On tangent sphere bundles with contact pseudo-metric structures
Narges Ghaffarzadeh Morteza Faghfouri
math.DG
































On tangent sphere bundles with  contact pseudo-metric structures]
 On tangent sphere bundles with  contact pseudo-metric structures




N. Ghaffarzadeh]Narges Ghaffarzadeh

Department of Pure Mathematics,
 Faculty of Mathematical Sciences,
 University of Tabriz, Tabriz, Iran.

n.ghaffarzadeh@tabrizu.ac.ir



M. Faghfouri]Morteza Faghfouri
Department of Pure Mathematics,
 Faculty of Mathematical Sciences,
 University of Tabriz, Tabriz, Iran.
faghfouri@tabrizu.ac.ir

53C15, 53C50, 53C07





To my boss



In this paper, we introduce a contact pseudo-metric structure on a tangent sphere bundle T_ε M.  we prove that the tangent sphere bundle T_εM is (κ, μ)-contact pseudo-metric manifold
 if and only if the manifold M is of constant sectional curvature. Also, we prove that this structure on the tangent sphere bundle is K-contact iff the base manifold has constant curvature ε.



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    January 1, 2004
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    January 1, 2004
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§ INTRODUCTION

In 1956, S. Sasaki <cit.> introduced a Riemannian metric on tangent bundle TM and tangent sphere bundle T_1M over a Riemannian manifold M. Thereafter, that metric was called the Sasaki metric. In 1962, Dombrowski <cit.> also showed at each Z∈ TM,  TM_Z=HTM_Z⊕ VTM_Z, where HTM_Z and VTM_Z orthogonal subspaces of dimension n, called horizontal and vertical distributions, respectively. He  defined an almost Kählerian structure on TM and proved that it is Kählerian manifold if M is flat. In the same year, Tachibana and Okumura <cit.> showed  that the tangent bundle
space TM of any non-flat Riemannian space M  always admits an almost Kählerian structure, which is not Kählerian.  Tashiro <cit.> introduced a contact metric structure on the unit tangent sphere bundle T_1M and he proved that  contact metric structure on T_1M is K-contact iff M has constant curvature 1, in which case the structure is Sasakian.

 Kowalski <cit.> computed  the curvature tensor of Sasaki metric. Thus, on T_1M,  R(X,Y)ξ can be computed by the formulas for the curvature of TM.

 In <cit.>, Blair et al. introduced (κ,μ)-contact Riemannian manifolds and proved that,  the tangent sphere bundle T_1M is a (κ, μ)-contact Riemannian manifold iff the base manifold M is of constant sectional curvature c.

 Takahashi <cit.> introduced contact pseudo-metric structures (η,g), where η is a contact one-form and g is a pseudo-Riemannian metric associated to it, are a natural generalization of contact metric
structures. Recently, contact pseudo-metric manifolds  have been studied by Calvaruso and Perrone <cit.> and authors of this paper <cit.> introduce and study (κ, μ)-contact pseudo-metric manifolds.

 In this paper, we suppose that (M,g) is pseudo-metric manifold and  define pseudo-metric on TM. Also, we introduce contact pseudo-metric structures (φ,ξ,η,g_cm)  on T_ε M and prove that 
    R̅(X,Y)ξ=c(4ε-(c+2)){η(Y)X-η(X)Y}-2ε c{η(Y)hX-η(X)hY}

 if and only if the base manifold M is of constant sectional curvature. That is, the tangent sphere bundle T_εM is a (κ, μ)-contact pseudo-metric manifold iff the base manifold M is of constant sectional curvature. Also, the contact pseudo-metric structure (φ,ξ,η,g_cm) on T_εM is K-contact if and only if the base manifold (M,g) has constant curvature ε.



§ PRELIMINARIES

Let (M,g) be a pseudo-metric manifold and  the associated Levi-Civita connection and  R=[,]-_[,] the curvature tensor. The tangent bundle of M, denoted by TM,  consists of pairs (x,u), where x∈ M and u∈ T_xM,( i.e.  TM=∪_x∈ MT_xM). The mapping π:TM→ M, π(x, u)=x is the natural projection.

The tangent space T_(x,u)TM splits into  the vertical subspace VTM_(x,u) is given by
VTM_(x,u):=π_*|_(x,u)
and the horizontal subspace HTM_(x,u) with respect to :

    T_(x,u)TM=VTM_(x,u)⊕ HTM_(x,u).

For every X∈ T_xM, there is a unique vector X^h∈ HTM_(x,u), such that π_*(X^h)=X. It is called the horizontal lift of X to (x,u). Also,  there is a unique vector X^v∈ VTM_(x,u), such that X^v(df)=Xf for all  f∈ C^∞(M). X^v is called the vertical lift of X to (x,u). The maps
X↦ X^h  between T_xM and HTM_(x,u), and   X↦ X^v between T_xM and VTM_(x,u) are isomorphisms.  Hence, every tangent vector Z̅∈ T_(x,u)TM can be decomposed Z̅=X^h+Y^v for uniquely determined vectors X,Y∈ T_xM. The horizontal ( respectively, vertical) lift of a vector field X on M to TM is the vector field X^h (respectively, X^v ) on M, whose value at the point (x,u) is the horizontal (respectively, vertical) lift of X_x to (x,u).

A system of local coordinate (x^1,…,x^n) on an open subset U of M induces on π^-1(U) of TM a system of local coordinate (x̅^1,…,x̅^n;u^1,…,u^n) as follows:

x̅^i(x,u)=(x^i∘π)(x,u)=x^i(x), u^i(x,u)=dx^i(u)=ux^i

for i=1,…,n and (x,u)∈π^-1(U). With respect to the induced  local coordinate system, the horizontal and vertical lifts of a vector field
X= X^i∂/∂ x^i
on U are given by

    X^h= (X^i∘π)∂/∂x̅^i-u^b((X^aΓ^i_ab)∘π)∂/∂ u^i,
    
    X^v=(X^i∘π)∂/∂ u^i,

where Γ^i_jk are the local components of , i.e., _∂/∂ x^j∂/∂ x^k=Γ^i_jk∂/∂ x^i. The span of the horizontal lifts at t∈ TM is called the horizontal subspace
of T_tTM.
For all t∈ TM, the connection map 𝒦:TTM→ TM is given by
𝒦X_t^h=0 and 𝒦X_t^v=X_π(t) <cit.>.

From (<ref>) and (<ref>), one can easily calculate the brackets of vertical and horizontal lifts:

    [X^h,Y^h]=[X,Y]^h-v{R(X,Y)u},
    
    [X^h,Y^v]=(_XY)^v,
    
    [X^v,Y^v]=0,

for all X,Y∈Γ(TM). We use some notation, due to M. Sekizawa (<cit.>). Let T be a tensor field of type (1,s) on M and X_1,…,X_s-1∈Γ(TM), the vertical vector field v{T(X_1,…,u,…,X_s-1)} on π^-1(U) is given by

    v{T(X_1,…,u,…,X_s-1)}:= u^a(T(X_1,…,∂/∂ x^a,…,X_s-1))^v.

Analogously, one defines  the horizontal vector field h{T(X_1,…,u,…,X_s-1)} and h{T(X_1,…,u,…,u,…,X_s-2)} and the vertical vector field v{T(X_1,… ,u,…,u,…,X_s-2)}.  Note that these vector fields do not depend on the choice of coordinates on U. Let (M,g) be a pseudo-metric manifold. On the tangent bundle TM, we can define a  pseudo-metric Tg to be

    Tg(X^h,Y^h)=Tg(X^v,Y^v)=g(X,Y)∘π,   Tg(X^h,Y^v)=0

for all X,Y∈Γ(TM). We call it Sasaki pseudo-metric.
  According (<ref>), If {E_1,…,E_n} is an orthonormal frame field on U then {E_1^v,…,E_n^v,E_1^h,…,E_n^h} is an orthonormal frame field on π^-1(U). So, we have the following:

  
If the index of g is ν then the index of the Sasaki pseudo-metric Tg is 2ν.
   

  Let  be  the Levi-Civita connection of Tg. It is easy to check that for X,Y∈Γ(TM) and (x,u)∈ TM(see <cit.> for more details):

    (_X^v Y^v)   =0,     (_X^v Y^h)=1/2h{R(u,X)Y},
    
    (_X^h Y^v)   =(_X Y)^v+1/2h{R(u,Y)X},
    
    (_X^h Y^h)   =(_X Y)^h-1/2v{R(X,Y)u}.



§ THE CURVATURE OF THE UNIT TANGENT SPHERE BUNDLE WITH PSEUDO-METRIC

Let (TM,Tg) be the tangent bundle of (M,g) endowed with its Sasaki pseudo-metric. We consider the hypersurface T_εM={(x,u)∈ TM| g_x(u,u)=ε}, which we call  the unit tangent sphere bundle.  A unit normal vector field N on T_xM is the (vertical) vector field N= u^i∂/∂ u^i=u^i(∂/∂ x^i)^v. N is independent of the choice of local coordinates and it is defined globally on TM.
We introduce some more notation. If X∈ T_xM, we define the tangential lift of X to (x,u)∈ T_εM by

    X^t_(x,u)=X^v_(x,u)-ε g(X,u)N_(x,u).

Clearly, the tangent space to T_εM at (x,u) is then spanned by vectors of the form X^h and X^t, where X∈ T_xM. Note that u^t_(x,u)=0. The tangential lift of a vector field X on M to T_εM is the vertical vector field X^t on T_εM, whose value at the point (x,u)∈ T_εM is the tangential lift of X_x to (x,u). For a tensor field T of type (1,s) on M and X_1,…,X_s-1∈Γ(TM), we define the vertical vector fields t{T(X_1,…,u,…,X_s-1)}
and t{T(X_1,…,u,…,u,…,X_s-2)} on T_εM in the obvious way.

In what follows, we will give explicit expressions for the brackets of vector fields on T_εM involving tangential lifts, the Levi-Civita connection and the associated curvature tensor of the induced metric g̅ on T_εM.

First, for the brackets of vector fields on T_εM involving tangential lifts, we obtain

    [X^h,Y^t]=(_XY)^t,
    
    [X^t,Y^t]=ε g(X,u)Y^t-ε g(Y,u)X^t.

Next, we denote by g̅  the pseudo-metric induced on T_εM from  Tg on TM.

The Levi-Civita connection  of (T_εM,g̅) is described completely by

    _X^t Y^t   =-ε g(Y,u)X^t,
    _X^t Y^h   =1/2h{R(u,X)Y},
    _X^h Y^t   =(_X Y)^t+1/2h{R(u,Y)X},
    _X^h Y^h   =(_X Y)^h-1/2t{R(X,Y)u}

for all vector fields X and Y on M.


This is obtained by an easy calculation using (<ref>) and the following equation

    _A̅B̅=_A̅B̅-ε Tg(_A̅B̅,N)N,

for vector fields A̅,B̅ tangent to T_εM.


The curvature tensor R̅ of (T_εM,g̅) is described completely by

    R̅(X^t,Y^t)Z^t=ε{-g̅(X^t,Z^t)Y^t+g̅(Z^t,Y^t)X^t},
    R̅(X^t,Y^t)Z^h=(R(X,Y)Z)^h-ε{g(Y,u)h(R(X,u)Z)+g(X,u)h(R(u,Y)Z)}
                  +1/4h{[R(u,X),R(u,Y)]Z},
    R̅(X^h,Y^t)Z^t=-1/2(R(Y,Z)X)^h+ε/2{g(Y,u)h(R(u,Z)X)+g(Z,u)h(R(Y,u)X)}
                  -1/4h{R(u,Y)R(u,Z)X},
    R̅(X^h,Y^t)Z^h=1/2(R(X,Z)Y)^t-ε/2g(Y,u)t{R(X,Z)u}
                  -1/4t{R(X,R(u,Y)Z)u}+1/2h{(_XR)(u,Y)Z},
    R̅(X^h,Y^h)Z^t=(R(X,Y)Z)^t-ε g(Z,u)t{R(X,Y)u}
                  +1/4t{R(Y,R(u,Z)X)u-R(X,R(u,Z)Y)u}
                  +1/2h{(_XR)(u,Z)Y-(_YR)(u,Z)X},
    R̅(X^h,Y^h)Z^h=(R(X,Y)Z)^h+1/2h{R(u,R(X,Y)u)Z}
                  -1/4h{R(u,R(Y,Z)u)X-R(u,R(X,Z)u)Y}
                  +1/2t{(_ZR)(X,Y)u}

for all vector fields X,Y and Z on M.


The proof is made by using the following equation and  equation (<ref>) for the covariant derivative, (<ref>), (<ref>) and (<ref>) for the brackets are explicitly calculated.


    R̅(A̅,B̅)C̅=_A̅_B̅C̅-_B̅_A̅C̅-_[A̅,B̅]C̅.





§ THE CONTACT PSEUDO-METRIC STRUCTURE OF THE UNIT TANGENT SPHERE BUNDLE

First, we give some basic facts on contact pseudo-metric structures. A pseudo-Riemannian manifold (M^2n+1,g) has a contact pseudo-metric structure if it admits a vector field ξ, a one-form η and a (1,1)-tensor field φ satisfying

    η (ξ)= 1,
       φ^2(X)=-X+η(X)ξ,
       g(φ X,φ Y)=g(X,Y)-εη(X)η(Y),
       dη(X,Y) =g(X,φ Y),

where ε=±1 and  X,Y∈Γ(TM).
In this case, (M,φ,ξ,η,g) is called a contact pseudo-metric manifold. In particular, the above conditions imply that the characteristic curves, i.e., the integral curves of the characteristic vector field ξ, are geodesics.

If ξ is in addition a Killing vector field with respect to g, then the manifold is said to be a K-contact (pseudo-metric) manifold. Another characterizing property of such contact pseudo-metric manifolds is the following:

 geodesics which are orthogonal to ξ at one point, always remain orthogonal to ξ. This yields a second special class of geodesics, the φ-geodesics.

Next, if (M^2n+1,φ,ξ,η,g) is a contact pseudo-metric manifold satisfying the additional condition
N_φ(X,Y)+2dη(X,Y)ξ=0 is said to be Sasakian,
where N_φ(X,Y)=φ^2[X,Y]+[φ X,φ Y]-φ[φ X,Y]-φ[X,φ Y] is the Nijenhuis torsion tensor of φ. Moreover, an almost contact pseudo-metric manifold (M^2n+1,φ,ξ,η,g) is a Sasakian manifold if and only if (_Xφ)Y=g(X,Y)-εη(Y)X.
 In particular, one can show that the characteristic vector field ξ is a Killing vector field. Hence, a Sasakian manifold is also a K-contact manifold(see <cit.> for more details). If a contact pseudo-metric manifold satisfying R(X,Y)ξ= εκ(η(Y )X -η(X)Y ) +εμ(η(Y )hX -η(X)hY ), we call  (κ, μ)-contact pseudo-metric manifold, where (κ ,μ)∈ℝ^2. (κ, μ)-contact pseudo-metric manifold is Sasakian if and only if κ=ε(see <cit.> for more details).


Take now an arbitrary pseudo-metric manifold (M,g). One can easily define an almost complex structure J on TM in the following way:

    JX^h=X^v,   JX^v=-X^h

for all vector fields X on M.
 From (<ref>), (<ref>) and (<ref>), we have the almost complex structure J is integrable if and only if (M,g) is flat.
 From the definition (<ref>) of the pseudo-metric Tg on TM, it follows immediately that (TM,Tg,J) is almost Hermitian. Moreover, J defines an almost Kählerian structure. It is a Kähler manifold only when (M,g) is flat<cit.>.

Next, we consider the unit tangent sphere bundle (T_εM,g̅), which is isometrically embedded as a hypersurface in (TM,Tg) with unit normal field N. Using the almost complex structure J on TM, we define a unit vector field ξ', a one-form η' and a (1,1)-tensor field φ' on T_εM by

    ξ'=-JN,   JX=φ' X+η'(X)N.

In local coordinates, ξ', η' and φ' are described by

    ξ'=u^i(∂/∂ x^i)^h,
       η'(X^t)=0,  η'(X^h)=ε g(X,u),
       φ'(X^t)=-X^h+ε g(X,u)ξ',  φ'(X^h)=X^t,
 where  X,Y∈Γ(TM).
It is easily checked that these tensors satisfy the conditions (<ref>) excepts or the last one, where we find g̅(X,φ' Y)=2 dη'(X,Y). So strictly speaking, (φ',ξ',η',g̅) is not a contact pseudo-metric structure. Of course, the difficulty is easily rectified and

    η=1/2η',  ξ=2ξ',  φ=φ',   g_cm=1/4g̅

is taken as the standard contact pseudo-metric structure on T_εM. In local coordinates,
with respect to induce the  local coordinates (x^i,u^i) on TM, the characteristic vector field is given by

    ξ=2u^i(∂/∂ x^i)^h,

the vector field u^i(∂/∂ x^i)^h is the well-known geodesic flow on T_εM. Before beginning our theorems, we explicitly obtain  the covariant derivatives of ξ and φ. For a horizontal tangent vector field, we may  use a horizontal lift again. Then

    _X^hξ=_X^hξ=-v{R(X,u)u}

and hence for any horizontal vector X^h at (x,u)∈ T_εM, we have

    _X^hξ=-v{R(X,u)u}=-t{R(X,u)u}.

For a vertical vector field X^v tangent to T_εM, we have

    _X^vξ=_X^vξ=-2φ X^v-h{R(X,u)u}.

Since J(∂/∂ x^i)^h=(∂/∂ x^i)^v, or in terms of tangential lifts of a vector X on M,

    _X^tξ=-2φ X^t-h{R(X,u)u}.

Comparing with _Xξ=-εφ X-φ hX  on T_εM for a vertical vector V and a horizontal vector X orthogonal to ξ, hV and hX are given by

    hV=(2-ε)V-v{R(𝒦V,u)u}  and   hX=-ε X+h{R(π_*X,u)u}.

Also for any tangent vector fields X and Y, we have

    (_Xφ)Y=   _XJY-(_Xη')(Y)N+η'(Y)AX
       -εg̅(X,Aφ Y)N-J(_XY)-εg̅(X,AY)ξ'.

We present two computations, one done in each manner.


As before, for X,Y horizontal vector fields, we suppose that they are horizontal lifts, and we have

    (_X^hφ)Y^h=1/2h{R(u,X)Y},

where we  used the first Bianchi identity.

For Y^v=Y^i∂/∂ u^i a vertical vector field tangent to T_εM and X^h a horizontal tangent vector, we have

    (_X^hφ)Y^v=1/2t{R(X,u)Y},

where we  used

    (_X^hη')(Y^v)=εg̅(Y^v,_X^hξ')=-ε/2g(Y,R(X,u)u)=ε/2Tg(N,(R(X,u)Y)^v).

Similarly, we obtain

    (_X^vφ)Y^h=t{R(X,u)Y}-2η(Y^h)X^v,


    (_X^vφ)Y^v=1/2h{R(X,u)Y}+2ε g_cm(X,Y)ξ.


Let (φ,ξ,η,g_cm) be a contact pseudo-metric structure on the tangent sphere bundle T_εM. Then

    R̅(X,Y)ξ=c(4ε-(c+2)){η(Y)X-η(X)Y}-2ε c{η(Y)hX-η(X)hY}

 if and only if the base manifold M is of constant sectional curvature c.


Assume that the manifold M is a pseudo-metric manifold of constant curvature c. Then from equations  (<ref>-<ref>), for X,Y orthogonal to ξ, we have  R̅(X, Y)ξ=0  and for a vertical vector V, we get R̅(V,ξ)ξ=c^2V and also, for a horizontal vector X orthogonal to ξ, we obtain R̅(X,ξ)ξ=(4ε c- 3c^2)X. Moreover, from equations (<ref>), hV=(2-ε (1+c))V and hX= ε (c- 1)X. Thus for all X,Y on T_εM, the curvature tensor on T_εM satisfies

    R̅(X,Y)ξ=c(4ε-(c+2)){η(Y)X-η(X)Y}-2ε c{η(Y)hX-η(X)hY}.

Conversely, if the contact pseudo-metric structure on T_εM satisfies the condition 

    R̅(X,Y)ξ=εκ{η(Y)X-η(X)Y}+εμ{η(Y)hX-η(X)hY},

where κ=c(4-ε(c+2)), μ= -2c then

    R̅(X,ξ)ξ=εκ X+εμ hX,

for any X orthogonal to ξ. Now, for a vector u on M, that g(u,u)=ε define a symmetric operator
ψ_u:⟨ u⟩^⊥→⟨ u⟩^⊥
 by ψ_uX= R(X, u)u. By placing the equation (<ref>) in (<ref>), we get
 
    R̅(V,ξ)ξ=ε(κ+μ(2-ε))V-εμ  v{ψ_u𝒦V}.

 Also using equations (<ref>-<ref>), we have
 
    R̅(V,ξ)ξ=-v(R(R(u,𝒦V)u,u)u)=v{ψ^2_u𝒦V}.

From a comparison of equations (<ref>) and (<ref>),  the operator ψ_u satisfies the equation

    ψ^2_u+εμψ_u-ε(κ+(2-ε)μ)I=0.

In a similar way, for a horizontal X orthogonal to ξ,

    R̅(X,ξ)ξ=(εκ-μ)X+εμ  h(ψ_uπ_*X),

and, from equations  (<ref>)-(<ref>), we get

    R̅(X,ξ)ξ=h(4ψ_uπ_*X-3ψ^2_uπ_*X),

giving

    3ψ^2_u+(εμ-4)ψ_u+(εκ-μ)I=0.

From  equations  (<ref>) and (<ref>), the eigenvalues a of ψ_u satisfy the two quadratic equations

    a^2+εμ a-(εκ+(2ε-1)μ)=0,   a^2+εμ-4/3a+εκ-μ/3=0.

Thus, the common root of the above quadratic equations is a=ε c.






Hence from ψ_uX= R(X, u)u=ε cX and g(u,u)=ε, M is of constant curvature c.

We can now rephrase Theorem <ref> as the following result.

 The tangent sphere bundle T_εM is (κ, μ)-contact pseudo-metric manifold
 if and only if the base manifold M is of constant sectional curvature c and κ=c(4-ε(c+2)), μ= -2c.

We now have the following theorem about the K-contact structure.

The contact pseudo-metric structure (φ,ξ,η,g_cm) on T_εM is K-contact if and only if the base manifold (M,g) has constant curvature ε.


Using Theorem 3.3 of <cit.>, since sectional curvature of all nondegenerate plane sections containing ξ equals ε, therefore  T_εM is K-contact and conversely.


Let M be an n-dimensional pseudo-metric manifold, n>2, of constant sectional curvature c. The tangent sphere bundle T_εM has constant φ-sectional curvature (4c(ε-1)+ε c^2) if and only if c=2ε±√(5).


Using Theorem 3.5 of <cit.>, (κ, μ)-contact pseudo-metric manifold M has constant φ-sectional curvature  -(κ+εμ) if and
only if μ=εκ+1. So, using Result <ref> and a straightforward calculation, the proof is completed.


 For ε=+1, the tangent sphere bundle T_1M is (κ, μ)-contact pseudo-metric manifold
 if and only if the base manifold M is of constant sectional curvature c and κ=c(2-c), μ= -2c. Also, T_1M is Sasakian if and only if c=1.


 For ε=-1, the tangent sphere bundle T_-1M is (κ, μ)-contact pseudo-metric manifold
 if and only if the base manifold M is of constant sectional curvature c and κ=c(6+c), μ= -2c. Also, T_-1M is Sasakian if and only if c=-3±2√(2).















siam




10

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