On metric connections with torsion on the cotangent bundle with modified Riemannian extension

0.19 MB·

by  Lokman Bilen

ON METRIC CONNECTIONS WITH TORSION ON THE
COTANGENT BUNDLE WITH MODIFIED RIEMANNIAN
6 EXTENSION
1
0
2 LOKMANBILENANDAYDINGEZER

n
a Abstract. LetMbeann−dimensionaldifferentiablemanifoldequippedwith
J atorsion-freelinearconnection∇andT∗M itscotangentbundle. Thepresent
8 paperaimstostudyametricconnection∇e withnonvanishingtorsiononT∗M
2 with modified Riemannian extension g∇,c. First, we give a characterization
offibre-preservingprojectivevector fieldson(T∗M,g∇,c)withrespecttothe
G] metric connection ∇e. Secondly, we study conditions for (T∗M,g∇,c) to be
e
semi-symmetric, Ricci semi-symmetric, Z semi-symmetric or locally conhar-
D monically flat with respect to the metric connection ∇e. Finally, we present
. some results concerning the Schouten-Van Kampen connection associated to
h
t theLevi-Civitaconnection ∇ofthemodifiedRiemannianextensiong∇,c.
a Mathematics subject classification 2010. 53C07,53C35,53A45.
m Keywordsandphrases. Cotangentbundle,fibre-preservingprojectivevec-
torfield,metricconnection, Riemannianextension,semi-symmetry.
[

1
v
6
1. Introduction
9
6
7 Let(M,∇)beann−dimensionaldifferentiablemanifoldequippedwithatorsion-
0 free linear connection ∇. Denote by T∗M the cotangent bundle of M and let π
1. be the natural projection T∗M → M. The vertical distribution V on T∗M (V is
0 the kernel of the submersion T∗M → M), which is the integrable distribution. If
6 M is a paracompactmanifold there exists a C∞−distribution H on T∗M which is
1
supplementary to the vertical distribution V, such as the Whitney sum TT∗M =
:
v HT∗M ⊕VT∗M holds.
Xi Forthetorsion-freelinearconnection∇onM,thecotangentbundleofM,T∗M,
r can be endowed with a pseudo-Riemannian metric g∇ of neutral signature, called
a the Riemannian extension of ∇, given by
g (HX,HY) = 0
∇
g (HX,V ω) = g (Vω,HX)=ω(X)
∇ ∇
g (Vω,V θ) = 0
∇
where HX and HY denote the horizontal lifts of the vector fields X and Y, and
Vω and Vθ denote the vertical lifts of the covectors (1−forms) ω and θ. Thus,
the Riemannian extension of (M,∇) is a pseudo-Riemannian manifold (T∗M,g ).
∇
Riemannian extensions were first defined and studied by Patterson and Walker
[16] and then investigated in Afifi [2]. Moreover, Riemannian extensions were also
considered in Garcia-Rio et al. [7] in relation to Osserman manifolds (see also
Derdzinski [5]). For further references relation to Riemannian extensions, see [1,
6, 10, 15, 21, 22, 23]. Classical Riemannian extensions have been generalized in
1
2 LOKMANBILENANDAYDINGEZER
several ways, see, as an example [13]. In [3, 4], the authors introduced another
generalization which is called modified Riemannian extension. For a symmetric
(0,2)-tensor field c on (M,∇), this metric is given by g =g +π∗c, that is,
∇,c ∇
g (HX,HY) = c(X,Y)
∇,c
g (HX,V ω) = g (Vω,HX)=ω(X)
∇,c ∇,c
g (Vω,V θ) = 0.
∇,c
In this paper, we consider a metric connection ∇ with nonvanishing torsion on
thecotangentbundleT∗M withmodifiedRiemannianextensiong . First,wegive
∇,c
a necessary and sufficient condition for a vector fieeld on (T∗M,g∇,c) to be fibre-
preservingprojectivevectorfieldonT∗M withrespectto themetricconnection∇.
This condition is representedby a set of relations involving certaintensor fields on
M. Secondly, we investigate the conditions for the cotangent bundle (T∗M,g∇,ec)
to be semi-symmetric, Ricci semi-symmetric, Z semi-symmetric and locally con-
harmonically flat with respect to the metric connection ∇. Finally, we show that
the Schouten-Van Kampen connection associateed to the Levi-Civita connection ∇
of the modified Riemannian extension g∇,c is equal to tehe horizontal lift H∇ of
the torsion-free linear connection ∇ to T∗M and present a result concerning the
curvature tensor of the Schouten-Van Kampen connection.
The manifolds, tensor fields and geometric objects we consider in this paper are
assumedtobe differentiableofclassC∞. Einstein’ssummationconventionisused,
the range of the indices h,i,j,k,l,m,r, being always {1,...,n}.
2. Preliminaries
We refer to [24] for further details concerning the material of this section. Let M
be an n−dimensional differentiable manifold with a torsion-free linear connection
∇ and denote by π : T∗M → M its cotangent bundle with fibres the cotangent
spaces to M. Then T∗M is a 2n−dimensional smooth manifold and some local
charts induced naturally from local charts on M, may be used. Namely, a system
of local coordinates U,xi , i = 1,...,n on M induces on T∗M a system of local
coordinates π−1(U)(cid:0), xi,(cid:1)xi =pi , i=n+i=n+1,...,2n, where xi =pi is the
components(cid:16)of covectors p in each(cid:17)cotangent space Tx∗M, x ∈ U with respect to
the natural coframe dxi .
Let X = Xi ∂ and ω = ω dxi be the local expressions in U of a vector field
∂xi (cid:8) (cid:9) i
X and a covector field ω on M, respectively. Then the vertical lift Vω of ω, the
horizontal lift HX of X are given, with respect to the induced coordinates, by
Vω =ω ∂ ,
i i
and
HX =Xi∂ +p ΓhXj∂
i h ij i
where ∂ = ∂ , ∂ = ∂ and Γh are the coefficients of ∇ on M.
i ∂xi i ∂xi ij
Next, we can introduce a frame field on each induced coordinate neighborhood
π−1(U) of T∗M. It is called the adapted frame and consists of the following 2n
ON METRIC CONNECTIONS WITH TORSION ON THE COTANGENT BUNDLE 3
linearly independent vector fields {E }= E ,E :
β j j
n o
E = ∂ +p Γa ∂
j j a hj h
E = ∂ .
(cid:26) j j
The indices α,β,γ,... = 1,...,2n indicate the indices with respect to the adapted
frame. The Lie brackets of the adapted frame of T∗M satisfy the following identi-
ties:
[E ,E ]=p R sE ,
i j s ijl l
(2.1)  Ei,Ej =−ΓjilEl,
 h E ,iE =0,
i j
where Rijls denote the coefficients ohf the cuirvature tensor R of ∇ on M.
With respect to the adaptedframe {E }, the vector fields Vω and HX on T∗M
β
has the components
0 Xj
Vω = and HX = .
ω 0
j
(cid:18) (cid:19) (cid:18) (cid:19)
3. The metric connection with nonvanishing torsion on the
cotangent bundle with respect to modified Riemannian extension
Let us consider T∗M equipped with the modified Riemannian extension g
∇,c
for a given torsion-free connection ∇ on M. In adapted frame {E }, the modified
β
Riemannianextension(g ) anditsinverse(g )βγ haveinthefollowingforms:
∇,c βγ ∇,c
c δj
(3.1) (g ) = ij i .
∇,c βγ δi 0
(cid:18) j (cid:19)
0 δi
(3.2) (g )βγ = j
∇,c δj −c
(cid:18) i ij (cid:19)
where c are the components of the symmetric (0,2)−tensor field c on (M,∇).
ij
For the Levi-Civita connection ∇ of the modified Riemannian extension g ,
∇,c
we get:
Proposition 1. Let ∇ be a torsion-free linear connection on M and T∗M be the
cotangentbundlewith themodified Riemannextension g over (M,∇). The Levi-
∇,c
Civita connection ∇ of (T∗M,g ) is given by
∇,c
∇ E =0, ∇ E =0,
Ei j Ei j
(3.3) ∇ E =−Γj E ,
 Ei j ih h
 ∇EiEj =ΓhijEh+{psRhji s+ 21(∇icjh+∇jcih−∇hcij)}Eh
with respectto the adapted frame {E }, where Γh and R s respectively denote
β ij hji
components of ∇ and its curvature tensor field R on M (see, [8]).
4 LOKMANBILENANDAYDINGEZER
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.
The Levi-Civita connection∇ of the modified Riemannian extensiong on T∗M
∇,c
istheuniqueconnectionwhichsatisfies∇ (g ) =0andhasazerotorsion. Now
α ∇,c βγ
we are interested in a metric connection ∇ of the modified Riemannian extension
g whose torsion tensor Tε is skew-symmetric in the indices γ and β. Metric
∇,c γβ
connectionwithnonvanishingtorsiononReiemannianmanifoldswereintroducedby
Hayden[9]. Wedenotecomeponentsofthemetricconnection∇byΓγ . Themetric
αβ
connection ∇ satisfies
e e
∇ (g ) =0 and Γγ − Γγ = Tγ .
e α ∇,c βγ αβ βα αβ
When the above equation is solved with respect to Γγ , one finds the following
e e e αβe
solution [9]
(3.4) Γγ =Γγ +Uγ , e
αβ αβ αβ
γ
where Γ is the components of the Levi-Civita connection ∇ of the modified
αβ e e
Riemannian extension g ,
∇,c
1
(3.5) U = (T +T +T )
αβγ αβγ γαβ γβα
2
and
e e e e
U =Uǫ (g ) , T =Tǫ (g ) .
αβγ αβ ∇,c ǫγ αβγ αβ ∇,c ǫγ
If we choose the torsion tensor T as
e e e e
Tr =−p R s,
(3.6) eij s ijr
otherwise=0,
(cid:26)
e
with the help of (3.6), from (3.5), we find non-zero component of Uγ as follows:
αβ
Uh=p R s
ij s jhi e
withrespecttotheadaptedframe. Inviewof(3.3)and(3.4),wehavethefollowing
e
proposition.
Proposition 2. Let ∇ be a torsion-free linear connection on M and T∗M be
the cotangent bundle with the modified Riemann extension g over (M,∇). The
∇,c
metric connection ∇ on T∗M with respect to the modified Riemannian extension
g satisfies
∇,c
e
∇ E =0, ∇ E =0,
Ei j Ei j
(3.7) ∇ E =−Γj E ,
 Ei j ih h
 ∇EiEj =ΓhiejEh+ 21(∇icjeh+∇jcih−∇hcij)Eh
e
with respect to the eadapted frame {Eβ}.
The horizontal lift H∇ of the torsion-free linear connection ∇ on M to T∗M is
characterizedthe following conditions:
H∇ Vθ =0, H∇ HY =0
Vω Vω
H∇ Vθ = V(∇ θ), H∇ HY = H(∇ Y)
HX X HX X
(cid:26)
ON METRIC CONNECTIONS WITH TORSION ON THE COTANGENT BUNDLE 5
forallvectorfieldsX,Y andcovectorfieldsω,θ onM ([24],p. 287). Intheadapted
frame, the followings satisfy (see, also [1])
H∇ E =0, H∇ E =0,
Ei j Ei j
H∇ E =−Γj E , H∇ E =ΓhE .
(cid:26) Ei j ih h Ei j ij h
From these formulas, we can readily deduce:
Proposition 3. Let ∇ be a torsion-free linear connection on M and T∗M be
the cotangent bundle with the modified Riemann extension g over (M,∇). The
∇,c
metric connection ∇ on T∗M of the modified Riemannian extension g coincides
∇,c
with thehorizontal lift H∇ of the torsion-free linear connection ∇ on M if and only
if the components ce of c satisfy the condition
ij
∇ c +∇ c −∇ c =0.
i jh j ih h ij
3.1. Projective vector fields on the cotangent bundle with respect to the
metric connection ∇. Given a linear connection ∇ on a manifold M, a vector
field V is said to be a projective vector field if there exists a 1−form θ such that
e
(L ∇)(X,Y)=θ(X)Y +θ(Y)X
V
for any pair of vector fields X and Y on M. In particular, if θ = 0, V is an affine
Killing vector field.
Let V be a vector field on T∗M and (vh,vh) its the components with respect to
the adapted frame {E }. The components vh and vh are said to be the horizontal
β
componeents and vertical components of V, respectively. As is known, a vector
fieldis calleda fibre-preservingvectorfield ifand only if its horizontalcomponents
depend only on the variables (xh). Hencee, one can easily say that every fibre-
preserving vector field V on T∗M induces a vector field V with components (vh)
on the base manifold M.
By straightforwardcealculations, we have the following.
Lemma 1. Let V be a fibre-preserving vector field on T∗M with components
(vh,vh). The Lie derivatives of the adapted frame satisfy
e
i)LVeEi = − Eivk Ek− vapsRiask+Eivk−vaΓika Ek,
ii)LVeEi = −(cid:0)vaΓa(cid:1)ki +Eiv(cid:16)k Ek, (cid:17)
(cid:16) (cid:17)
where Le denotes the Lie derivation with respect to V.
V
The general forms of fibre-preserving projective vector fields on T∗M with re-
e
spect to the metric connection ∇ are given by:
Theorem 1. Let ∇ be a torsion-free linear connection on M and T∗M be the
e
cotangent bundle with the modified Riemannian extension g over (M,∇). Then
∇,c
a vector field V is a fibre-preserving projective vector field with associated 1−form
θ on T∗M with respect to the metric connection ∇ if and only if the vector field V
is defined by e
e(3.8) X =HV +V B+γeA, e
where the vector field V =(vh), the covector field B =(B ), the (1,1)−tensor field
h
e
A=(Ah) and the associated 1−form θ satisfy
i
e
6 LOKMANBILENANDAYDINGEZER
(i) θ =θ dxi,
i
(ii) ∇ Ai =θ δk−vaR i ,
j k j i jak
(iii)eL Γh =θ δh+θ δh
V ij i j j i
(iv) ∇ ∇ B +R aB + 1va∇ M + 1(∇ va)M
i j l lji a 2 a ijl 2 j ial
+1(∇ va)M −AaM =0 (M :=∇ c +∇ c −∇ c )
2 i ajl l ija ijl i jl j il l ij
(v) ∇ ∇ As+R aAs −R sAa+va∇ R s+(∇ va)R s =0.
i j l lji a aji l i jal i jal
Proof. Afibre-preservingvectorfieldV =vhE +vhE onT∗M isafibre-preserving
h h
projectivevectorfieldifandonlyifthereexista1−formθ withcomponents(θ ,θ )
i i
on T∗M such that e
e e e
(3.9) (LVe ∇)(Y,Z) = LVe(∇YeZ)−∇Ye(LVeZ)−∇(LVeYe)Z
= θ(Y)Z+θ(Z)Y
e e e e e e e e e
for any vector fields Y˜ and Z on T∗M.
e e e e e e
Putting Y =Ei,Z =E¯j in (3.9), we get
e
(3.10) e e Ei E¯jvk¯ Ek¯ =θiE¯j +θ¯jEi.
(cid:16) (cid:17)
Putting Y =E ,Z =E in (3.9), we find
i j
(3.11) θ =0
i
and
(3.12) vaR i +E E vk¯ − E va¯ Γ a =θ δk.
jak i j i jk j i
In view of (3.11), (3.10) reduce(cid:16)s to (cid:17) (cid:0) (cid:1)
Ei E¯jvk¯ Ek¯ =0
from which it follows that (cid:16) (cid:17)
(3.13) vk¯ =p As +B
s k k
where As and B are certain functions which depend only on the variables (xh).
k k
The coordinate transformation rule implies that A is a (1,1)−tensor field with
components (As) and B is a covector field with components (B ). Hence, the
k k
fibre-preserving projective vector field V on T∗M can be written in the form:
V = vkE +vkE =vkE +{p As +B }E
k ke k s k k k
= HV +V B+γA
e
where γ is an operator applied to the (1,1)−tensor field A and expressed locally
γA=(p As)E (for details related to the operator γ, see [24], p.12−13).
s k k
Substitution (3.13) into (3.12) gives
(3.14) vaR i +∇ Ai =θ δk.
jak j k j i
Contracting i and k in (3.14), we have
1
θ = ∇ Ak.
j n j k
Finally, putting Y =E ,Z =E in (3.9), we obtain
i j
L Γh =θ δh+θ δh,
V ij i j j i
ON METRIC CONNECTIONS WITH TORSION ON THE COTANGENT BUNDLE 7
1
∇ ∇ B −R aB + va∇ M
i j l ijl a 2 a ijl
1 1
+ (∇ va)M + (∇ va)M −AaM
2 j ial 2 i ajl l ija
= 0
and
∇ ∇ As+R aAs −R sAa+va∇ R s+(∇ va)R s =0,
i j l lji a aji l i jal i jal
where M =∇ c +∇ c −∇c .
ijl i jl j il l ij
Conversely,ifB ,vh,θ andAh aregivensothattheysatisfy(i)−(v),reversing
h h i
the above steps, we see that V = HV +V B+γA is a fibre-preserving projective
vector field on T∗M with respect to the metric connection ∇. This completes the
proof. e (cid:3)
e
The below result follows immediately from Theorem 1 and its Proof.
Corollary 1. Let ∇ be a torsion-free linear connection on M and T∗M be the
cotangent bundle with the modified Riemannian extension g over (M,∇). Every
∇,c
fibre-preserving projective vector field V with respect to the metric connection ∇ is
of the form (3.8) and it naturally induces a projective vector field V on M.
e e
3.2. Semi-Symmetry properties of the cotangent bundle with respect to
the metric connection ∇. Given a manifold M (dim(M) ≥ 3) endowed with a
linear connection ∇ whose curvature tensor is marked as R, for any tensor field of
S of type (0,k),k ≥1, theetensor field R(X,Y).S is expressed in the form:
(R(X,Y).S)(X ,X ,...,X ) = −S(R(X,Y)X ,X ,...,X )
1 2 k 1 2 k
−...−S(X1,X2,...,Xk−1,R(X,Y)Xk)
forany vectorfields X ,X ,...,X ,X,Y onM,where R(X,Y) acts asa derivation
1 2 k
on S. If R(X,Y).S =0, then the manifold M is said to be S semi-symmetric with
respect to the linear connection ∇. A (pseudo-) Riemannian manifold (M,g) such
that its curvature tensor R satisfies the condition
R(X,Y).R=0
iscalledasemi-symmetricspace. Also,notethatlocallysymmetricspacesaresemi-
symmetric, but in general the converse is not true. The semi-symmetric space was
firststudiedbyCartan. Nevertheless,Sinjukovfirstusedthename”semi-symmetric
spaces” for manifolds satisfying the above curvature condition [18]. Later, Szabo
gave the full local and global classification of semi-symmetric spaces [19, 20]. A
(pseudo-)Riemannianmanifold(M,g)iscalledRiccisemi-symmetricifthefollowing
condition is satisfied:
R(X,Y).Ric=0,
where Ric is the Ricci tensor of (M,g). It is obvious that any semi-symmetric
manifold is Ricci semi-symmetric.
The curvature tensor R of the metric connection ∇ on T∗M is obtained from
the formula
e e
R(E ,E )E =∇ ∇ E −∇ ∇ E −∇ E
α β γ Eα Eβ γ Eβ Eα γ [Eα,Eβ] γ
with respect to the adapted frame. For the curvature tensor R of the metric con-
e e e e e e
nection ∇, with the help of (2.1) and (3.7), we have:
e
e
8 LOKMANBILENANDAYDINGEZER
Proposition 4. Let ∇ be a torsion-free linear connection on M and T∗M be
the cotangent bundle with the modified Riemann extension g over (M,∇). The
∇,c
curvature tensor R of the metric connection ∇ on T∗M satisfies the following con-
ditions:
e e
R(E ,E )E = R hE
i j k ijk h
1
+ {∇ (∇ c −∇ c )−∇ (∇ c −∇ c )
e i k jh h jk j k ih h ik
2
−R mc −R mc }E
ijk mh ijh km h
R(E ,E )E = R kE ,
i j k jih h
R(E ,E )E = 0, R(E ,E )E = 0, R(E ,E )E =0,
e i j k i j k i j k
R(E ,E )E = 0, R(E ,E )E =0, R(E ,E )E =0
e i j k e i j k e i j k
e e e
with respect to the adapted frame {E }.
β
Let X and Y be vector fields of T∗M. The curvature operator R(X,Y) is a
differential operator on T∗M. Similarly, for vector fields X and Y of M, R(X,Y)
is a diffeerentialeoperator on M. Now, we operate the curvature operaetoreRe(X,Y)
to the curvature tensor R. That is, for all Z,W and U, we consider the condition
(R(X,Y)R)(Z,W)U = 0. In the case, we shall call the cotangent bundle Te∗Me eas
semi-symmetric with respeect to the metric ceonfnectione∇.
eInethee aedapetefdfreame {E }, the tensor (R(X,Y)R)(Z,W)U is locally expressed
β
as follows: e
e e e e e f e
(3.15) ((R(X,Y)R)(Z,W)U) ε
αβγθσ
= R εR τ −R τR ε−R τR ε−R τR ε.
αeβτeγeθσe eαβfγ eτθσ αβθ γτσ αβσ γθτ
e e e e e e e e
Similarly, in local coordinates,
((R(X,Y)R)(Z,W)U) n
ijklm
= R nR p−R pR n−R pR n−R pR n.
ijp klm ijk plm ijl kpm ijm klp
Theorem2. Let∇beatorsion-freelinearconnectiononM andT∗M bethecotan-
gent bundle with the modified Riemannian extension g over (M,∇). Under the
∇,c
assumptionthat∇ (∇ c −∇ c )−∇ (∇ c −∇ c )−R mc −R mc =
i k jh h jk j k ih h ik ijk mh ijh km
0, where R is the curvature tensor of ∇, the cotangent bundle T∗M is semi-
symmetric with respect to the metric connection ∇ if and only if the base manifold
M is semi-symmetric with respect to ∇.
e
Proof. We consider the conditions (R(X,Y)R)(Z,W)U = 0 for all vector fields
X,Y,Z,W and U on T∗M.
Forallcasesα=(i,i),β =(j,j),γe=e(k,ek),eθ =e(fl,l)e,σ =(m,m)andε=(h,h)
ien (e3.1e5),fthe none-zero components of the tensor ((R(X,Y)R)(Z,W)U) ε are
αβγθσ
e e e e e f e
ON METRIC CONNECTIONS WITH TORSION ON THE COTANGENT BUNDLE 9
as follows:
(3.16) i) ((R(X,Y)R)(Z,W)U) h
ijklm
= R hR p+R hR p−R pR h−R pR h
iejp e kelme eijfp eklm ijk plm ijk plm
−R pR h−R pR h−R pR h−R pR h
e ijle kpm e ijle kpm e ijme klp e ijme klp
= R hR p−R pR h−R pR h−R pR h
iejp kelm iejk pelm iejl kepm iejm kelp
= ((R(X,Y)R)(Z,W)U) h.
ijklm
ii) ((R(X,Y)R)(Z,W)U) h
ijklm
= R hR p+R hR p−R pR h−R pR h
ijp kelme e iejp ekflm e ijk plm ijk plm
−R pR h−R pR h−R pR h−R pR h
e ijle kpm e ijle kpm e ijme klp e ijme klp
= −R mR p+R pR m+R pR m+R pR m
ijp klh ijh klp ijk plh ijl kph
e e e e e e e e
= −((R(X,Y)R)(Z,W)U) m.
ijklh
iii) ((R(X,Y)R)(Z,W)U) h
ijklm
= R hR p+R hR p−R pR h−R pR h
ijp eklme e eijp e kflme ijk plm ijk plm
−R pR h−R pR h−R pR h−R pR h
e ijle kpm e ijle kpm e ijme klp e ijme klp
If we assume that
e e e e e e e e
R h = ∇ (∇ c −∇ c )−∇ (∇ c −∇ c )
ijk i k jh h jk j k ih h ik
−R mc −R mc =0,
ijk mh ijh km
e
thenitfollowsfrom(3.16)that(R(X,Y)R)(Z,W)U =0ifandonlyif(R(X,Y)R)(Z,W)U =
0. This completes the proof. (cid:3)
e e e e e f e
Denote by R = R σ the contracted curvature tensor (Ricci tensor) of the
αβ σαβ
metricconnection∇. Theonlynon-zerocomponentofR isasfollows: R =R ,
αβ ij ij
where R deneote theecomponents of the Ricci tensor of ∇ on M. Now we prove
ij
the following theoreem. e e
Theorem 3. Let ∇ be a torsion-free linear connection on M and T∗M be the
cotangent bundle with the modified Riemannian extension g over (M,∇). The
∇,c
cotangent bundle T∗M is Ricci semi-symmetric with respect to the metric connec-
tion ∇ if and only if the base manifold M is Ricci semi-symmetric with respect to
∇.
e
Proof. The tensor (R(X,Y)Ric)(Z,W) has the components
(3.17) ((R(X,Y)Ric)(Z,W)) =R εR +R εR
e e e g e fαβγθ αβγ εθ αβθ γε
with respect to the adapted frame {E }.
β
e e e g e f e e e e
Choosing α=i,β =j,γ =k,θ =l in (3.16), we find
((R(X,Y)Ric)(Z,W)) = R pR +R pR
ijkl ijk pl ijl kp
= R pR +R pR
ijk pl ijl kp
e e e g e f e e e e
= ((R(X,Y)Ric)(Z,W)) ,
ijkl
all the others being zero. This finishes the proof. (cid:3)
10 LOKMANBILENANDAYDINGEZER
For the scalarcurvature r of the metric connection∇ with respectto g , with
∇,c
the help of (3.2) we find
e r=Rαβ(g∇,c)αβ =0. e
Thus we have the following theorem.
Theorem 4. Let ∇ be a torseion-efree linear connection on M and T∗M be the
cotangent bundle with the modified Riemannian extension g∇,c over (M,∇). The
scalar curvature of the cotangent bundle T∗M with the metric connection ∇ with
respect to g∇,c vanishes. e
e
Next, weshallapplythe differentialoperatorR(X,Y)to the torsiontensorT of
e
the metric connection ∇.
e e e e
Theorem 5. Let ∇ be a torsion-free linear connection on M and T∗M be the
e
cotangent bundle with the modified Riemannian extension g over (M,∇). Then
∇,c
R(X,Y).T =0for all vector fields X andY onT∗M,where T is thetorsion tensor
of the metric connection ∇ if and only if the base manifold M is semi-symmetric
weithereespeect to ∇. e e e
e
Proof. ThedifferentialoperatorR(X,Y)appliedthetorsiontensorT ofthemetric
connection ∇ is in the form:
((R(X,eY)eT)e(Z,W)) ε e
αβγθ
e
= R εT τ −R τT ε−R τT ε
αeβτeγθe e eαβfγ τθ αβθ γτ
with respect to the adapted frame {E }. It follows immediately that
β
e e e e e e
((R(X,Y)T)(Z,W)) h =R hT m+R hT m
ijkl ijm kl ijm kl
 −RijkmTmhl −RijkmTmhl −RijlmTkmh−RijlmTkmh
 e=eeps=(eRe−iejphms(Re(eRkflm(sX+,eYR)iRjkm)(RZeem,Wlsh)+eUeR)ijijkllmheRes,kmshee)
otherwise=0
which finishesthe proof. (cid:3)
Onan n−dimensionalRiemannianmanifold (M,g), it wasdefined a generalized
(0,2)−symmetric Z tensor given by [14]
Z(X,Y)=Ric(X,Y)+φg(X,Y)
for all vector fields X and Y on M, where where φ is an arbitrary scalar func-
tion. Analogous to this definition, it may be locally define generalizedZ tensor on
(T∗M,g ) with respect to the metric connection ∇ as follows:
∇,c
e
Z =R +φ(g ) .
αβ αβ ∇,c αeβ
Putting the values of R and g in the above equation, we have the non-zero
αβ e ∇,ce e
components
e
(3.18) Z = R +φc ,
ij ij ij
Z = φδi,
eij j e
Z = φδi.
eij e j
We can state the following theorem.
e e