Research article

-Ricci tensor on (κ,μ)-contact manifolds

  • Received: 30 December 2021 Revised: 07 March 2022 Accepted: 10 March 2022 Published: 13 April 2022
  • MSC : 53C21, 53D10, 53D15

  • We introduce the notion of semi-symmetric -Ricci tensor and illustrate that a non-Sasakian (κ,μ)-contact manifold is -Ricci semi-symmetric or has parallel -Ricci operator if and only if it is -Ricci flat. Then we find that among the non-Sasakian (κ,μ)-contact manifolds with the same Boeckx invariant IM, only one is -Ricci flat, so we can think of it as the representative of such class. We also give two methods to construct -Ricci flat (κ,μ)-contact manifolds.

    Citation: Rongsheng Ma, Donghe Pei. -Ricci tensor on (κ,μ)-contact manifolds[J]. AIMS Mathematics, 2022, 7(7): 11519-11528. doi: 10.3934/math.2022642

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  • We introduce the notion of semi-symmetric -Ricci tensor and illustrate that a non-Sasakian (κ,μ)-contact manifold is -Ricci semi-symmetric or has parallel -Ricci operator if and only if it is -Ricci flat. Then we find that among the non-Sasakian (κ,μ)-contact manifolds with the same Boeckx invariant IM, only one is -Ricci flat, so we can think of it as the representative of such class. We also give two methods to construct -Ricci flat (κ,μ)-contact manifolds.



    In differential geometry, curvature tensor R has a very important influence on the properties of manifolds. The authors of this paper have investigated some properties of Lorentzian generalized Sasakian space-forms which are related to the curvature tensor R in [1]. We know many manifolds have special curvature properties. For instance, if (M2n+1,ϕ,ξ,η,g) is a Sasakian manifold, then it satisfies

    R(X,U)ξ=η(U)Xη(X)U,

    where X,U are vector fields on M. So we can classify manifolds according to their special curvature properties. As the meaningful generalization of the curvature condition of Sasakian manifold, D. E. Blair, T. Koufogiorgos and B. J. Papantoniou introduced (κ,μ)-contact manifold in [2]. If a contact manifold (M2n+1,ϕ,ξ,η,g) satisfies

    R(X,U)ξ=κ(η(U)Xη(X)U)+μ(η(U)hXη(X)hU), (1.1)

    where (κ,μ)R2, h is half of the Lie derivative of structure tensor ϕ along Reeb vector field ξ. This curvature condition of a non-Sasakian contact manifold is attractive to mathematicians not only because it determines the curvature tensor R completely but also remains unchange under Da-homothetic deformation while κ and μ vary. Moreover (κ,μ)-contact manifolds are examples of H-contact manifolds (see [3]) or other remarkable contact manifolds. In Eq 1.1, when κ and μ are non constant smooth functions, it is a generalized (κ,μ)-contact manifold and this condition can only occur in dimension three (see [4]).

    Many mathematicians have studied (κ,μ)-contact manifolds from many aspects. The most excellent work was done by E. Boeckx who showed that non-Sasakian (κ,μ)-contact manifolds are locally homogeneous and locally strongly ϕ-symmetric in [5]. Moreover E. Boeckx introduced an invariant IM which called Boeckx invariant and he used it to give a full classification of non-Sasakian (κ,μ)-contact manifolds (see [6]). When IM1, the classification of (κ,μ)-contact manifolds was given by E. Loiudice and A. Lotta in [7]. X. Chen investigated (κ,μ)-contact manifolds admit weakly Einstein metrics and gave the classification of them. D. S. Patra and A. Ghosh prove that if a non-Sasakian (κ,μ)-contact manifold is a non-trivial complete Einstein-type manifold, then it is flat for n=1 or locally isometric to En+1×Sn(4) for n>1 in [8].

    We know the Ricci tensor is an important tensor on the manifold induced by curvature tensor. In contact geometry, there is another important tensor, that is the -Ricci tensor Ric. In 1959, S. Tachibana gave the notion of -Ricci tensor in complex geometric (see [9,10]). In 2002, T. Hamada extended this notion to almost contact manifolds in [11]. Like Ricci tensor, -Ricci tensor is also a trace but may not be symmetric. We know that the asymmetry tensor makes no much geometric or physical meaning. So when the -Ricci tensor is symmetric, we can study it directly; when it is not symmetric, we first want to know how can it be symmetric or we study the symmetric part of it.

    After the -Ricci tensor of contact manifold being proposed, it attracts great interest of mathematicians. The notion of -Ricci solitons was intrduced by G. Kaimakamis and K. Panagiotidou and it was first defined on real hypersurfaces in complex space form (see [12]). Since then many mathematicians have studied -Ricci solitons in many aspects, and obtain important and meaningful results (see [13,14]). Moreover some of the latest connected studies can be seen in [15,16,17]. For example, in [18], we know that if there is a -Ricci soliton on three-dimensional Kenmotsu manifold, then the manifold is of constant curvature 1. In [19], it has been proved that if a three-dimensional Sasakian manifold admits a -Ricci soliton on it, then it is a manifold with constant curvature. A. Ghosh and D. S. Patra give a complete classification of -Ricci soliton of non-Sasakian (κ,μ)-contact manifolds (see [20]). We give the classification of trans-Sasakian three-manifolds with Reeb invariant -Ricci operator in [21].

    We investigate the -Ricci tensor and -Ricci operator on (κ,μ)-contact manifolds in this paper. Analogue to the notion of semi-symmetric Ricci tensor, we propose the definition of semi-symmetric -Ricci tensor which is:

    0=(R(X,U)Ric)(V,W)=Ric(R(X,U)V,W)Ric(V,R(X,U)W), (1.2)

    where X,U,V and W are arbitrary vector fields of the manifold. If the -Ricci tensor is semi-symmetric, we also say that the manifold is -Ricci semi-symmetric. We give the necessary and sufficient condition that the non-Sasakian (κ,μ)-contact manifold is -Ricci semi-symmetric and has parallel -Ricci operator. We find that if some (κ,μ)-contact manifolds sharing the same Boeckx invariant IM, -Ricci flat manifold can be considered as their representative, because the -Ricci flat manifold is unique among them. Using Da-homothetic deformation (see [22]) which has been investigated by Y. Wang and H. Wu that they study invariant vector fields under it (see [23]), we also show how to construct -Ricci flat (κ,μ)-contact manifolds.

    Firstly we recall some basic notations about contact metric manifolds. If a Riemannian manifold (M2n+1,g) admits a triple (ϕ,ξ,η) which satisfies (see [24]):

    ϕξ=0,ηϕ=0,
    ϕ2X=X+η(X)ξ,η(ξ)=1,
    η(X)=g(ξ,X),
    g(ϕX,ϕU)=g(X,U)η(X)η(U),

    where we call ϕ structure tensor field, ξ Reeb vector field, η is a 1-form dual to ξ, for arbitrary vector fields X and U on M2n+1, then it is an almost contact manifold and the triple (ϕ,ξ,η) called almost contact metric structure. Moreover if the metric g and the 2-form dη satisfy:

    dη(X,U)=g(X,ϕU),

    then the manifold M is a contact manifold and denoted by (M2n+1,ϕ,ξ,η,g).

    The -Ricci tensor Ric of an almost contact manifold is half of the trace (see [11]):

    Ric(X,U)=12trace{VR(X,ϕU)ϕV}, (2.1)

    it is equivalent to

    Ric(X,U)=12trace{ϕR(X,ϕU)}.

    For a contact structure (ϕ,ξ,η), the induced tensor field h is defined by h=12Lξϕ, where L is Lie differentiation. h vanishes if and only if ξ is a Killing vector field, that is the manifold M2n+1 is K-contact manifold. For metric g, h is self-adjoint and enjoys many properties such as:

    hξ=0,hϕ=ϕh,
    trace(h)=trace(hϕ)=0,

    and there holds

    Xξ=ϕXϕhX.

    For the pair (κ,μ)R2, we can define a distribution on a contact metric manifold that

    N(κ,μ):pNp(κ,μ)={VTpMR(X,U)V=(κ+μh)(g(U,V)Xg(X,V)U)},

    which is called (κ,μ)-nullity distribution, where the mapping N(κ,μ) assigns to each point p of M a subspace Np(κ,μ) of TpM. If the Reeb vector field ξ of a contact metric manifold M2n+1 belongs to the (κ,μ)-nullity distribution, then it is a (κ,μ)-contact manifold. That is the curvature tensor R of M2n+1 satisfies Eq 1.1.

    For a (κ,μ)-contact manifold M2n+1, κ1 and when k=1 the manifold is Sasakian. In our paper, we consider the non-Sasakian situation, that is κ<1. The tensor fields h and ϕ satisfy the relations:

    h2=(k1)ϕ2,

    and

    (Xϕ)U=g(X+hX,U)ξη(U)(X+hX).

    The (κ,μ)-nullity distribution completely determines the curvature tensor of a contact manifold such that the (0,4) curvature tensor is (see [5]):

    g(R(X,U)V,W)=(1μ2)(g(U,V)g(X,W)g(X,V)g(U,W))+g(U,V)g(hX,W)g(X,V)g(hU,W)g(U,W)g(hX,V)+g(X,W)g(hU,V)+1μ/21κ(g(hU,V)g(hX,W)g(hX,V)g(hU,W))μ2(g(ϕU,V)g(ϕX,W)g(ϕX,V)g(ϕU,W))+κμ/21κ(g(ϕhU,V)g(ϕhX,W)g(ϕhX,V)g(ϕhU,W))+μg(ϕX,U)g(ϕV,W)+η(X)η(W)((κ1+μ2)g(U,V)+(μ1)g(hU,V))η(X)η(V)((κ1+μ2)g(U,W)+(μ1)g(hU,W))+η(U)η(V)((κ1+μ2)g(X,W)+(μ1)g(hX,W))η(U)η(W)((κ1+μ2)g(X,V)+(μ1)g(hX,V)). (2.2)

    Lemma 2.1. Let (M2n+1,ϕ,ξ,η,g) be a non-Sasakian (κ,μ)-contact manifold. Then its -Ricci tensor Ric and -Ricci operator Q are

    Ric(X,U)=g(κ+μn)g(ϕX,ϕU), (2.3)

    and

    QX=(κ+μn)ϕ2X, (2.4)

    where X,UΓ(TM).

    Proof. Let {e1,,e2n+1} be the local orthonormal basis of M. From the definition of -Ricci tensor (Eq 2.1), we have:

    Ric(X,U)=122n+1i=1g(R(X,ϕU)ϕei,ei)=12((2μ)g(ϕX,ϕU)+2g(ϕU,ϕhX)+2g(ϕX,hϕU)+2μ1κg(hϕU,ϕhX)μg(ϕ2X,ϕ2U)+2κμ1κg(ϕhϕU,ϕ2hX)2nμg(ϕX,ϕU))=g(κ+μn)g(ϕX,ϕU).

    Since Ric(X,U)=g(QX,U), we have

    QX=(κ+μn)ϕ2X.

    From the expression of -Ricci tensor, we can see that

    Lemma 2.2. A non-Sasakian (κ,μ)-contact manifold M2n+1 is -Ricci flat, that is

    Ric(X,U)=0,

    if and only if κ+μn=0.

    Theorem 3.1. A non-Sasakian (κ,μ)-contact manifold M2n+1 is -Ricci semi-symmetric if and only if it is -Ricci flat.

    Proof. Putting Eqs 2.3 and 2.2 in Eq 1.2, we have

    (R(X,U)Ric)(V,W)=Ric(R(X,U)V,W)Ric(V,R(X,U)W)=(κ+μn)(g(ϕR(X,U)V,ϕW)+g(ϕR(X,U)W,ϕV))=(κ+μn)(g(R(X,U)ϕ2V,W)g(ϕR(X,U)V,ϕ2W))=(κ+μn)((1μ2)(η(U)η(V)g(X,W)η(X)η(W)g(U,V)η(X)η(V)g(U,W)+η(U)η(W)g(X,V))+η(U)η(V)g(hX,W)η(X)η(V)g(hU,W)+η(U)η(W)g(hX,V)η(X)η(W)g(hU,V)(κ1+μ2)η(X)η(W)g(U,V)(μ1)η(X)η(W)g(hU,V)+(κ1+μ2)η(U)η(W)g(X,V)+(μ1)η(U)η(W)g(hX,V)(κ1+μ2)η(X)η(V)g(U,W)(μ1)η(X)η(V)g(hU,W)+(κ1+μ2)η(U)η(V)g(X,W)+(μ1)η(U)η(V)g(hX,W))=(κ+μn)(κ(η(U)η(V)g(X,W)η(X)η(W)g(U,V)η(X)η(V)g(U,W)+η(U)η(W)g(X,V))+μ(η(U)η(V)g(X,W)η(X)η(W)g(U,V)η(X)η(V)g(U,W)+η(U)η(W)g(X,V))).

    If (R(X,U)Ric)(V,W)=0, putting X=V=ξ, we have

    0=(R(ξ,U)Ric)(ξ,W)=(κ+μn)(κ(η(U)η(W)g(U,W))μg(hU,W))=(κ+μn)(κg(ϕU,ϕW)μg(hU,W))=(κ+μn)g(κϕ2UμhU,W).

    Now we suppose that κ+μn0, then there must be κϕ2UμhU=0, from which we will have κϕ3U=μϕhU=μhϕU=μϕhU. So μϕhU=0 and κϕU=κϕ3U=μϕhU=0. Thus κ=0 and μhU=0. So μ=0 since h0. We get κ+μn=0, which is contradict to our assumption κ+μn0. So if M2n+1 is -Ricci semi-symmetric, then κ+μn=0, it is -Ricci flat.

    Conversely, if M2n+1 is -Ricci flat, then

    (R(X,U)Ric)(V,W)=Ric(R(X,U)V,W)Ric(V,R(X,U)W)=0,

    it is -Ricci semi-symmetric. Thus we have completed the proof of the theorem.

    Theorem 3.2. The -Ricci operator of a non-Sasakian (κ,μ)-contact manifold is parallel if and only if the manifold is -Ricci flat.

    Proof. Since

    X(QU)=(XQ)U+Q(XU)=(κ+μn)((Xϕ)(ϕU)+ϕ((Xϕ)U)+ϕ2(XU)),

    thus we have

    (XQ)U=(κ+μn)((Xϕ)(ϕU)+ϕ((Xϕ)U))=(κ+μn)(g(X+hX,ϕU)ξη(U)(ϕX+ϕhX)).

    If XQ=0, putting U=ξ, we have

    (XQ)ξ=(κ+μn)(ϕXϕhX)=0,

    Suppose κ+μn0, there must be ϕX+ϕhX=0. Then we have

    ϕ2X=ϕ2hX=hX,

    and

    ϕ2X=ϕhϕX=hϕ2X=hX.

    From above two equations we have h=0, M2n+1 is Sasakian. This is a contradiction since we have assumed M2n+1 is non-Sasakian manifold. So we have κ+μn=0, then M2n+1 is -Ricci flat.

    Conversely if M2n+1 is -Ricci flat, then Q=0. Obviously Q is parallel. The proof is completed.

    From Theorems 3.1 and 3.2, we have the following theorem:

    Theorem 3.3. For non-Sasakian (κ,μ)-contact manifold M, the following are equivalent:

    (1)M is -Ricci flat.

    (2)M is -Ricci semi-symmetric.

    (3)M has parallel -Ricci operator.

    The classification of three dimensional non-Sasakian complete (κ,μ)-contact manifold M3 can be found in [2]. According to the values of c2=1λμ2 and c3=1+λμ2, where λ=1κ is the eigenvalue of h such that hX=λX and η(X)=0,g(X,X)=1,M3 can be putted in five classes. That is if c2>0 and c3>0, then M3 is locally isometric to SU(2) or SO(3); if c2<0,c3>0 or c2<0,c3<0, then M3 is locally isometric to SL(2,R) or O(1,2); if c2=0 and μ<2, then M3 is locally isometric to E(2), where E(2) is the group of rigid motions of the Euclidean 2-space, when κ=μ=0 then it is flat; if c3=0 and μ>2, then M3 is locally isometric to E(1,1), where E(1,1) is the group of rigid motions of the Minkowski 2-space. All of these Lie groups are equipped with left invariant metric.

    Applying the above classification theorem, if the non-Sasakian (κ,μ)-contact manifold M3 is -Ricci flat, we have classification theorem in the following:

    Theorem 3.4. Let M3 be a -Ricci flat non-Sasakian complete (κ,μ)-contact manifold. Then we can classify M3 according to the value of μ:

    (1)1<μ<0, M3 is locally isometric to SU(2) or SO(3).

    (2)μ=0, M3 is flat.

    (3)μ>0,μ8, M3 is locally isometric to SL(2,R) or O(1,2).

    (4)μ=8, M3 is locally isometric to E(1,1) (the group of rigid motions of the Minkowski 2-space).

    Proof. From Lemma 2.2, if M3 is -Ricci flat, then κ+μ=0. Since κ<1, then μ>1.

    CaseI. If 1<μ<0, then c2=11κμ2>0 and c3=1+1κμ2>0, M3 is locally isometric to SU(2) or SO(3).

    CaseII. If μ=0, then κ=0, M3 is flat.

    CaseIII. If 0<μ<8, then c2=11κμ2<0 and c3=1+1κμ2>0; if μ>8, then c2=11κμ2<0 and c3=1+1κμ2<0, both of these cases M3 is locally isometric to SL(2,R) or O(1,2).

    CaseIV. If μ=8, then c3=1+1κμ2=0, M3 is locally isometric to E(1,1).

    If the dimension of a (0,0)-contact manifold is bigger then 3, then it is locally isometric to En+1×Sn(4) (see [24]), in additional of above theorem, we have

    Theorem 3.5. A -Ricci flat non-Sasakian complete (κ,μ)-contact manifold M2n+1 with κ=0 or μ=0 is flat for n=1 or locally isometric to En+1×Sn(4) for n>1.

    In the following we give two ways to construct -Ricci flat non-Sasakian (κ,μ)-contact manifolds.

    Example 1. Let Mn+1(n>1) be a manifold of constant curvature c. T1M is the unit tangent sphere bundle, then it is a (c(2c),2c)-contact manifold with the standard contact metric. If Mn+1 is of constant curvature c=2(1n), then T1M is a -Ricci flat (4n(1n),4(1n))-contact manifold.

    Example 2. For contact metric structure, we recall the notion of Da-homothetic deformation. If there is a contact metric structure (ϕ0,ξ0,η0,g0) on a manifold M2n+1, then

    ϕ=ϕ0,ξ=1aξ0,η=aη0,g=ag0+a(a1)η0η0,

    where a is a positive constant, is also a contact metric structure on M2n+1. From [6], we know that a Da-homothetic deformation preserve the condition of Eq 1.1 but change (κ,μ) to (¯κ,¯μ) where

    ¯κ=κ+a21a2and¯μ=μ+2a2a.

    Now the question is, given a (κ,μ)-contact manifold, whether there is a positive number a so that the (¯κ,¯μ)-contact manifold obtained after the Da-homothetic deformation is -Ricci flat. The answer is positive since if (¯κ,¯μ)-contact manifold is -Ricci flat then ¯κ+¯μn=0, putting above two equations in it we will have the quadratic equation

    (2n+1)a2+(μ2)na+κ1=0,

    the unique positive solution is

    a=(μ1)2n2+4(2n+1)(1κ)(μ2)n2(2n+1).

    In this way, we know that using the appropriate Da-homothetic deformation, we can transform any (κ,μ)-contact manifold into -Ricci flat manifold. Moreover this Da-homothetic deformation is unique.

    In [6], for (κ,μ)-contact manifold, E.Boeckx introduced an invariant IM:

    IM=1μ21κ,

    and IMR. We call it Boeckx invariant. He proved that the Da-homothetic deformation preserved the invariant IM. Then E.Boeckx gave a full classification of (κ,μ)-contact manifolds that two (κ,μ)-contact manifolds (M2n+1i,ϕi,ξi,ηi,gi), i=1,2, are locally isometric as contact metric spaces if and only if IM1=IM2 up to a Da-homothetic deformation.

    Theorem 3.6. For arbitrary IR, there exists a unique -Ricci flat (κ,μ)-contact manifold M2n+1 such that the Boeckx invariant IM=I.

    Proof. We just need to prove the flowing binary equations have a unique solution:

    {κ+μn=01μ21κ=IM=I.

    Putting 1κ=t>0, then we have κ=1t2 and μ=2(1tI). The above binary equations come to

    t2+2nIt2n1=0,

    the unique positive solution is t=(4n2I2+4(2n+1)2nI)/2. So the binary equations have a unique solution.

    From above theorem, we can regard a -Ricci flat (¯κ,¯μ)-contact manifold as the representation of some (κ,μ)-contact manifolds sharing the same Boeckx invariant IM.

    Actually from above theorem and Example 2, we have:

    Corollary 3.7. A (κ,μ)-contact manifold M2n+1 is locally isometric to a unique -Ricci flat (¯κ,¯μ)-contact manifold.

    Firstly we propose the definition of -Ricci semi-symmetry and prove that a non-Sasakian (κ,μ)-contact manifold is -Ricci semi-symmetric if and only if it is -Ricci flat. Then we give the classification of three dimension -Ricci flat non-Sasakian (κ,μ)-contact manifolds. We find that the -Ricci flat non-Sasakian (κ,μ)-contact manifolds are representatives of each Boeckx invariant IM class.

    The authors wish to express their sincere thanks to the referee for helpful comments to improve the original manuscript. The second author was supported by National Natural Science Foundation of China (Grant No. 11671070).

    All authors declare no conflicts of interest in this paper.



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  • This article has been cited by:

    1. Rongsheng Ma, Donghe Pei, The ∗-Ricci Operator on Hopf Real Hypersurfaces in the Complex Quadric, 2022, 11, 2227-7390, 90, 10.3390/math11010090
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