In this article, we characterize a Lorentzian manifold $\mathcal{M}$ with a
semi-symmetric metric connection. At first, we consider a semi-symmetric metric
connection whose curvature tensor vanishes and establish that if the associated
vector field is a unit time-like torse-forming vector field, then $\mathcal{M}$
becomes a perfect fluid spacetime. Moreover, we prove that if $\mathcal{M}$
admits a semi-symmetric metric connection whose Ricci tensor is symmetric and
torsion tensor is recurrent, then $\mathcal{M}$ represents a generalized
Robertson-Walker spacetime. Also, we show that if the associated vector field
of a semi-symmetric metric connection whose curvature tensor vanishes is a $f-$
Ric vector field, then the manifold is Einstein and if the associated vector
field is a torqued vector field, then the manifold becomes a perfect fluid
spacetime. Finally, we apply this connection to investigate Ricci solitons

De, Krishnendu

De, Uday Chand

Güler, Sinem

Publicationes Mathematicae Debrecen

English

arXiv

In this article, we characterize a Lorentzian manifold M with a semisymmetric metric connection. At first, we consider a semi-symmetric metric connection whose curvature tensor vanishes and establish that if the associated vector
field is a unit time-like torse-forming vector field, then M becomes a perfect fluid
spacetime. Moreover, we prove that if M admits a semi-symmetric metric connection whose Ricci tensor is symmetric and torsion tensor is recurrent, then M
represents a generalized Robertson-Walker spacetime. Also, we show that if the
associated vector field of a semi-symmetric metric connection whose curvature
tensor vanishes is a f− Ric vector field, then the manifold is Einstein and if
the associated vector field is a torqued vector field, then the manifold becomes
a perfect fluid spacetime. Finally, we apply this connection to investigate Ricci
solitons

Chand De, Uday

İstanbul Sabahattin Zaim University Institutional Repository

arXiv:2406.16108v1  [math.DG]  23 Jun 2024CHARACTERIZATIONS OF A LORENTZIAN MANIFOLD WITHA SEMI-SYMMETRIC METRIC CONNECTIONUDAY CHAND DE , KRISHNENDU DE AND SİNEM GÜLER ∗Abstract. In this article, we characterize a Lorentzian manifold M with a semi-symmetric metric connection. At first, we consider a semi-symmetric metric con-nection whose curvature tensor vanishes and establish that if the associated vectorfield is a unit time-like torse-forming vector field, then M becomes a perfect fluidspacetime. Moreover, we prove that if M admits a semi-symmetric metric con-nection whose Ricci tensor is symmetric and torsion tensor is recurrent, then Mrepresents a generalized Robertson-Walker spacetime. Also, we show that if theassociated vector field of a semi-symmetric metric connection whose curvaturetensor vanishes is a f− Ric vector field, then the manifold is Einstein and ifthe associated vector field is a torqued vector field, then the manifold becomesa perfect fluid spacetime. Finally, we apply this connection to investigate Riccisolitons.1. IntroductionIn this study, we investigate several vector fields and solitons on a Lorentzianmanifold M obeying a semi-symmetric metric connection (briefly, SSMC). The no-tion of a semi-symmetric linear connection was first introduced by Friedman andSchouten on a differentiable manifold many years ago. The concept of metric con-nection with torsion tensor T was then presented by Hayden [16] in 1932 on aRiemannian manifold. Yano [23] made a methodical investigation of SSMC in 1970.On a Lorentzian manifold M, a linear connection ∇ is named semi-symmetric ifT described byT (U1, V1) = ∇U1V1 −∇V1U1 − [U1, V1]obeys(1.1) T (U1, V1) = π(V1)U1 − π(U1)V1,0The Mathematics subject classification 2020: 83C05, 53C05, 53C20, 53C50.Lorentzian manifolds; semi-symmetric metric connection; perfect fluid spacetimes.∗ Corresponding author12where π is defined by π(U1) = g(U1, ξ), for a fixed vector field ξ (also, namedassociated vector field ). If we replace U1 and V1, respectively, by φU1 and φV1 in theright side of (1.1), then the SSMC ∇ changes into a quarter-symmetric connection[13], φ being a (1,1)- tensor field.Again, if a SSMC ∇ on M fulfills(1.2) (∇U1g)(V1, Z1) = 0,then ∇ is named metric [23], if not, it is non-metric [16].We know [23] that if ∇ is a SSMC and D be a Levi-Civita connection, then(1.3) ∇U1V1 = DU1V1 + η(V1)U1 − g(U1, V1)ξ.Further, it is also known [23] that if R and R denote the curvature tensors of ∇ andD, respectively, thenR(U1, V1)Z1 = R(U1, V1)Z1 − L1(V1, Z1)U1 + L1(U1, Z1)V1−g(V1, Z1)L2U1 + g(U1, Z1)L2V1,(1.4)where L1, a (0,2) tensor and L2, a (1,1) tensor are defined by(1.5) L1(U1, V1) = (DU1π)(V1)− π(U1)π(V1) +12π(ξ)g(U1, V1)and(1.6) g(L2U1, V1) = L1(U1, V1)for any vector fields U1 and V1.In a local coordinate system, equations (1.4) and (1.5) can be written as follows:(1.7) Rhijk = Rhijk − ghkπij + gikπhj − gijπhj + ghkπik,(1.8) πik = Divk − vivk +12gikv, v = vhvhin which Rhijk and Rhijk denote the curvature tensor of ∇ and D, respectively.The idea of the SSMC was investigated by many researchers (see, [3], [23], [25],[26], [27], [28]).The current paper concerns with n-dimensional spacetimes (that is, a connectedtime-oriented Lorentz manifold) (M, g). For ψ > 0, a smooth function (also calledscale factor, or warping function), if M = −I×ψ,M, where I is the open intervalof R , Mn−1 denotes the Riemannian manifold, then M is named a generalizedRobertson-Walker (briefly, GRW ) spacetime [1]. When the dimension of M is three3and of constant sectional curvature, this spacetime represents a Robertson-Walker(briefly, RW ) spacetime. In this context we may mention the work of Ünal andShenawy on warped product manifolds ([21], [22]).Due to the absence of a stress tensor and heat conduction terms correspondingto viscosity, the fluid is referred to as perfect and the energy momentum tensor Thkis written by(1.9) Thk = (σ + p)vhvk + pghk,in which p and σ stand for the perfect fluid’s isotropic pressure and energy density,respectively [20].For a gravitational constant κ, the Einstein’s field equations without a cosmolog-ical constant is given by(1.10) Rhk −R2ghk = κThk,where Rhk = Rihki and R = ghkRhk are the Ricci tensor and the scalar curvature,respectively.A spacetime M is named a perfect fluid (briefly PF ) spacetime if the non-vanishing Ricci tensor Rhk fulfills(1.11) Rhk = αghk + βvhvkwhere α and β indicate smooth functions on M. In the last equation, g is theLorentzian metric of M and vh, the velocity vector is defined by ghkvhvk = −1 andvh = ghkvk.Combining the equations (1.9), (1.10) and (1.11), we acquire(1.12) β = k2(p + σ), α =k2(p − σ)2− n.In [20], O’Neill established that a Robertson-Walker spacetime is a PF spacetime.Every 4-dimensional generalized Robertson-Walker spacetime is a PF spacetime ifand only if it is a Robertson-Walker spacetime [12]. For additional insights about thePF and GRW spacetimes, we refer( [2], [4], [5], [8], [9], [11], [19]) and its references.Definition 1.1. For a non vanishing 1-form ωk and a scalar function φ if therelation ∇kuh = ωkuh + φgkh holds, then the vector field u is called torse-forming.4This notion was introduced by Yano [24] on a Riemannian manifold. It is notedthat the foregoing torse-forming condition becomes ∇kuh = φ(ukuh + gkh), for aunit time-like vector.In [19], Mantica and Molinari and in [4], Chen have proven the subsequent theo-rems:Let M be a Lorentzian manifold of dimension n (n ≥ 3).Theorem 1.1. [19] M is a GRW spacetime if and only if it admits a unit time-liketorse-forming vector field: ∇kvj = ϕ(gij + vjvi), which is also an eigenvector of theRicci tensor.Theorem 1.2. [4] M is a GRW spacetime if and only if it admits a time-likeconcircular vector field.In this article, we consider a SSMC and prove the following:Theorem 1.3. If the associated vector field of M obeying a SSMC whose curvaturetensor vanishes is a unit time-like torse-forming vector field, then M becomes a PFspacetime.Definition 1.2. In a semi-Riemannian manifold with a SSMC ∇, the torsion tensorT is called recurrent if it obeys ∇kThij = AkThij, Ak being a covariant vector.Here, we choose a SSMC whose torsion tensor is recurrent and establish thesubsequent result:Theorem 1.4. Let M admit a SSMC whose Ricci tensor is symmetric and torsiontensor is recurrent. Then the manifold M represents a GRW spacetime.Definition 1.3. If ∇kvh = fRkh, then vh is named a f -Ric vector field, where f isa constant.The foregoing idea of f -Ric vector field was presented by Hinterleitner and Kiosak[17] on a Riemannian manifold.Definition 1.4. [6]A vector field is called a torqued vector field if ∇kvh = fgkh +ωkvh and ωkvk = 0, where f is a real constant and 1-form ω is named the torquedform of v.5In this article, we consider a SSMC whose associated vector field is f− Ric andtorqued vector field, respectively and prove the following results:Theorem 1.5. If M admits a SSMC whose curvature tensor vanishes and theassociated vector field is f− Ric, then M is Einstein.Theorem 1.6. Let M admit a SSMC whose curvature tensor vanishes. If theassociated vector field is torqued, then M becomes a PF spacetime.Hamilton [15] introduces the concept of Ricci flow as a solution to the challengeof obtaining a canonical metric on a smooth manifold. Let M be satisfied by anevolution equation ∂∂tghk(t) = −2Rhk, then it is called Ricci flow [15]. If the metricof M obeys the relation(1.13) Lvghk + 2Rhk + 2λghk = 0,then it is called a Ricci soliton [14], where Lv is the Lie derivative operator andλ denotes a real constant. Here, v is called the potential vector field of the solitons.The Ricci soliton is named shrinking, expanding or steady, if λ is negative, positive,or zero, respectively.Many mathematicians were fascinated in the geometry of Ricci solitons over thepast few years. Some interesting findings on this solitons have been investigated in([2], [7], [10], [18]) and also by others.The above investigations motivate us to SSMC and establish the subsequent:Theorem 1.7. Let us suppose that M admit a SSMC whose curvature tensor van-ishes and the associated vector field is a unit time-like torse-forming vector field. IfM admits a Ricci soliton (g, v, λ), then the soliton is shrinking if φ < 1, expandingif φ > 1, or steady if φ = 1, respectively. Additionally, the integral curves that vgenerates are geodesics.2. Semi-symmetric metric connectionAgreement: Throughout this article, we assume that the associated vector vi isa unit time-like vector, that is, vivi = −1.Then, the equation (1.8) becomes(2.1) πik = Divk − vivk −12gik.6Multiplying (1.7) with ghk, we acquire(2.2) Rij = Rij − (n− 2)πij − πgij ,where Rij and Rij are the Ricci tensor of ∇ and D, respectively and π = ghkπhk.Also, we have(2.3) ∇hvk = Dhvk − vhvk − ghk.Proposition 2.1. Rij is symmetric if and only if vi is closed.Proof. From the equation (2.2), we obtainRij −Rji = −(n− 2){πij − πji}= −(n− 2){Divj − vivj −12gij −Djvi + vjvi +12gji}= −(n− 2){Divj −Djvi}.(2.4)If vi is closed, then the foregoing equation reduces to(2.5) Rij −Rji = 0,that is, Rij is symmetric and conversely. 2.1. Proof of the Theorem 1.3. Since by hypothesis, Rhijk = 0, from (1.7), weget(2.6) Rhijk = ghkπij − gikπhj + gijπhj − ghkπik.Let(2.7) ∇kvj = φ(gkj + vjvk).Then from (2.3) it follows that(2.8) Dkvj = (φ+ 1)(gkj + vjvk).Hence, using the foregoing equation in (2.1), we obtain(2.9) πik = (φ−12)gik + φvivk).Now, making use of the previous equation in (2.6), we acquire(2.10) Rhijk = (2φ− 1){ghkgij − ghjgik}+φ{ghkvivj + gijvhvk− gikvhvj − ghjvivk}.Multiplying with gij the above equation yieldsRhk = (2φ− 1)(n − 1)ghk + φ{−ghk + nvhvk − vhvk − vhvk},7which implies(2.11) Rhk = {(n − 1)(2φ − 1)− φ}ghk + (n− 2)φvhvk.The foregoing equation entails that M is a PF spacetime.Hence, the proof is finished.Now, using Theorem 1.1, we may state:Corollary 2.1. A GRW-spacetime admitting a SSMC whose curvature tensor van-ishes is a PF spacetime.Remark 2.1. In [19], Mantica and Molinari established that a GRW spacetimesatisfying divC = 0 represents a PF spacetime, where C indicates the conformalcurvature tensor. Here, curvature condition divC = 0 is replaced by a SSMC whosecurvature tensor vanishes and acquires the same result.For n = 4, comparing the equations (1.11) and (2.11), we have(2.12) αghk + βvhvk = (5φ − 3)ghk + 2φvhvk.Making use of (1.12), the foregoing equation yields(2.13) k2(3p − 2σ) = 3,which implies(2.14) (3p − 2σ) = c (say).Corollary 2.2. A PF spacetime admitting a SSMC satisfies the state equationσ = 32p+ constant.2.2. Proof of the Theorem 1.4. It is known [3] that if a Riemannian manifoldadmits a SSMC ∇ whose torsion tensor T is recurrent, then the curvature tensorRhijk of ∇ is given byRhijk = Rhijk +Ak[viδhj − gijvh]−Aj [vkδhi − gikvh] + vlvl[gjkδhi − gikδhj ].(2.15)8For a Lorentzian manifold the above relation becomesRhijk = Rhijk +Ak[viδhj − gijvh]−Aj[vkδhi − gikvh]− [gjkδhi − gikδhj ],(2.16)since vlvl = −1.First we prove the following lemma:Lemma 2.1. If the torsion tensor T is given by(2.17) ∇kThij = AkThij ,then Ak is gradient.Proof. Let(2.18) f2 = ThijThij,where Thij = ghkTkij .From (2.18), we get2f∇lf = ∇lThijThij + Thij∇lThij= 2Alf2,(2.19)which implies that(2.20) ∇lf = fAl.Therefore, from the above equation, we obtain(2.21) ∇l∇mf −∇m∇lf = ∇mfAl −∇lfAm + f(∇mAl −∇lAm).Since f is a scalar, using the equation (2.20) in (2.21), we acquire(2.22) f(∇mAl −∇lAm) = 0.Since f 6= 0, Al is gradient. From T hij = δhi vj − δhj vi, we get(2.23) T hij = (n− 1)vj ,which implies(2.24) ∇kThij = (n− 1)vkvj .9Again from ∇kThij = AkThij, we obtain(2.25) ∇kThij = AkThij = (n− 1)Akvj .Contracting h and k in (2.16) yields(2.26) Rij = Rij + [Ajvi − gijAhvh].By hypothesis Rij is symmetric. Hence, we have from (2.26)(2.27) Aivj = Ajvi.Multiplying the foregoing equation by vi gives(2.28) Aj = avj ,where a = −Aivi.From (2.24) and (2.25) it follows that(2.29) ∇kvj = Akvj .Using (2.3) in the previous equation yields(2.30) Dhvk = ghk + µhvk,where µh = Ah + vh.From the above equation we conclude that vk is a torseforming vector field withα = β = 1.Here, vh is closed, since by hypothesis the Ricci tensor of the SSMC is symmetric.Also by Lemma 2.1, Ah is closed and hence µh is closed. Therefore, vh is concircular.Hence, using Theorem 1.2 we state that M represents a GRW spacetime.This completes the proof.2.3. Proof of the Theorem 1.5. Since the associated vector field is f− Ric, thenwe have(2.31) ∇kvh = fRhk,f is a constant.Then from (2.3), we acquire(2.32) Dhvk = fRhk + ghk + vhvk.10Hence, using the previous equation in (2.1), we infer(2.33) πik = fRik +12gik.Since by assumption Rhijk = 0. From (1.7), we obtain(2.34) Rhijk = ghkπij − gikπhj + gijπhj − ghkπik.Now, using equation (2.33) in (2.34), we get(2.35) Rhijk = f{ghkRij − gikRhj + gijRhk − ghjRik}+ {ghkgij − ghjgik}.Multiplying with gij the foregoing equation yieldsRhk = f{nRhk −Rhk +Rghk −Rhk}+ (n− 1)ghk,which entails(2.36) Rhk =fR+ (n− 1)1− (n− 2)fghk.The above equation states that M is Einstein.This ends the proof.Remark 2.2. From the equation (2.35), we also get the spacetime is of constantcurvature 2f2R+nf+11−(n−2)f .If R = −nf+1f2, then...2.4. Proof of the Theorem 1.6. By hypothesis, v is a torqued vector field. There-fore, we get(2.37) ∇kvh = fghk + ωkvjand ωkvk = 0. Then from (2.3), we obtain(2.38) Dhvk = (f + 1)ghk + (ωh + vh)vk.Hence, using the above equation in (2.1), we acquire(2.39) πik = (f +32)gik + ωivk.Since Rhijk = 0, from (1.7), we get(2.40) Rhijk = ghkπij − gikπhj + gijπhj − ghkπik.11Now, using equation (2.39) in (2.40), we have(2.41) Rhijk = 2(f +32){ghkgij − ghjgik}+ {ghkωi − gikωh}vj + {gijωh − ghjωi}vk.Multiplying with gij , we acquireRhk = 2n(f +32)ghk − 2(f +32)ghk − (n− 1)ωhωk + {ghkωhvk − ωhvk}.(2.42) Rhk = (2n − 2)(f +32)ghk + (n− 2)ωhvk.Again, multiplying with vh yields(2.43) Rhkvh = (2n− 2)(f +32)vk,since ωivi = 0.Also, from (2.42), we get ωhvk = ωkvh. Multiplying the foregoing relation by vk,we have ωh = avh, where a = ωkvk. Hence, the equation (2.42) becomes(2.44) Rhk = (2n − 2)(f +32)ghk + (n− 2)avhvk.Foregoing equation tells that M is a PF spacetime.This accomplishes the proof.3. Applications to Ricci solitons3.1. Proof of the Theorem 1.7. Now, suppose that M with SSMC admits aRicci soliton. Then from equation (1.13), we get(3.1) ∇hvk +∇kvh + 2Rhk + 2λghk = 0.Multiplying by vhvk, we obtain(3.2) Rhkvhvk − λ = 0,since ghkvhvk = −1 and vhvk∇hvk = 0. Again, multiplying the equation (2.11) byvhvk, we infer(3.3) Rhkvhvk = −{(n− 1)(2φ − 1)− φ}+ (n− 2)φ.12Comparing the last two equations, we acquire λ = (1 − n)(φ − 1). This showsthat the Ricci soliton is shrinking if φ < 1, expanding if φ > 1, or steady if φ = 1,respectively.Again, multiplying (3.1) by vh, we find(3.4) vh∇kvh + 2Rhkvh + 2λvk = 0,since vh∇hvk = 0.Again, multiplying (2.11) by vh yields(3.5) Rhkvh = {(n− 1)(2φ − 1)− φ} − (n− 2)φ.Using the above equation in (3.4), we have vh∇kvh = 0, since λ = (1− n)(φ− 1).This ends the proof.References[1] Alias, L., Romero, A. and Sánchez M., Uniqueness of complete spacelike hypersurfaces ofconstant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativ. Gravit. 27(1995), 71-84.[2] Blaga, A.M., Solitons and geometrical structures in a perfect fluid spacetime, Rocky MountainJ. Math. 50 (2020), 41-53.[3] Chaki, M. C. and Konar, A., On a type of semi-symmetric metric on a Riemannian manifolds,J. Pure Math. 1 (1981), 77-80.[4] B. Y. Chen, Pseudo-Riemannian Geometry, δ-invariants and Applications, World Scientific,2011.[5] , A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relativ.Gravit. 46 (2014), 1833 (5 pages).[6] , Rectifying submanifolds of Riemannian manifolds and torqued vector fields, KragujevacJ. Math. 41(1) (2017), 93-103.[7] Chen, B.Y. and Deshmukh, S., Ricci solitons and concurrent vector fields, Balkan J. Geom.Appl., 20 (2015), 14-25.[8] De, U.C., Chaubey, S.K. and Shenawy, S., Perfect fluid spacetimes and Yamabe solitons, JMath Phys. 62, 032501 (2021); https://doi.org/10.1063/5.0033967[9] De, K., De, U.C., Syied, A. A., Turki N. B. and Alsaeed, S., Perfect fluid spacetimes andgradient solitons, Journal of Nonlinear Mathematical Physics https://doi.org/10.1007/s44198-022-00066-5.[10] Deshmukh, S., Alodan, H. and Al-Sodais, H., A Note on Ricci Soliton, Balkan J. Geom. Appl.16, 1 (2011), 48-55.[11] Duggal, K. L. and Sharma, R.,Symmetries of spacetimes and Riemannian manifolds, Mathe-matics and its Applications 487 (Kluwer Academic Press, Boston, London 1999).[12] Gutiérrez, M. and Olea, B., Global decomposition of a Lorentzian manifold as a generalizedRobertson-Walker space, Differ. Geom. Appl. 27 (2009), 146-156.[13] Golab, S., On semi-symmetric and quarter-symmetric linear connections, Tensor (N. S.) 29(1975), 249-254.[14] Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Differ. Geom. 17 (1982),255-306.13[15] Hamilton, R. S., The Ricci flow on surfaces, Contemp. Math. 71 (1988), 237-261.[16] Hayden, H.A., Subspace of space with torsion, Proc. Lon. Math. Soc. 34 (1932), 27-50.[17] Hinterleitner, I. and Volodymyr, A. K., φ(Ric)-vector fields in Riemannian spaces, ArchivumMath. (Brno) 44 (2008), 385-390. 00[18] Karaca, F. and Ozgur, C., Gradient Ricci solitons on multiply warped product manifolds, Filo-mat, 32 (2018), 4221-4228.[19] Mantica, C.A. and Molinari, L.G., Generalized Robertson-Walker spacetimes-A survey, Int. J.Geom. Methods Mod. Phys. 14 (2017), 1730001 (27 pages).[20] O’Neill, B., Semi-Riemannian Geometry with Applications to the Relativity, Academic Press,New York-London, 1983.[21] Ünal, B. and Shenawy, S., 2-Killing vector fields on warped product manifolds, InternationalJournal of Mathematics. 26 (2015). 1550065. 10.1142/S0129167X15500652.[22] Ünal, B. and Shenawy, S., The W2 curvature tensor on warped product manifolds and applica-tions, Int. J. Geom. Methods Mod. Phys. 13 (2016). 10.1142/S0219887816500997.[23] Yano, K., On semi-symmetric metric connection, Revue Roumaine de Math. Pures et Ap-pliques. 15 (1970), 1579-1586.[24] K. Yano, On torse forming direction in a Riemannian space, Proc. Imp. Acad. Tokyo 20 (1944),340–345.[25] Zengin, F.O., Demirbag, S. A., Uysal, S. A. and Yilmaz, H. B., Some vector fields on a Rie-mannian manifold with semi-symmetric metric connection, Bull. Iranian Math. Soc., 38(2012),479-490.[26] Zengin F.O., Uysal S. A. and Demirbag S. A., On sectional curvature of a Riemannian manifoldwith semi-symmetric metric connection, Ann. Polon. Math., 101(2011), 131-138.[27] Zhao, P.B., Fengyun, F. and Yang, X., Some Properties of S-Semi-Symmetric Metric Connec-tions, Int. J. Contemp. Math. Sciences, 6 (2011), 1817-1827.[28] Zhao, P.B., Han, Y. and Fengyun, F., On semi-symmetric metric connection in sub-Riemannian manifolds, Tamkang Journal of Mathematics, 47 (2016), 373-384.Department of Pure Mathematics, University of Calcutta, West Bengal, India.ORCID iD: https://orcid.org/0000-0002-8990-4609Email address: uc−de@yahoo.comDepartment of Mathematics, Kabi Sukanta Mahavidyalaya, The University of Bur-dwan. Bhadreswar, P.O.-Angus, Hooghly, Pin 712221, West Bengal, India., ORCID iD:https://orcid.org/0000-0001-6520-4520Email address: krishnendu.de@outlook.inDepartment of Industrial Engineering, Istanbul Sabahattin Zaim University, Hal-kali, Istanbul, Turkey.Email address: sinem.guler@izu.edu.tr

Characterizations of a Lorentzian Manifold With a Semi-Symmetric Metric Connection

In this article, we characterize a Lorentzian manifold $\mathcal{M}$ with a semi-symmetric metric connection. At first, we consider a semi-symmetric metric connection whose curvature tensor vanishes and establish that if the associated vector field is a unit time-like torse-forming vector field, then $\mathcal{M}$ becomes a perfect fluid spacetime. Moreover, we prove that if $\mathcal{M}$ admits a semi-symmetric metric connection whose Ricci tensor is symmetric and torsion tensor is recurrent, then $\mathcal{M}$ represents a generalized Robertson-Walker spacetime. Also, we show that if the associated vector field of a semi-symmetric metric connection whose curvature tensor vanishes is a $f-$ Ric vector field, then the manifold is Einstein and if the associated vector field is a torqued vector field, then the manifold becomes a perfect fluid spacetime. Finally, we apply this connection to investigate Ricci solitons

Characterizations of a Lorentzian Manifold with a semi-symmetric metric connection

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