In this note we show that Lorentzian Concircular Structure manifolds (LCS)_n
coincide with Generalized Robertson-Walker space-times.Comment: 2 page

Mantica, Carlo Alberto

Molinari, Luca Guido

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arXiv

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

AIR Universita degli studi di Milano

Commun. Korean Math. Soc. 34 (2019), No. 2, pp. 633–635https://doi.org/10.4134/CKMS.c180138pISSN: 1225-1763 / eISSN: 2234-3024A NOTE ON CONCIRCULAR STRUCTURE SPACE-TIMESCarlo Alberto Mantica and Luca Guido MolinariReprinted from theCommunications of the Korean Mathematical SocietyVol. 34, No. 2, April 2019c©2019 Korean Mathematical SocietyCommun. Korean Math. Soc. 34 (2019), No. 2, pp. 633–635https://doi.org/10.4134/CKMS.c180138pISSN: 1225-1763 / eISSN: 2234-3024A NOTE ON CONCIRCULAR STRUCTURE SPACE-TIMESCarlo Alberto Mantica and Luca Guido MolinariAbstract. In this note we show that Lorentzian Concircular Structuremanifolds (LCS)n coincide with Generalized Robertson-Walker space-times.Generalized Robertson-Walker (GRW) space-times were introduced in 1995by 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 Lorentzianmanifolds characterised by a metricgijdxidxj = −(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 widegeneralization of space-times for cosmological models.In 2003 A. A. Shaikh [12] introduced the notion of Lorentzian ConcircularStructure (LCS)n. It is a Lorentzian manifold endowed with a unit time-likeconcircular 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)nmanifolds [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 arethe 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 scalarfunction.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 Society633634 C. A. MANTICA AND L. G. MOLINARIFialkow 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 manifoldis 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-formingproperty by Yano becomes precisely Eq. (2), with generic scalar field ϕ. Basedon Chen’s theorem we proved:Proposition 4 (Mantica and Molinari, [7,8]). A n > 3 dimensional Lorentzianmanifold 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)nspace-times) we evaluateRjklmum = [∇j ,∇k]ul = (hkl∇j − hjl∇k)ϕ− ϕ2(ujgkl − ukgjl),where hkl = ukul + gkl. Contraction with gjl givesRkmum = uk[um∇mϕ+ (n− 1)ϕ2]− (n− 2)∇kϕ.(4)If (3) holds, then uk is an eigenvector of the Ricci tensor, and we conclude thata (LCS)n manifold is a GRW space-time.If Rkmum = 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 ofconstant mean curvature in generalized Robertson-Walker spacetimes, Gen. RelativityGravitation 27 (1995), no. 1, 71–84.[2] K. Arslan, R. Deszcz, R. Ezentas, M. Hotlos´, and C. Murathan, On generalizedRobertson-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 generalizedRobertson-Walker space-times, J. Math. Phys. 57 (2016), no. 10, 102502, 6 pp.[8] , Generalized Robertson-Walker spacetimes—a survey, Int. J. Geom. MethodsMod. 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 andKilling 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. Tokyo20 (1944), 340–345.Carlo Alberto ManticaI.I.S. LagrangeVia L. Modignani 65, 20161, Milano, ItalyEmail address: carloalberto.mantica@libero.itLuca Guido MolinariPhysics DepartmentUniversita` degli Studi di Milano and I.N.F.N. sez. MilanoVia Celoria 16, 20133 Milano, ItalyEmail address: luca.molinari@unimi.it

A note on concircular structure space-times

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A note on Concircular Structure space-times

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