? 0. Thus, under certain conditions, he concludes the non-existence of harmonic tensor and consequently the vanishing of the Betti numbers by virtue of a theorem of Hodge [6]. In a recent paper [11], the present author has proved an integral formula

257 citations

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TL;DR: In this article, the authors considered the possibility that the universe is made of a dark fluid described by a quadratic equation of state, where the energy density is the sum of two terms: a rest-mass term and an internal energy term, and they provided a simple analytical solution of the Friedmann equations for a universe undergoing a stiff matter era, a dark matter era and a dark energy era due to the cosmological constant.

Abstract: We consider the possibility that the Universe is made of a dark fluid described by a quadratic equation of state $P=K{\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the rest-mass density and $K$ is a constant. The energy density $\ensuremath{\epsilon}=\ensuremath{\rho}{c}^{2}+K{\ensuremath{\rho}}^{2}$ is the sum of two terms: a rest-mass term $\ensuremath{\rho}{c}^{2}$ that mimics ``dark matter'' ($P=0$) and an internal energy term $u=K{\ensuremath{\rho}}^{2}=P$ that mimics a ``stiff fluid'' ($P=\ensuremath{\epsilon}$) in which the speed of sound is equal to the speed of light. In the early universe, the internal energy dominates and the dark fluid behaves as a stiff fluid ($P\ensuremath{\sim}\ensuremath{\epsilon}$, $\ensuremath{\epsilon}\ensuremath{\propto}{a}^{\ensuremath{-}6}$). In the late universe, the rest-mass energy dominates and the dark fluid behaves as pressureless dark matter ($P\ensuremath{\simeq}0$, $\ensuremath{\epsilon}\ensuremath{\propto}{a}^{\ensuremath{-}3}$). We provide a simple analytical solution of the Friedmann equations for a universe undergoing a stiff matter era, a dark matter era, and a dark energy era due to the cosmological constant. This analytical solution generalizes the Einstein--de Sitter solution describing the dark matter era, and the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model describing the dark matter era and the dark energy era. Historically, the possibility of a primordial stiff matter era first appeared in the cosmological model of Zel'dovich where the primordial universe is assumed to be made of a cold gas of baryons. A primordial stiff matter era also occurs in recent cosmological models where dark matter is made of relativistic self-gravitating Bose-Einstein condensates (BECs). When the internal energy of the dark fluid mimicking stiff matter is positive, the primordial universe is singular like in the standard big bang theory. It expands from an initial state with a vanishing scale factor and an infinite density. We consider the possibility that the internal energy of the dark fluid is negative (while, of course, its total energy density is positive), so that it mimics anti-stiff matter. This happens, for example, when the BECs have an attractive self-interaction with a negative scattering length. In that case, the primordial universe is nonsingular and bouncing like in loop quantum cosmology. At $t=0$, the scale factor is finite and the energy density is equal to zero. The universe first has a phantom behavior where the energy density increases with the scale factor, then a normal behavior where the energy density decreases with the scale factor. For the sake of generality, we consider a cosmological constant of arbitrary sign. When the cosmological constant is positive, the Universe asymptotically reaches a de Sitter regime where the scale factor increases exponentially rapidly with time. This can account for the accelerating expansion of the Universe that we observe at present. When the cosmological constant is negative (anti--de Sitter), the evolution of the Universe is cyclic. Therefore, depending on the sign of the internal energy of the dark fluid and on the sign of the cosmological constant, we obtain analytical solutions of the Friedmann equations describing singular and nonsingular expanding, bouncing, or cyclic universes.

139 citations

01 Jan 1970

TL;DR: In this paper, the Curvature tensor has been defined and its properties have been elaborated in terms of physical and geometric properties, including its properties and properties of curvature tensors.

Abstract: In this paper we have defined the Curvature tensor and elaborated its various physical and geometric properties.

110 citations

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TL;DR: In this article, the Bianchi identity of the new "Codazzi deviation tensor" is shown to be equivalent to a Bianchi tensor on the Riemann tensor.

Abstract: Derdzinski and Shen’s theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new “Codazzi deviation tensor”, with a geometric significance. The general properties are studied, with their implications on Pontryagin forms. Examples are given of manifolds with Riemann-compatible tensors, in particular those generated by geodesic mappings. Compatibility is extended to generalized curvature tensors, with an application to Weyl’s tensor and general relativity.

54 citations