In this note we show that Lorentzian Concircular Structure manifolds (LCS)_n coincide with Generalized Robertson-Walker space-times

C.A. Mantica

L.G. Molinari

English

AIR Universita degli studi di Milano

Commun. Korean Math. Soc. 34 (2019), No. 2, pp. 633–635
https://doi.org/10.4134/CKMS.c180138
pISSN: 1225-1763 / eISSN: 2234-3024
A NOTE ON CONCIRCULAR STRUCTURE SPACE-TIMES
Carlo Alberto Mantica and Luca Guido Molinari
Reprinted from the
Communications of the Korean Mathematical Society
Vol. 34, No. 2, April 2019
c©2019 Korean Mathematical Society
Commun. Korean Math. Soc. 34 (2019), No. 2, pp. 633–635
https://doi.org/10.4134/CKMS.c180138
pISSN: 1225-1763 / eISSN: 2234-3024
A NOTE ON CONCIRCULAR STRUCTURE SPACE-TIMES
Carlo Alberto Mantica and Luca Guido Molinari
Abstract. In this note we show that Lorentzian Concircular Structure
manifolds (LCS)n coincide with Generalized Robertson-Walker space-
times.
Generalized Robertson-Walker (GRW) space-times were introduced in 1995
by Al´ıas, Romero and Sa´nchez [1] as the warped product −1 ×q2 M∗, where
(M∗, g∗) is a Riemannian submanifold. In other terms, they are Lorentzian
manifolds characterised by a metric
gijdx
idxj = −(dt)2 + q(t)2g∗µν(x1, . . . , xn−1)dxµdxν .(1)
They are interesting not only for geometry [2, 5, 9–11], but also for physics:
they include relevant space-times such as Robertson-Walker, Einstein-de Sitter,
static Einstein, de-Sitter, the Friedmann cosmological models. They are a wide
generalization of space-times for cosmological models.
In 2003 A. A. Shaikh [12] introduced the notion of Lorentzian Concircular
Structure (LCS)n. It is a Lorentzian manifold endowed with a unit time-like
concircular vector field, i.e., uiui = −1 and
∇kuj = ϕ(ukuj + gkj),(2)
where ϕ 6= 0 is a scalar function obeying
∇jϕ = µuj(3)
being µ a scalar function. Various authors studied the properties of (LCS)n
manifolds [6, 13–15].
We show that GRW and (LCS)n are the same space-times.
We recall few definitions that will be used in this note. The first ones are
the definitions of “torse-forming” and “concircular” vector fields, by Yano:
Definition 1 (Yano, [16, 17]). A vector field Xj is named torse-forming if
∇kXj = ωkXj + ϕgkj , being ϕ a scalar function and ωk a non vanishing one-
form. It is named concircular if ωk is a gradient or locally a gradient of a scalar
function.
Received April 5, 2018; Accepted May 16, 2018.
2010 Mathematics Subject Classification. Primary 53B30; Secondary 53C25.
Key words and phrases. Generalized Robertson-Walker space-time, Lorentzian concircu-
lar structure, torse-forming vector, concircular vector.
c©2019 Korean Mathematical Society
633
634 C. A. MANTICA AND L. G. MOLINARI
Fialkow gave a definition different from Yano’s:
Definition 2 (Fialkow [4]). A vector field Xj is named concircular if it satisfies
∇kXj = ρgkj , being ρ a scalar function.
The following simple but deep result was recently proven:
Theorem 3 (Bang-Yen Chen, [3]). A n > 3 dimensional Lorentzian manifold
is a GRW space-time if and only if it admits a time-like concircular vector field
(in the sense of Fialkow).
It is worth noticing that for a unit time-like vector field, the torse-forming
property by Yano becomes precisely Eq. (2), with generic scalar field ϕ. Based
on Chen’s theorem we proved:
Proposition 4 (Mantica and Molinari, [7,8]). A n > 3 dimensional Lorentzian
manifold is a GRW space-time if and only if it admits a unit time-like torse-
forming vector, (2), that is also an eigenvector of the Ricci tensor.
Now comes the equivalence: from (2) (holding either for GRW and (LCS)n
space-times) we evaluate
Rjkl
mum = [∇j ,∇k]ul = (hkl∇j − hjl∇k)ϕ− ϕ2(ujgkl − ukgjl),
where hkl = ukul + gkl. Contraction with g
jl gives
Rk
mum = uk[u
m∇mϕ+ (n− 1)ϕ2]− (n− 2)∇kϕ.(4)
If (3) holds, then uk is an eigenvector of the Ricci tensor, and we conclude that
a (LCS)n manifold is a GRW space-time.
If Rkmu
m = ξuk it is (n − 2)∇kϕ = αuk for some scalar field α, i.e., (3)
holds. Then we conclude that a GRW space-time is a (LCS)n manifold.
References
[1] L. Al´ıas, A. Romero, and M. Sa´nchez, Uniqueness of complete spacelike hypersurfaces of
constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativity
Gravitation 27 (1995), no. 1, 71–84.
[2] K. Arslan, R. Deszcz, R. Ezentas, M. Hotlos´, and C. Murathan, On generalized
Robertson-Walker spacetimes satisfying some curvature condition, Turk. J. Math. 38
(2014) 353–373.
[3] B.-Y. Chen, A simple characterization of generalized Robertson-Walker spacetimes,
Gen. Relativity Gravitation 46 (2014), no. 12, Art. 1833, 5 pp.
[4] A. Fialkow, Conformal geodesics, Trans. Amer. Math. Soc. 45 (1939), no. 3, 443–473.
[5] M. Gutie´rrez and B. Olea, Global decomposition of a Lorentzian manifold as a general-
ized Robertson-Walker space, Differential Geom. Appl. 27 (2009), no. 1, 146–156.
[6] S. K. Hui, On φ-pseudo symmetries of (LCS)n-manifolds, Kyungpook Math. J. 53
(2013), no. 2, 285–294.
[7] C. A. Mantica and L. G. Molinari, On the Weyl and Ricci tensors of generalized
Robertson-Walker space-times, J. Math. Phys. 57 (2016), no. 10, 102502, 6 pp.
[8] , Generalized Robertson-Walker spacetimes—a survey, Int. J. Geom. Methods
Mod. Phys. 14 (2017), no. 3, 1730001, 27 pp.
A NOTE ON CONCIRCULAR STRUCTURE SPACE-TIMES 635
[9] A. Romero, R. M. Rubio, and J. J. Salamanca, Uniqueness of complete maximal hyper-
surfaces in spatially parabolic generalized Robertson-Walker spacetimes, Classical Quan-
tum Gravity 30 (2013), no. 11, 115007, 13 pp.
[10] M. Sa´nchez, On the geometry of generalized Robertson-Walker spacetimes: geodesics,
Gen. Relativity Gravitation 30 (1998), no. 6, 915–932.
[11] , On the geometry of generalized Robertson-Walker spacetimes: curvature and
Killing fields, J. Geom. Phys. 31 (1999), no. 1, 1–15.
[12] A. A. Shaikh, On Lorentzian almost paracontact manifolds with a structure of the con-
circular type, Kyungpook Math. J. 43 (2003), no. 2, 305–314.
[13] , Some results on (LCS)n-manifolds, J. Korean Math. Soc. 46 (2009), no. 3,
449–461.
[14] A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes, J. Math. Stat. 1
(2005), no. 2, 129–132.
[15] , On concircular structure spacetimes II, American J. Appl. Sci. 3 (2006), no. 4,
1790–1794.
[16] K. Yano, Concircular geometry I-IV, Proc. Imp. Acad. Tokyo 16 (1940) 195–200, 354–
360, 442–448, 505–511.
[17] , On the torse-forming directions in Riemannian spaces, Proc. Imp. Acad. Tokyo
20 (1944), 340–345.
Carlo Alberto Mantica
I.I.S. Lagrange
Via L. Modignani 65, 20161, Milano, Italy
Email address: carloalberto.mantica@libero.it
Luca Guido Molinari
Physics Department
Universita` degli Studi di Milano and I.N.F.N. sez. Milano
Via Celoria 16, 20133 Milano, Italy
Email address: luca.molinari@unimi.it


A note on concircular structure space-times

https://air.unimi.it/bitstream/2434/684944/2/14%29%20A%20note%20on%20concircular%20structure%20spt.pdf

A note on concircular structure space-times

Abstract

Similar works

Full text

Available Versions

AIR Universita degli studi di Milano