Volume 1 No. ? pp. 000–000 (2014) c⃝IEJG
CURVATURE PROPERTIES OF RIEMANNIAN METRICS OF THE FORM Sg
f+Hg ON THE TANGENT BUNDLE OVER A RIEMANNIAN MANIFOLD (M, g)
AYDIN GEZER, LOKMAN B˙ILEN, CAGRI KARAMAN, AND MURAT ALTUNBAS (Communicated by Kazım ˙Ilarslan )
Abstract. In this paper, we define a special new family of metrics which rescale the horizontal part by a nonzero differentiable function on the tangent bundle over a Riemannian manifold. We investigate curvature properties of the Levi-Civita connection and another metric connection of the new Riemannian metric.
1. Introduction
The research in the topic of differential geometry of tangent bundles over Rie-mannian manifolds has begun with S. Sasaki. In his original paper [17] of 1958, he constructed a Riemannian metricSg on the tangent bundle T M of a
Riemann-ian manifold (M, g), which depends closely on the base metric g. Although the Sasaki metric is naturally defined, it was shown in many papers that the Sasaki metric presents a kind of rigidity. In [10], O. Kowalski proved that if the Sasaki metricSg is locally symmetric, then the base metric g is flat and therefore Sg is
also flat. In [12], E. Musso and F. Tricerri demonstrated an extreme rigidity of
Sg in the following sense: if (T M,Sg) is of constant scalar curvature, then (M, g)
is flat. They also defined a new Riemannian metric gCG on the tangent bundle
T M which they called the Cheeger Gromoll metric. Given a Riemannian metric g on a differentiable manifold M , there are well known classical examples of metrics on the tangent bundle T M which can be constructed from a Riemannian metric g, namely the Sasaki metric, the horizontal lift and the vertical lift. The three classical constructions of metrics on tangent bundles are given as follows:
(a) The Sasaki metricSg is a (positive definite) Riemannian metric on the tan-gent bundle T M which is derived from the given Riemannian metric on M as
Date: July 3, 2014 and, in revised form, June 00, 0000.
2000 Mathematics Subject Classification. Primary 53C07; Secondary 53C21.
Key words and phrases. Metric connection, Riemannian metric, Riemannian curvature tensor,
tangent bundle, Weyl curvature tensor.
follows: S g(HX,HY) = g (X, Y ) Sg(HX,V Y) = Sg(VX,HY)= 0 Sg(VX,V Y) = g (X, Y ) for all X, Y ∈ ℑ1 0(M ).
(b) The horizontal lift Hg of g is a pseudo-Riemannian metric on the tangent
bundle T M with signature (n, n) which is given by
Hg(HX,HY) = 0,
Hg(HX,V Y) = Hg(VX,HY)= g (X, Y ) , Hg(VX,V Y) = 0
for all X, Y ∈ ℑ10(M ).
(c) The vertical lift Vg of g is a degenerate metric of rank n on the tangent bundle T M which is given by
Vg(HX,HY) = 0
Vg(HX,V Y) = Vg(VX,HY)= 0 Vg(VX,V Y) = g (X, Y )
for all X, Y ∈ ℑ1 0(M ).
Another classical construction is the complete lift of a tensor field to the tangent bundle. It is well known that the complete liftCg of a Riemannian metric g coincides
with the horizontal lift Hg given above. A ”nonclassical” example is the
Cheeger-Gromoll metric gCG on the tangent bundle T M . Other metrics on the tangent
bundle T M can be constructed by using the three classical liftsSg,Hg and Vg of
the metric g (for example, see [7, 19]).
V. Oproiu and his collaborators constructed natural metrics on the tangent bun-dles of Riemannian manifolds possessing interesting geometric properties ([13, 14, 15, 16]). All the preceding metrics belong to a wide class of the so-called g-natural metrics on the tangent bundle, initially classified by O. Kowalski and M. Sekizawa [11] and fully characterized by M.T.K Abbassi and M. Sarih [1, 2, 3] (see also [9] for other presentation of the basic result from [11] and for more details about the concept of naturality).
In [20](see also [21, 22], B. V. Zayatuev introduced a Riemannian metricSg on
the tangent bundle T M given by
Sg f (H X,HY) = f g (X, Y ) , Sg f (H X,V Y) = Sgf (V X,HY)= 0, Sg f (V X,V Y) = g (X, Y ) ,
where f > 0, f ∈ C∞(M ) (see also, [5, 18]). For f = 1, it follows thatSg f =S g,
i.e. the metric Sgf is a generalization of the Sasaki metric Sg. For the rescaled
Sasaki type metric on the cotangent bundle, see [6].
Our purpose is to study some properties of a special new family of metrics on the tangent bundle constructed from the base metric, and generated by positive functions on M, which the metric is in the form fG =e Sg
f+Hg. The paper can
family of metrics on the tangent bundle. It is worth mentioning that a metric from this new family is g-natural only if the generating function is constant. So the considered family is far from being a subfamily of the class of g-natural metrics, and its study could be of interest in some sense.
The present paper is organized as follows: In section 2, we review some intro-ductory materials concerning with the tangent bundle T M over an n-dimensional Riemannian manifold M and also introduce the adapted frame in the tangent bun-dle T M . In section 3, we present a Riemannian metric of the formfG =e Sg
f+Hg
defined by
fGe(HX,HY) = f g (X, Y )
fGe(HX,V Y) = fGe(VX,HY)= g (X, Y ) fGe(VX,V Y) = g (X, Y )
for all X, Y ∈ ℑ10(M ), where f > 1, f ∈ C∞(M ) and compute the Christoffel symbols of the Levi-Civita connectionf∇ ofe fG with respect to the adapted frame.e In section 4 and 5, we compute all kinds of curvatures of the metricfG with respecte to the adapted frame and give some geometric results concerning them. In section 5, we give conditions for which the metricfG is locally conformally flat. Section 6e
deals with another metric connection with torsion of the metricfG.e
Throughout this paper, all manifolds, tensor fields and connections are always assumed to be differentiable of class C∞. Also, we denote by ℑp
q(M ) the set of
all tensor fields of type (p, q) on M , and by ℑpq(T M ) the corresponding set on the
tangent bundle T M .
2. Preliminaries
2.1. The tangent bundle. Let T M be the tangent bundle over an n-dimensional Riemannian manifold (M, g), and π be the natural projection π : T M → M. Let the manifold M be covered by a system of coordinate neighborhoods (U, xi), where
(xi), i = 1, ..., n is a local coordinate system defined in the neighborhood U . Let
(yi) be the Cartesian coordinates in each tangent space T
PM at P ∈ M with
respect to the natural basis { ∂ ∂xi|P
}
, where P is an arbitrary point in U with coordinates (xi). Then we can introduce local coordinates (xi, yi) on the open set
π−1(U )⊂ T M. We call such coordinates as induced coordinates on π−1(U ) from (U, xi). The projection π is represented by (xi, yi)→ (xi). The indices I, J, ... run
from 1 to 2n, while i, j, ... run from n + 1 to 2n. Summation over repeated indices is always implied.
Let X = Xi ∂
∂xi be the local expression in U of a vector field X on M . Then
the vertical liftVX and the horizontal liftHX of X are given, with respect to the
induced coordinates, by (2.1) VX = Xi∂i, and (2.2) HX = Xi∂i− ysΓiskX k∂ i, where ∂i= ∂x∂i, ∂i= ∂ ∂yi and Γ i
jkare the coefficients of the Levi-Civita connection
Explicit expressions for the Lie bracket [, ] of T M are given by Dombrowski in [4]. The bracket operation of vertical and horizontal vector fields is given by the formulas (2.3) [H X,HY]=H[X, Y ]−V (R(X, Y )u) [H X,VY]=V(∇ XY ) [V X,V Y]= 0
for all vector fields X and Y on M , where R is the Riemannian curvature of g defined by R (X, Y ) = [∇X,∇Y]− ∇[X,Y ] (for details, see [19]).
2.2. The adapted frame. We insert the adapted frame which allows the tensor calculus to be efficiently done in T M. With the connection∇ of g on M, we can introduce adapted frames on each induced coordinate neighborhood π−1(U ) of T M . In each local chart U⊂ M, we write X(j)=
∂
∂xj, j = 1, ..., n. Then from (2.1) and
(2.2), we see that these vector fields have, respectively, local expressions
HX
(j)= δhj∂h+ (−ysΓhsj)∂h VX
(j)= δjh∂h
with respect to the natural frame{∂h, ∂h
}
, where δh
j denotes the Kronecker delta.
These 2n vector fields are linearly independent and they generate the horizontal distribution of ∇g and the vertical distribution of T M , respectively. We call the
set{HX(j),VX(j) }
the frame adapted to the connection∇ of g in π−1(U )⊂ T M. By denoting
Ej = HX(j), (2.4)
Ej = VX(j), we can write the adapted frame as{Eβ} =
{ Ej, Ej
} . Using (2.1), (2.2) and (2.4), we have
(2.5) VX = ( 0 Xh ) = ( 0 Xjδh j ) = Xj ( 0 δh j ) = XjEj, and (2.6) HX = ( Xjδhj −XjΓh sjys ) = Xj ( δjh −Γh sjys ) = XjEj
with respect to the adapted frame{Eβ} (see [19]).
3. The Riemannian metric and its Levi-Civita connection Let (M, g) be a Riemannian manifold. A Riemannian metric fG is defined one
T M by the following three equations
fG(e HX,HY ) = f g(X, Y ), (3.1) fG(e HX,VY ) = fG(e VX,HY ) = g(X, Y ), fG(e VX,VY ) = g(X, Y ) for all X, Y ∈ ℑ1 0(M ), where f > 1 and f∈ C∞(M ).
From the equations (3.1), by virtue of (2.5) and (2.6), The metricfG and its in-e versefGe−1respectively have the following components with respect to the adapted
frame{Eβ}: (3.2) fG = (e fGeαβ) = ( f gij gij gij gij ) and (3.3) fGe−1= (fGeαβ) = ( 1 f−1g ij − 1 f−1g ij − 1 f−1g ij f f−1g ij ) . We now consider local 1-forms ωλ in π−1(U ) defined by ωλ= ˜Aλ
BdxB, where (3.4) A−1= ˜AλB= ( ˜ Ahj A˜h¯j ˜ A¯h j A˜ ¯ h ¯ j ) = ( δh j 0 ysΓh sj δjh )
is the inverse matrix of the matrix
(3.5) A = AβA= ( A h j A h¯ j A¯jh A ¯h ¯ j ) = ( δhj 0 −ysΓh sj δhj )
of the transformation Eβ= AβA∂A. We easily see that the set{ωλ} is the coframe
dual to the adapted frame{Eβ}, e.i. ωλ(Eβ) = ˜AλBA B β = δ
λ β.
Since the adapted frame field{Eβ} is non-holonomic, we put
[Eα, Eβ] = ΩαβγEγ
from which we have
Ωγβα= (EγAβA− EβAγA) ˜AαA.
According to (2.4), (3.4) and (3.5), the components of non-holonomic object Ωγβα are given by (3.6) { Ω k ij =−Ω k ji = Γ k ji Ω k ij =−Ωjik =−y s R k ijs
all the others being zero, where R k
ijs are local components of the Riemannian
curvature tensor R of the Riemannian manifold (M, g).
Let f∇ be the Levi-Civita connection of the Riemannian metrice fG. Puttinge
f∇e
EαEβ =
feΓγ
αβEγ, from the equation f∇eXeYe − f ∇e e YX =e [ e X, eY ] , ∀ eX, eY ∈ ℑ1 0(T M ), we have (3.7) feΓαγβ−feΓαβγ= Ωγβα. The equation (f∇e e X fG)( ee Y , eZ) = 0,∀ eX, eY , eZ∈ ℑ1
0(T M ) has the form (3.8) Eα fGeγβ− feΓ
ε f
δγ Geεβ− feΓεδβ fGe
γε= 0
with respect to the adapted frame{Eβ}. Thus we have from (3.7) and (3.8)
(3.9) feΓαβγ= 1 2 fGeαε(E β fGeεγ+ Eγ fGeβε− EεfGeβγ) + 1 2(Ω α βγ + Ω α βγ+ Ω α γβ),
where Ωαγβ=fGeαε fGeδβΩεγδ,fGeαεare the contravariant components of the metric fG with respect to the adapted frame.e
Taking account of (3.3), (3.6) and (3.9), for various types of indices, we find the following relations (3.10) feΓk ij= Γkij+ 1 2(f−1)y p(R k pij + Rpjik) + 1 2(f−1) fAk ij feΓk ij= 1 2(f−1)y pR k pij feΓk ij= 1 2(f−1)y pR k pji feΓk ij=− 1 2(f−1) fAk ij− 1 2y pR k ijp − 1 2(f−1)y p(R k pij + Rpjik) feΓk ij=− 1 2(f−1)y pR k pij feΓk ij= Γ k ij− 1 2(f−1)y pR k pji feΓk ij= 0 feΓk ij= 0
with respect to the adapted frame, wherefAkij is a tensor field of type (1, 2) defined byfAk
ij= (fiδkj + fjδik− f.kgji), fi= ∂if .
4. The Riemannian curvature tensor
The Riemannian curvature tensor R of the connection ∇ is obtained from the well-known formula
R (X, Y ) Z =∇X∇YZ− ∇Y∇XZ− ∇[X,Y ]Z for all X, Y ∈ ℑ1
0(M ). With respect to the adapted frame{Eβ}, we writef∇eEαEβ=
feΓγ
αβEγ, wherefeΓγαβ denote the Levi-Civita connection constructed by
fG. Thene
the Riemannian curvature tensorfR has the componentse fRe σ
αβγ = Eα feΓσβγ− Eβ feΓσαγ+ feΓσαϵfeΓϵβγ− feΓσβϵ feΓϵαγ− Ωαβ ϵfeΓσϵγ.
From (3.6) and (3.10), we obtain the components of the Riemannian curvature tensorfR of the metrice fG as follows:e
fRe k mij = 0, fRe k mij = 0, fRe k mij = 1 f−1R k mij +4(f−1)1 2y pys(R k pmh R h sij − RpihkR h smj ), fRe k mij =− 1 f−1R k mij −4(f−1)1 2y pys(R k pmh R h sij − RpihkR h smj ), fRe k mij = 1 2(f−1)R k mji + 1 4(f−1)2y pysR k pmhR h sji , fRe k mij =− 1 2(f−1)R k mji − 1 4(f−1)2y pysR k pmhR h sji ,
(4.1) fRe k mij =− 1 2(f−1)R k ijm −4(f−1)1 2y pysR k pih R h sjm, fRe k mij = 1 2(f−1)R k ijm +4(f−1)1 2y pysR k pih R h sjm, fRe k mij = − 1 2R k ijm −2(f1−1)(R k mij + Rmjik) +2(f1−1)y p∇ iRpmjk − 1
4(f−1)2ypys[Rpmhk(Rsijh+ Rsjih)− Rsmjh(Rpihk+ (f− 1)Rihpk)]
− 1 4(f−1)2yp[2fiRpmjk+ Rpmhk fA h ij−Rpmjh fA k ih], fRe k mij = 1 2(f−1)(R k mij + Rmjik)− 1 2(f−1)y p∇ iRpmjk +4(f−1)1 2y pys[R k pmh(R h sij + R h sji )− R h smjR k pih ] +4(f−1)1 2y p[2f iRpmjk+ R k pmh fAh ij−Rpmjh fAk ih], fRe k mij = 1 2R k mji + 1 2(f−1)(R k imj + R k ijm)− 1 2(f−1)y p∇ mRpijk − 1 4(f−1)2y pys[R h sij ((f− 1)R k mhp+ R k pmh)− R k pih (R h smj + R h sjm)] +4(f−1)1 2y p[2f mRpijk− (R h pij fAk mh−R k pih fAh mj)], fRe k mij =− 1 2(f−1)(R k imj + Rijmk) + 1 2(f−1)y p∇ mRpijk +4(f−1)1 2y p[−2f mRpijk+ R h pij fAk mh−R k pih fAh mj] − 1 4(f−1)2ypys[Rpihk(Rsmjh+ Rsjmh)− RsijhRpmhk] fRe k mij = 1 2(f−1)y p(∇ mRpjik− ∇iRpjmk) + 1 4(f−1)2y pys(R h sji R k pmh− R h sjmR k pih ) +4(f−1)1 2y p[2f iRpjmk− 2fmRpjik+ R h pji fAk mh−Rpjmh fAk ih], fRe k mij = 12y p(∇ iRmjpk− ∇mRijpk) +2(f1−1)y p[∇ i(Rpmjk+ Rpjmk) −∇m(Rpijk+ Rpjik)] + 1 4(f−1)2yp[2fm(Rpijk+ Rpjik) −2fi(Rpmjk+ Rpjmk) + (Rpmjh+ Rpjmh)fA k ih− (Rpijh+ Rpjih)fA k mh +R k pih fA h mj− RpmhkfA h ij+ (f− 1)(RihpkfA h mj− RmhpkfA h ij)]
+4(f−1)1 2ypys[Rpihk(Rsmjh+ Rsjmh)− Rpmhk(Rsijh+ Rsjih)
+(f− 1)(Rihpk(Rsmjh+ Rsjmh)− Rmhpk(Rsijh+ Rsjih) +RphmkRijsh− RphikRmjsh− 2RmishRphjk)]
+4(f−1)1 2[2fmfA k ij−2fifA k mj+fAkih fAmjh − fAkmhfAhij + 2(f− 1)(∇ifA k mj− ∇mfA k ij)], fRe k mij = R k mij + 1 2(f−1)y p(∇ iRpjmk− ∇mRpjik) + 1 4(f−1)2yp[2fmRpjik− 2fiRpjmk −R h pji fAkmh+ R h pjmfAkih] + 1 4(f−1)2y pys[(f− 1)R k ihp R h sjm+ RphikR h sjm −(f − 1)R k mhpRsjih− RkpmhRsjih] fRe k mij = R k mij + 1 2(f−1)y p[∇ m(Rpijk+ R k pji )− ∇i(Rpmjk+ R k pjm)] +4(f1
−1)2yp[−2fm(Rpijk+ Rpjik) + 2fi(Rpmjk+ Rpjmk) + (Rpijh+ Rpjih)fAkmh
−(R h pmj+ R h pjm ) fAk ih+ R k pmh fAh ij− R k pih fAh mj] +4(f−1)1 2y pys[R k pmh(R h sij + R h sji )− (f − 1)R k phmR h ijs − R k pih (R h smj + R h sjm ) +(f− 1)R k phi Rmjsh+ 2(f− 1)RmishRphjk]− 1 4(f−1)2[ 2fmfAkij− 2fifAkmj +fAk ih fAh mj −fAkmh fAh ij+ 2(f− 1)(∇ifAkmj− ∇mfAkij)]
with respect to the adapted frame{Eβ} .
We now compare the geometries of the Riemannian manifold (M, g) and its tangent bundle T M equipped with the Riemannian metricfG.e
Theorem 4.1. Let (M, g) be a Riemannian manifold and T M be its tangent bundle
with the Riemannian metricfG. Then T M is flat if M is flat ande
2fmfAkij− 2fifAkmj+ fAk ih fAh mj − fAk mh fAh ij+ 2(f− 1)(∇ifAkmj− ∇mfAkij) = 0.
Proof. It follows from the equations (4.1) that if 2fmfAkij− 2fifAkmj+ fAk ih fAh mj − fAk mh fAh ij+ 2(f− 1)(∇ifAkmj− ∇mfAkij) = 0, then R≡ 0 impliesfRe≡ 0.
Corollary 4.1. Let (M, g) be a Riemannian manifold and T M be its tangent bundle
with the Riemannian metric fG. Assume that f = C(const.). In the case, T M ise
flat if and only if M is flat.
5. The scalar curvature
We now turn our attention to the Ricci tensor and scalar curvature of the Rie-mannian metric fG. Lete fRe
αβ =fReσαβ σ and
fer =f Geαβ fRe
αβ denote the Ricci
tensor and scalar curvature of the Riemannian metricfG, respectively. From (4.1),e the components of the Ricci tensorfReαβ are characterized by
(5.1) fRe ij =− 1 4(f−1)2y pysR m pih R h sjm, fRe ij =− 1 2(f−1)Rij+ 1 2(f−1)y p(∇ pRij− ∇iRpj)−4(f−1)1 2y pysR m pih R h sjm +4(f−1)1 2yp(n− 4)fmRpijm, fRe i j =− 1 2(f−1)Rji+ 1 2(f−1)y p(∇ pRji− ∇jRpi)−4(f−1)1 2y pysR h sjmRpihm +4(λ−1)1 2y p(n− 4)f mRpjim, fRe ij= ff−2−1Rij+2(f1−1)yp(2∇pRij− ∇iRpj− ∇jRpi) + 1 4(f−1)2y p[(n− 4)f m(Rpijm+ R m pji )] + 1 4(f−1)2y pys[−R m pih R h sjm +(f− 1)R m
phi Rmjsh+ 2(f− 1)RmishRphjm+ (f− 1)RihpmRsmjh]
− 1
4(f−1)2[2fmfAmij − 2fifAmmj−fAmmhfAhij +fAmihfAhmj
+2(f− 1)(∇ifAmmj− ∇mfAmij)]
with respect to the adapted frame{Eβ}. From (3.3) and (5.1), the scalar curvature
of the Riemannian metricfG is given bye fer= 1 f−1r− 1 2(f−1)2ypysRphikRshik− 1 4(f−1)3gij[2fmfAmij − 2fifAmmj −fAm mh fAh ij+fAmih fAh mj+ 2(f− 1)(∇ifAmmj− ∇mfAmij)].
Thus we have the result as follows.
Theorem 5.1. Let (M, g) be a Riemannian manifold and T M be its tangent bundle
with the metricfG. Let r be the scalar curvature of g ande fer be the scalar curvature
of fG. Then the following equation holds:e fer= 1 f− 1r− 1 2(f− 1)2y pysR phikRshik− fL, where fL = 1 4(f−1)3g ij[2f mfAmij − 2fifAmmj−fAmmh fAh ij +fAm ihfAhmj+ 2(f− 1)(∇ifAmmj− ∇mfAmij)].
Corollary 5.1. Let (M, g) be a Riemannian manifold and T M be its tangent bundle
with the metricfG. Ife fer= 0, thenfL = 0 implies r = 0.
Let (M, g), n > 2, be a Riemannian manifold of constant curvature κ, i.e. Rphim= κ(δpmghi− δhmgpi)
and
r = n(n− 1)κ
where δ is the Kronecker’s. By virtue of Theorem 5.1, we have
fer = 1 f − 1r− 1 2(f− 1)2y pysR phikRshik− fL = 1 f − 1r− 1 2(f− 1)2y pysg kmRphimghlgitRsltk−fL = 1 f − 1n(n− 1)κ − fL − 1 2(f− 1)2y pysg km(κ(δpmghi− δhmgpi))ghlgit(κ(δskglt− δlkgst)) = 1 f − 1n(n− 1)κ − fL − 1 2(f− 1)2κ 2 ypys(gkpδli− gpiδkl)(δ k sδ i l− δ k lδ i s) = 1 f − 1n(n− 1)κ − 1 2(f− 1)22(n− 1)κ 2 gpsypys−fL = (n− 1)κ f− 1 (n− κ f− 1∥y∥ 2 )− fL. Hence we have the theorem below.
Theorem 5.2. Let (M, g), n > 2, be a Riemannian manifold of constant curvature
κ. Then the scalar curvaturefer of (T M,fG) ise fer = (n− 1)κ f− 1 (n− κ f− 1∥y∥ 2 )− fL. where∥y∥2= gpsypys and
fL = 1 4(f−1)3g ij[2f mfAmij − 2fifAmmj− fAm mh fAh ij +fAm ihfAhmj+ 2(f− 1)(∇ifAmmj− ∇mfAmij)].
6. Locally conformally flat tangent bundles
In this section we investigate locally conformally flatness property of T M equipped with the Riemannian metricfG.e
Theorem 6.1. Let M be an n-dimensional Riemannian manifold with the
Rie-mannian metric g and let T M be its tangent bundle with the RieRie-mannian metric
fG. The tangent bundle T M is locally conformally flat if and only if M is locallye
Proof. The tangent bundle T M with the Riemannian metricfG is locally confor-e mally flat if and only if the components of the curvature tensor of T M satisfy the following equation: (6.1) fRe αγβσ =− fer 2(2n−1)(n−1) { fGe αβfGeγσ− fGeασfGeγβ } +2(n1−1)(fGeγσ fReαβ−fGeασfReγβ+fGeαβfReγσ−fGeγβ fReασ), wherefReαγβσ =fGeσϵ fReαγβ ϵ.
From (6.1), we have the following special cases:
fRe mijk = − fer 2(2n− 1)(n − 1)(gmjgik− gmkgij) + 1 2(n− 1)(gik fRe mj (6.2) −gmk fReij+ gmj fReik− gij fRemk) and fRe mijk = − fer 2(2n− 1)(n − 1)(gmjgik− gmkgij) + 1 2(n− 1)(gik fRe mj (6.3) −gmk fReij+ gmj fReik− gij fRemk).
By the first and second equation in (4.1) and (3.2), fromfRe
αγβσ =fGeσϵ fReαγβ ϵ,
we obtainfRemijk= 0 and fRemijk= 0. Hence from (6.2) and (6.3), we obtain
(6.4) fer (2n− 1)(gmjgik− gmkgij) = gik fRe mj− gmk fReij+ gmj fReik− gij fRemk and (6.5) fer (2n− 1)(gmjgik− gmkgij) = gik fRe mj− gmk fReij+ gmj fReik− gij fRemk,
it follows that fReik =fReik. By means of the first and second equations in (5.1), we get Rij= 0, fm= 0, i.e. f = C(constant) and (6.6) fReij=− 1 4(f− 1)2y pysR m pih R h sjm. Transvecting (6.5) by gik, we obtain (6.7) (n− 1) fer (2n− 1) gmj= (n− 2) fRe mj+ g ikg mj fReik. Transvecting (6.7) by gmj, we get (6.8) n(n− 1) (2n− 1) fer = 2(n − 1)gik fRe ik.
On the other hand, from (6.6), we have gik fReik = − 1 4(f− 1)2y pysgik R m pih R h skm (6.9) = 1 4(f− 1)2y pysR pilhRs ilh = −1 2 fer.
Thus by (6.8) and (6.9), we obtainfer = 0, then it follows R
pilhRs ilh= 0 by using
f = C(constant). This shows Rpilh= 0. This completes the proof.
7. Curvature properties of another metric connection of the Riemannian metricfGe
Let ∇ be a linear connection on an n−dimensional differentiable manifold M. The connection ∇ is symmetric if its torsion tensor vanishes, otherwise it is non-symmetric. If there is a Riemannian metric g on M such that ∇g = 0, then the connection∇ is a metric connection, otherwise it is non-metric. It is well known that a linear connection is symmetric and metric if and only if it is the Levi-Civita connection. In section 4, we have considered the Levi-Levi-Civita connection
f∇ of the Riemannian metrice fG on the tangent bundle T M over (M, g). Thee
connection is the unique connection which satisfies f∇eαfGeβγ = 0 and has a zero
torsion. H. A.Hayden [8] introduced a metric connection with a non-zero torsion on a Riemannian manifold. Now we are interested in a metric connection (M )∇ofe the Riemannian metricfG whose torsion tensore (M )∇
Tε
γβ is skew-symmetric in the
indices γ and β. We denote components of the connection (M )∇ bye (M )eΓ. The metric connection(M )∇ satisfiese
(7.1) (M )∇eαfGeβγ= 0 and (M )eΓ γ αβ− (M )eΓγ βα= (M )∇ Tαβγ .
On the equation (7.1) is solved with respect to (M )eΓγαβ, one finds the following solution [8]
(7.2) (M )eΓγαβ= feΓγαβ+ eUαβγ , wherefeΓγ
αβis components of the Levi-Civita connection of the Riemannian metric fG,e (7.3) Ueαβγ = 1 2( (M )∇ Tαβγ+ (M )∇ Tγαβ+ (M )∇ Tγβα) and e Uαβγ = Uαβϵ fGe ϵγ, (M )∇ Tαβγ = Tαβϵ fGe ϵγ. If we put (7.4) (M )∇Tijr = ypRijpr all other(M )∇Tαβγ not related to(M )∇Tr
ij being assumed to be zero. We choose this (M )∇
Tαβγ in T M which is skew-symmetric in the indices γ and β as torsion tensor and determine a metric connection in T M with respect to the Riemannian metric
fG (see also, [16, p.151-155]. By using (7.3) and (7.4), we get non-zero componentse of eUαβγ as follows: e Ukij = −1 2(f− 1)y p(R k pij + R k pji ), e Ukij = 1 2y pR k ijp + 1 2(f− 1)y p(R k pij + R k pji ), e Ukij = −1 2(f− 1)y pR k pij , e Ukij = 1 2(f− 1)y pR k pij , e Ukij = −1 2(f− 1)y pR k pji , e Ukij = 1 2(f− 1)y pR k pji
with respect to the adapted frame. From (7.2) and (3.10), we have components of the metric connection(M )∇ with respect toe fG as follows:e
(M )eΓk ij= Γkij+2(f1−1) fAk ij, (M )eΓk ij=−2(f1−1) fAk ij, (M )eΓk ij= Γ k ij, (M )eΓk ij= 0, (M )eΓk ij= 0 (M )eΓk ij= 0, (M )eΓk ij = 0, (M )eΓk ij = 0
with respect to the adapted frame, where Rhjisare the local coordinate components of the curvature tensor field R of g.
Remark 7.1. The metric connection (M )∇ and he Levi-Civita connectione f∇ one
T M of the Riemannian metric fG coincide if and only if the base manifold M ise
flat.
The non-zero components of the curvature tensor(M )R of the metric connectione (M )∇ are given as follows:e
(M )Re k mij =Rmijk− 1 4(f−1)2[ 2fmfAkij− 2fifAkmj +fAkihfAhmj −fAkmhfAhij+ 2(f− 1)(∇ifAkmj− ∇mfAkij)] (M )Re k mij = 1 4(f−1)2[ 2fmfAkij− 2fifAkmj +fAk ihfAhmj −fAkmhfAhij+ 2(f− 1)(∇ifAkmj− ∇mfAkij)] (M )Re k mij =R k mij
with respect to the adapted frame.
The non-zero component of the contracted curvature tensor field (Ricci tensor field)(M )Re
γβ=(M )Re α
αβγ of the metric connection(M )∇ is as follows:e
(M )Re ij=Rij−4(f−1)1 2[ 2fmfAmij − 2fifAmmj +fAm ih fAh mj −fAmmh fAh ij+ 2(f− 1)(∇ifAmmj− ∇mfAmij)]
For the scalar curvature(M )er of the metric connection (M )∇ with respect toe fG ,e we obtain (M )er = 1 f− 1r− fL where fL = 1 4(f− 1)3g ij[2f mfAmij − 2fifAmmj− fAm mh fAh ij+ fAm ih fAh mj +2(f− 1)(∇ifAmmj− ∇mfAmij)].
Thus we have the following theorem.
Theorem 7.1. Let M be an n-dimensional Riemannian manifold with the
Rie-mannian metric g and let T M be its tangent bundle with the RieRie-mannian metric
fG. Then the tangent bundle T M with the metric connectione (M )∇ has a vanishinge scalar curvature with respect to the Riemannian metric fG if the scalar curvaturee
r of the Levi-Civita connection of g is zero andfL = 0.
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Department of Mathematics, University of Ataturk, Erzurum-TURKEY
E-mail address: agezer@atauni.edu.tr
Igdir Vocational School, University of Igdir, Igdir-TURKEY
E-mail address: lokman.bilen@igdir.edu.tr
Department of Mathematics, University of Ataturk, Erzurum-TURKEY
E-mail address: cagri.karaman@ogr.atauni.edu.tr
Department of Mathematics, University of Erzincan, Erzincan-TURKEY
