# On Compact Trans-Sasakian Manifolds

Authors: Ibrahim Al-Dayel; Sharief Deshmukh
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

We study 3-dimensional compact and simply connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. The first two results deal with finding necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to an Einstein Sasakian manifold and in the third result deals with finding necessary and sufficient condition on a compact and simply connected trans-Sasakian manifold to be homothetic to a Sasakian manifold.


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## Body

## 1. Introduction

It is well known that for an almost contact metric manifoldM,F,ζ,η,g (cf. [1]), the product M¯=M×R has an almost complex structure J, which with product metric g¯ makes M¯,g¯ an almost Hermitian manifold. The properties of the almost Hermitian manifold M¯,J,g¯ control the properties of the almost contact metric manifold M,F,ζ,η,g and provide several structures on M such as a Sasakian structure and a quasi-Sasakian structure (cf. [1–3]). There are known sixteen different types of structures on M¯,J,g¯ (cf. [4]), and using the structure in the class W4 on M¯,J,g, a structure F,ζ,η,g,α,β was introduced on M, which is called trans-Sasakian structure (cf. [5]), that generalizes Sasakian structure, Kenmotsu structure, and cosymplectic structure on a contact metric manifold (cf. [2, 3]), where αandβ being the real functions defined on M.Recall that a trans-Sasakian manifoldM,F,ζ,η,g,α,β is called a trans-Sasakain manifold of type α,β, and trans-Sasakian manifolds of type 0,0, α,0, and 0,β are called a cosymplectic, a α-Sasakian, and a β-Kenmotsu manifolds, respectively. It is on account of a result proved in [6] that a trans-Sasakian manifold of dimension five or greater than five reduces to a cosymplectic manifold, a α-Sasakian manifold, or a β-Kenmotsu manifold, so there is an emphasis on studying three-dimensional trans-Sasakian manifolds.Among other questions, finding conditions under which a compact 3-dimensional trans-Sasakian manifoldM,F,ζ,η,g is homothetic to a Sasakian manifold is of prime importance. The geometry of 3-dimensional trans-Sasakian manifold is also important owing to Thurston’s conjecture (cf. [7]), and fetching conditions on a 3-dimensional trans-Sasakian manifold M,F,ζ,η,g in matching it among Thurston’s eight geometries becomes more interesting. It is worth noting that in Thurston’s eight geometries, the first place is occupied by the spherical geometry S3.In ([8–13]), the authors have studied compact 3-dimensional trans-Sasakian manifolds with some suitable restrictions on functions αandβ appearing in the definition of a trans-Sasakian manifold for getting conditions under which a trans-Sasakian manifold is homothetic to a Sasakian manifold. In particular, it is known that a 3-dimensional compact simply connected trans-Sasakian manifold M,F,ζ,η,g,α,β satisfying Poisson equations Δα=β and Δα=α2β, respectively, is necessarily homothetic to a Sasakian manifold (cf. [10]).An interesting work on 3-dimensional trans-Sasakian manifolds is found in [14, 15], where the authors have considered other aspects in Thurston’s eight geometries. In [10], it is asked whether the function β on a 3-dimensional compact trans-Sasakian manifold M,F,ζ,η,g,α,β satisfying gradβ=ζβζ necessitates the trans-Sasakian manifold to be homothetic to a Sasakian manifold. In [15], it is shown that this question has negative answer.Einstein Sasakian manifolds are very important due to their geometric importance (cf. [16]). In this paper, in our first two results, we find necessary and sufficient conditions on a compact simply connected 3-dimensional trans-Sasakian manifold M,F,ζ,η,g,α,β to be homothetic to an Einstein Sasakian manifold, and in the third, we find a necessary and sufficient condition on a compact simply connected 3-dimensional trans-Sasakian M,F,ζ,η,g,α,β to be homothetic to a Sasakian manifold.In the first result, we consider a compact and simply connected trans-Sasakian manifoldM,F,ζ,η,g,α,β of positive constant scalar curvature τ, the function β satisfying Fischer-Marsden equation shows that the functions αandβ are related to τ by the inequality βα2−β2−τ/4≥0, and the Ricci operator Q satisfying Codazzi-type equation with respect to vector field ζ necessarily implies that M,F,ζ,η,g,α,β is homothetic to an Einstein Sasakian manifold. In the second result, we show that a compact simply connected trans-Sasakian manifold with function α constant along the integral curves of ζ, scalar curvature τ satisfying the inequality α6α2−τ≥0, and the Ricci operator Q satisfying Codazzi-type equation with respect to vector field ζ necessarily imply that M,F,ζ,η,g,α,β is homothetic to an Einstein Sasakian manifold. Finally, in the last result, we show that on a compact and simply connected trans-Sasakian manifold, the function β satisfies the differential inequality ζβ2≤−2β3, and vector fields ∇Qgradα,ζ,ζ are orthogonal, which necessarily imply that M,F,ζ,η,g,α,β is homothetic to a Sasakian manifold, where the covariant derivative ∇QU,ζ=∇UQζ−Q∇Uζ for a smooth vector field U on M.


## 2. Preliminaries

LetM,F,ζ,η,g be an almost contact metric manifold dimM=3, where F being a 1,1-tensor field, ζ a unit vector field, and η smooth 1-form dual to ζ with respect to the Riemannian metric g satisfying
(1)F2=−I+η⊗ζ,Fζ=0,η∘F=0,gFU1,FU2=gU1,U2−ηU1ηU2,U1,U2∈ΓTM,where ΓTM is the space of smooth sections of the tangent bundle TM (cf. [1]). If there exist functions αandβ on an almost contact metric manifold M,F,ζ,η,g satisfying
(2)∇FU1,U2=αgU1,U2ζ−ηU2U1+βgFU1,U2ζ−ηU2FU1,then M,F,ζ,η,g,α,β is said to be a trans-Sasakian manifold, where ∇FU1,U2=∇U1FU2−F∇U1U2, U1,U2∈ΓTM, and ∇ is the Levi-Civita connection with respect to the metric g (cf. [8–15]). Using equations (1) and (2), it follows that
(3)∇Uζ=−αFU+βU−ηUζ,U∈ΓTM.Using the Ricci tensorRic of a Riemannian manifold M,g, the Ricci operator Q is defined by RicU1,U2=gQU1,U2 and U1,U2∈ΓTM. We have the following for a 3-dimensional trans-Sasakian manifold M,F,ζ,η,g,α,β:
(4)ζα=−2αβ,(5)Qζ=Fgradα−gradβ+2α2−β2ζ−ζβζ.Note that equation (3) implies
(6)divζ=2β,and using this equation together with equation (4), we have
(7)divαkζ=kαk−1ζα+αkdivζ=−2kαkβ+2αkβ=−2k−1αkβ.Thus, on compact 3-dimensional trans-Sasakian manifoldM,F,ζ,η,g,α,β, using equation (6) and the above equation, we have
(8)∫Mβ=0,∫Mαkβ=0fork≠1.Now, we state the following result of Okumura.Theorem 1.
[17] Let M,g be a Riemannian manifold. If M admits a Killing vector field ζ of constant length satisfying
(9)c2∇U1∇U2ζ−∇∇U1U2ζ=gU2,ζU1−gU1,U2ζfor nonzero constant c and any vector fields U1 and U2, then M is homothetic to a Sasakian manifold.For a smooth functionh on the Riemannian manifold M,g, then the operator Ah defined by
(10)AhU=∇Ugradh,U∈ΓTMis called the Hessian operator of h, and it is a symmetric operator. Moreover, the Hessian Hessh of h is defined by
(11)HesshU1,U2=gAhU1,U2,U1,U2∈ΓTM.The Laplace operatorΔ on M,g is defined by Δh=divgradh, and we also have
(12)Δh=trAh.Fischer-Marsden differential equation on a Riemannian manifoldM,g is (cf. [18])
(13)Δhg+hRic=Hessh.


## 3. Trans-Sasakian Manifolds Homothetic to Einstein Sasakian Manifolds

In this section, we find necessary and sufficient conditions for a compact and simply connected 3-dimensional trans-Sasakian manifoldM,F,ζ,η,g,α,β to be homothetic to an Einstein Sasakian manifold.Theorem 2.
A compact and simply connected 3-dimensional trans-Sasakian manifoldM,F,ζ,η,g,α,β with positive constant scalar curvature τ and the function β a solution of Fischer-Marsden equation satisfying
(14)βα2−β2−τ4≥0is homothetic to an Einstein Sasakian manifold of positive scalar curvature, if and only if, the Ricci operator Q satisfies
(15)∇QU,ζ=∇Qζ,U,U∈ΓTM.Proof.
SupposeM,F,ζ,η,g,α,β is a compact simply connected 3-dimensional trans-Sasakian manifold satisfying the hypothesis. Then, equation (13) gives
(16)Δβg+βRic=Hessβ,and taking trace in above equation and using equation (12), we have
(17)Δβ=−τ2β.Note that by equation (3), we have ∇ζζ=0, and therefore, Hessβζ,ζ=ζζβ. Using this equation and equation (17) in equation (16), we get
(18)−τ2β+βRicζ,ζ=ζζβ.Now, using equation (5), we have Ricζ,ζ=2α2−β2−ζβ. Thus, the above equation becomes
(19)−τ2β+2βα2−β2−ζβ=ζζβ.Using equation (6), we have divζβζ=ζζβ+2βζβ, and inserting it in the above equation, we conclude
(20)−τ2β+2βα2−β2=divζβζ.Integrating the above equation, we get(21)∫Mβα2−β2−τ4=0.Using the inequality in the statement, we conclude(22)βα2−β2−τ4=0.SinceM is simply connected, it is connected, and therefore equation (22) implies either (i) β=0 or (ii) α2−β2−τ/4=0. Suppose (ii) holds, then as τ is a constant, we get ζα2=ζβ2, which in view of equation (4) implies βζβ=−2α2β; that is, 3β2ζβ=−6α2β2. Thus, we have
(23)ζβ3=−6α2β2.Using equation (6), we have divβ3ζ=ζβ3+2β4, and inserting it in above equation, we get
(24)divβ3ζ=2β2β2−3α2.Integrating the above equation, we get(25)∫Mβ23α2−β2=0.Now, using (ii) in above integral, we have(26)∫Mβ22α2+τ4=0, and since the scalar curvature τ>0, through above integral, we conclude that β=0. Thus, using equations (2), (3), (4), and (5), take the forms
(27)∇FU1,U2=αgU1,U2ζ−ηU2U1,∇Uζ=−αFU,(28)ζα=0,Qζ=Fgradα+2α2ζ.Taking the covariant derivative in the second equation of equation (28), we get
(29)∇QU,ζ+Q∇Uζ=∇FU,gradα+FAαU+4αUαζ+2α2∇Uζ,and using equation (27) in above equation, we arrive at
(30)∇QU,ζ−αQFU=5αUαζ+FAαU−2α3FU,U∈ΓTM.Now, using the Codazzi equation type condition onQ in the hypothesis, we get
(31)∇Qζ,U−αQFU=5αUαζ+FAαU−2α3FU,U∈ΓTM.Using the second equation in equation (27), we compute the Lie derivative of g with respect to ζ to conclude
(32)£ζgU1,U2=−αgFU1,U2−αgFU2,U1=0,that is, ζ is a Killing vector field and that the flow of ζ consists of isometries of the Riemannian manifold M. Thus, we have
(33)£ζQU=0,U∈ΓTM,and using equation (27), we conclude
(34)∇Qζ,U=αQFU−αFQU,U∈ΓTM.Combining the above equation with equation (31), we have
(35)−αFQU=5αUαζ+FAαU−2α3FU,U∈ΓTM.Taking the inner product withζ in above equation, we conclude
(36)αUα=0,U∈ΓTM.We claim thatM being simply connected, α≠0; for if α=0, then by equation (27), we see that ζ is parallel and that η is closed, which implies η is exact; that is, η=df for a smooth function f on M. This implies ζ=gradf, and M being compact, there is a point q∈M such that gradfq=0, and we get ζq=0, a contradiction to the fact that ζ is a unit vector field. Hence, α≠0, and equation (36) implies Uα=0,U∈ΓTM; that is, α is a nonzero constant.Now, equation (28) gives Qζ=2α2ζ, and taking the covariant derivative in this equation yields
(37)∇QU,ζ−αQFU=−2α3FU.Using the condition in the hypothesis and equation (34) with α≠0, in above equation, we get
(38)−FQU=−2α2FU,U∈ΓTM.OperatingF on above equation while using equation (1) and Qζ=2α2ζ, we conclude
(39)QU=2α2U,U∈ΓTM.This proves thatM is an Einstein manifold. Finally, using equation (27), with α a nonzero constant, we compute
(40)∇U1∇U2ζ−∇∇U1U2ζ=−α∇U1FU2−F∇U1U2=−α∇FU1,U2=α2gU2,ζU1−gU1,U2ζ.Hence, by Theorem1, we conclude that M is homothetic to a compact simply connected Einstein Sasakian manifold of positive scalar curvature. The converse is trivial.Theorem 3.
A compact and simply connected 3-dimensional trans-Sasakian manifoldM,F,ζ,η,g,α,β satisfying ζα=0 and the scalar curvature τ satisfying
(41)α6α2−τ≥0is homothetic to an Einstein Sasakian manifold, if and only if, the Ricci operator Q satisfies
(42)∇QU,ζ=∇Qζ,U,U∈ΓTM.Proof.
SupposeM,F,ζ,η,g,α,β is a compact and simply connected 3-dimensional trans-Sasakian manifold satisfying the hypothesis. Then, using equation (4) and the condition ζα=0, we get αβ=0. However, if α=0, then equation (3) implies that η is closed, and as seen in the proof of Theorem 2, we get a contradiction owing to simply connectedness of M. Thus, α≠0, and on connected M, equation αβ=0 implies that β=0. Therefore, equations (27) and (28) hold. Now, using equation (28), we have Qζ=Fgradα+2α2ζ, which gives
(43)−gradα=FQζ.Taking covariant derivative in above equation, we have(44)−AαU=∇FU,Qζ+F∇QU,ζ+FQ∇Uζ,U∈ΓTM,and using equation (27), we get
(45)−AαU=αRicU,ζζ−Ricζ,ζU+F∇QU,ζ−αFQFU,U∈ΓTM.Using condition in the hypothesis, we have(46)−AαU=αRicU,ζζ−Ricζ,ζU+F∇Qζ,U−αFQFU,U∈ΓTM.Also, equation (27) implies that ζ is a Killing vector field, and therefore, using its outcome equation (34) as well as equation (28), we conclude
(47)−AαU=αRicU,ζζ−2α2U+FαQFU−αFQU−αFQFU,U∈ΓTM.That is, on using equation (1), we have
(48)AαU=α2α2U−QU,U∈ΓTM.Thus, we have(49)Δα=α6α2−τ,and integrating the above equation, we get
(50)∫Mα6α2−τ=0.Using the condition in the hypothesis, we haveα6α2−τ=0, and as α≠0, we conclude τ=6α2. Consequently, equation (49) implies that Δα=0, and M being compact, we conclude that α is a constant. Thus, α being nonzero constant, equation (48) gives
(51)QU=2α2U,U∈ΓTM.This proves thatM is an Einstein manifold, and using equation (27), we see that the unit Killing vector field ζ satisfies
(52)∇U1∇U2ζ−∇∇U1U2ζ=α2gU2,ζU1−gU1,U2ζ,U1,U2∈ΓTM.Hence, in view of Theorem1, we conclude that M is homothetic to an Einstein Sasakian manifold. The converse is trivial.Theorem 4.
A compact and simply connected 3-dimensional trans-Sasakian manifoldM,F,ζ,η,g,α,β satisfying
(53)ζβ2≤−2β3is homothetic to a Sasakian manifold, if and only if, the vector fields ∇Qgradα,ζ and ζ are orthogonal.Proof.
SupposeM,F,ζ,η,g,α,β is a compact and simply connected 3-dimensional trans-Sasakian manifold satisfying the hypothesis. Then, using equation (5), we have
(54)gradβ=Fgradα−Qζ+2α2−β2ζ−ζβζ,and taking covariant derivative in above equation with respect to U∈ΓTM, while using equation (3), we get
(55)AβU=∇FU,gradα+FAαU−∇QU,ζ−Q−αFU+βU−βηUζ+2Uα2−β2ζ+2α2−β2−αFU+βU−βηUζ−Uζβζ−ζβ−αFU+βU−βηUζ.Taking trace in above equation and noting the following outcome of equation (2),
(56)∑i=13∇Fei,ei=2αζ, for a local orthonormal frame e1,e2,e3 on M, we get
(57)Δβ=−2αζα−12ζτ−βτ+βRicζ,ζ+2ζα2−β2+4βα2−β2−ζζβ−2βζβ,where we have used TrFAα=0, TrQF=0, TrF=0, and well known formula
(58)∑i=13∇Qei,ei=12gradτ.Now, using equation (4) and Ricζ,ζ=2α2−β2−ζβ in equation (57), we have
(59)Δβ=−12ζτ−βτ−6β3+10α2β−8βζβ−ζζβ,that is,
(60)Δβ=−12ζτ−βτ−6β3+10α2β−ζζβ+4β2.Note that on using equation (6), we have 1/2divτζ=1/2ζτ+βτ and
(61)divζβ+4β2ζ=ζζβ+4β2+2βζβ+4β2.Inserting these equations in equation (60), we arrive at
(62)Δβ=−12divτζ−6β3+10α2β+2βζβ+4β2−divζβ+4β2ζ,and integrating the above equation while keeping in mind equation (8), we get
(63)∫Mζβ2+2β3=0.Using the condition in the hypothesis, we concludeζβ2=−2β3 and βζβ=−β3; that is, 3β2ζβ=−3β4. We get ζβ3=−3β4, which in view of equation (6) implies
(64)divβ3ζ=−β4.Integrating the above equation yieldsβ=0. Thus, equations (27) and (28) are now available to us. Taking covariant derivative in second equation of equation (28) with respect to U∈ΓTM, we get
(65)∇QU,ζ−αQFU=∇FU,gradα+FAαU+4αUαζ−2α3FU,where we have used the second equation in equation (27). Now, using equation (27) and noting that ζα=0 in above equation, we conclude
(66)∇QU,ζ−αQFU=5αUαζ+FAαU−2α3FU.Taking the inner product withζ in above equation, we get
(67)5αUα=g∇QU,ζ,ζ−αgFU,Qζ.Now, using equation (28), we have
(68)gFU,Qζ=gFU,Fgradα+2α2ζ=Uα,and inserting it in equation (67), we get
(69)6αUα=g∇QU,ζ,ζ,U∈ΓTM.TakingU=gradα in above equation and using the condition in the hypothesis, we conclude
(70)αgradα2=0.Note thatM being compact and simply connected, α is not allowed to be zero. Hence, the above equation implies that α is nonzero constant. Thus, we have by virtue of equation (27) that
(71)∇U1∇U2ζ−∇∇U1U2ζ=α2gU2,ζU1−gU1,U2ζ,U1,U2∈ΓTM.This proves thatM is homothetic to a Sasakian manifold. The converse is trivial.



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