We prove theorems about the Ricci and the Weyl tensors on Generalized Robertson-Walker space-times of dimension n ≥ 3. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihilated by Chen’s vector, and any of the two conditions is necessary and sufficient for the Generalized Robertson-Walker (GRW) space-time to be a quasi-Einstein (perfect fluid) manifold. Finally, the general structure of the Riemann tensor for Robertson-Walker space-times is given, in terms of Chen’s vector. In n = 4, a GRW space-time with harmonic Weyl tensor is a Robertson-Walker space-time.
REFERENCES
1.
B.-Y.
Chen
, “A simple characterization of generalized Robertson-Walker spacetimes
,” Gen. Relativ. Gravitation
46
, 1833
(2014
).2.
A.
Fialkow
, “Conformal geodesics
,” Trans. Am. Math. Soc.
45
(3
), 443
–473
(1939
).3.
K.
Yano
, “On the torse-forming directions in Riemannian spaces
,” Proc. Imp. Acad.
20
, 340
–345
(1944
).4.
L. J.
Alías
, A.
Romero
, and M.
Sánchez
, “Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker space-times
,” Gen. Relativ. Gravitation
27
(1
), 71
–84
(1995
).5.
M.
Sánchez
, “On the geometry of generalized Robertson-Walker spacetimes: Geodesics
,” Gen. Relativ. Gravitation
30
, 915
–932
(1998
).6.
R.
Deszcz
and M.
Kucharski
, “On curvature properties of certain generalized Robertson-Walker space-times
,” Tsukuba J. Math.
23
(1
), 113
–130
(1999
), http://www.jstor.org/stable/43686121.7.
M.
Gutíerrez
and B.
Olea
, “Global decomposition of a Lorentzian manifold as a Generalized Robertson-Walker space
,” Differ. Geom. Appl.
27
, 146
–156
(2009
).8.
A.
Gȩbarowski
, “On nearly conformally symmetric warped product spacetimes
,” Soochow J. Math.
20
(1
), 61
–75
(1994
), http://163.14.246.20/mp/pdf/S20N17.pdf.9.
M.
Sánchez
, “On the geometry of generalized Robertson-Walker spacetimes: Curvature and Killing fields
,” Gen. Relativ. Gravitation
31
, 1
–15
(1999
).10.
C. A.
Mantica
, L. G.
Molinari
, and U. C.
De
, “A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time
,” J. Math. Phys.
57
, 022508
(2016
);Erratum,
C. A.
Mantica
, L. G.
Molinari
, and U. C.
De
, J. Math.Phys.
57
, 049901
(2016
).11.
C. A.
Mantica
, Y. J.
Suh
, and U. C.
De
, “A note on generalized Robertson-Walker space-times
,” Int. J. Geom. Methods Mod. Phys.
13
, 1650079
(2016
).12.
C. A.
Mantica
and L. G.
Molinari
, “Riemann compatible tensors
,” Colloq. Math.
128
(2
), 197
–210
(2012
).13.
C. A.
Mantica
and L. G.
Molinari
, “Weyl compatible tensors
,” Int. J. Geom. Methods Mod. Phys.
11
, 1450070
(2014
).14.
D.
Lovelock
and H.
Rund
, Tensors, Differential Forms and Variational Principles
, reprint ed. (Dover
, 1988
).15.
R.
Deszcz
and M.
Hotloś
, “Remarks on Riemannian manifolds satisfying a certain curvature condition imposed on the Ricci tensor
,” Prace Nauk. Pol. Szczec.
11
, 23
–34
(1989
).16.
Chojnacka-Dulas
, R.
Deszcz
, M.
Głogowska
, and M.
Prvanović
, “On warped product manifolds satisfying some curvature conditions
,” J. Geom. Phys.
74
, 328
–341
(2013
).17.
M.
Brozos-Vázquez
, E.
García-Río
, and R.
Vázquez-Lorenzo
, “Some remarks on locally conformally flat static space-times
,” J. Math. Phys.
46
, 022501
(2005
).18.
B.-Y.
Chen
and K.
Yano
, “Hypersurfaces of conformally flat spaces
,” Tensor (N.S.)
26
, 318
–322
(1972
).© 2016 Author(s).
2016
Author(s)
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