ISSN 20779879
Eurasian
Mathematical Journal
2017, Volume 8, Number 4
Founded in 2010 by
the L.N. Gumilyov Eurasian National University in cooperation with
the M.V. Lomonosov Moscow State University the Peoples' Friendship University of Russia
the University of Padua
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c
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MUKHTARBAY OTELBAEV (to the 75th birthday)
On October 3, 2017 was the 75th birthday of Mukhtarbay Otel- baev, Doctor of Physical and Mathematical Sciences (1978), Pro- fessor (1983), academician of the National Academy of Sciences of the Republic of Kazakhstan (2004), Honored Worker of the Repub- lic of Kazakhstan (2012), laureate of the State Prize of the Republic of Kazakhstan in the eld of science and technology (2007), Direc- tor of the Eurasian Mathematical Institute (since 2001), Professor of the Department â»Fundamental Mathematicsâê of the L.N.
Gumilyov Eurasian National University, the editor-in-chief of the Eurasian Mathematical Journal (together with V.I. Burenkov and V.A. Sadovnichy).
M. Otelbaev was born in the village of Karakemer of the Kurdai district, Zhambyl region. He graduated from the M.V. Lomonosov Moscow State University (1969) and then completed his postgraduate studies at the same uni- versity (1972). There he defended his doctor of sciences thesis (1978).
Professor Otelbaev's scientic interests are related to functional analysis, dierential equa- tions, computational mathematics, and theoretical physics.
He introduced the q-averaging, which is now called the Otelbaev function; using it he ob- tained a number of fundamental results. For embedding of the Sobolev weighted spaces and the resolvent of the Schr¨odinger operator, he established criterions for the compactness and niteness of the type, as well as estimates of the eigenvalues of the Schr¨odinger and Dirac operators that are exact in order. He was the rst to establish that there is no universal asymptotic formula for the distribution function of the Sturm-Liouville operator. He obtained eective conditions for the separation of the dierential operators with nonsmooth and oscillating coecients, he developed an abstract theory of extension and contraction of operators which are not necessarily linear in linear topological spaces. M. Otelbaev proposed a new numerical method for solving boundary value problems, and a method for approximate calculation of eigenvalues and eigen- vectors of compact operators. He obtained the fundamental results in the theory of nonlinear evolution equations and in theoretical physics.
He has published more than 70 scientic papers in leading international journals entering the rating lists of Thomson Reuters and Scopus. Under his supervision 70 postgraduate students have defended their candidate of sciences theses, 9 of them became doctors of sciences. In 2006 and 2011 he was awarded the state grant "The best university teacher".
The Editorial Board of the Eurasian Mathematical Journal congratulates Mukhtarbay Otel- baev on the occasion of his 75th birthday and wishes him good health and new achievements in mathematics and mathematical education.
Award for the Eurasian Mathematical Journal
Dear readers, authors, reviewers and members of the Editorial Board of the Eurasian Mathematical Journal,
we are happy to inform you that in November 2017 the Eurasian Mathematical Journal was awarded the title "Leader of Science 2017" by the National Center of State Scientic-Technical Expertise of the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan in the nomination "Leader of Kazakhstan Scientic Publications" for the high level of publication activities and high level of citations in Web of Science Core Collection in 2014-2016.
Recall that the Eurasian Mathematical Journal was founded by the L.N. Gumilyov Eurasian National University in 2010 in co-operation with the M.V. Lomonosov Moscow State University, the Peoples' Friendship University of Russia and the University of Padua (see [1]).
The journal pulishes carefully selected original research papers in all areas of mathematics, survey papers, and short communications. It publishes 4 issues in a year. The language of the paper must be English only. Papers accepted for publication are edited from the point of view of English.
More than 280 papers were published written by mathematicians from more than 40 countries representing all continents.
In 2014 the journal was registered in Scopus and in September 2014 the Elsevier-Kazakhstan Research Excellence Forum was held at the L.N. Gumilyov Eurasian National University dedi- cated to this occasion in which the Elsevier Chairman Professor Y.S. Chi participated (see [3]
for details).
In 2015 the Eurasian Mathematical Journal was included in the list of Scopus mathematical journals, quartile Q4, and it is on the way to entering quartile Q3 (see [3]).
Attached is the invitation letter to the Rector of the L.N. Gumilyov Eurasian National University Professor E.B. Sydykov to the ceremony of awarding, which took place in Almaty on November 8, 2017.
On behalf of the Editorial Board of the EMJ V.I. Burenkov, E.D. Nursultanov, T.Sh. Kalmenov, R. Oinarov, M. Otelbaev, T.V. Tararykova, A.M. Temirkhanova
References
[1] B. Abdrayim, Opening address by the rector of the L.N. Gumilyov Eurasian National University, Eurasian Math. J. 1 (2010), no. 1, p. 5.
[2] Eurasian Mathematical Journal is indexed in Scopus, Eurasian Math. J. 5 (2014), no. 3, 6-8.
[3] V.I. Burenkov, E.D. Nursultanov, T.Sh. Kalmenov, R. Oinarov, M. Otelbaev, T.V. Tararykova, A.M. Temirkhanova, EMJ: from Scopus Q4 to Scopus Q3 in two years?!, Eurasian Math. J. 7 (2016), no.
3, p. 6.
EURASIAN MATHEMATICAL JOURNAL ISSN 2077-9879
Volume 8, Number 4 (2017), 18 34
SOME RESULTS ON RIEMANNIAN g− NATURAL METRICS
GENERATED BY CLASSICAL LIFTS ON THE TANGENT BUNDLE L. Bilen, A. Gezer
Communicated by J.A. Tussupov
Key words: ane Killing and Killing vector elds, conformal curvature tensor, Riemannian g−natural metric, metric connection, tangent bundle.
AMS Mathematics Subject Classication: 53C07, 53B20, 53C21.
Abstract. Let (M, g) be an n-dimensional Riemannian manifold and T M its tangent bundle equipped with Riemanniang−natural metrics which are linear combinations of the three classical lifts of the base metric with constant coecients. The purpose of the present paper is three-fold.
Firstly, to study conditions for the tangent bundleT M to be locally conformally at. Secondly, to dene a metric connection on the tangent bundleT M with respect to the Riemanniang−natural metric and study some its properties. Finally, to classify ane Killing and Killing vector elds on the tangent bundle T M.
1 Introduction
Let (M, g) be a Remannian manifold. On the tangent bundle T M of M one can construct several (pseudo-) Riemannian metrics obtained by lifting the metric g from the base manifold M to the tangent bundle T M. The rst known Riemannian metric which is called the Sasaki metric on the tangent bundle was constructed by S. Sasaki in [16]. It was shown in many papers that the study of some geometric properties of the tangent bundle endowed with the Sasaki metric led to the atness of the base manifold (for recent survey related to the Sasaki metric, see [7]). In the next years, using, in particular, various kinds of classical lifts of the metric g from M to T M, some authors were interested in nding other lifted metrics on the tangent bundles, with quite interesting properties (e.g. [4, 5, 6, 15, 18, 20]). According to this concept of lift, the Sasaki metric is no other than the diagonal lift of the base metric. V. Oproiu and his collaborators constructed natural metrics on the tangent bundles of Riemannian manifolds possessing interesting geometric properties ([11, 12, 13, 14]). All metrics mentioned above belong to a wide class of the so-called g-natural metrics on the tangent bundle, initially classied by O.
Kowalski and M. Sekizawa [10] 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 [10]).
In this paper we consider a tangent bundle T M equipped with a Riemanian g−natural metrics of the form G =aSg +bHg +cVg, where a, b and c are constants satisfying a >0 and a(a+c)−b2 > 0, generated by the classical lifts: the Sasaki metric Sg, the horizontal lift Hg and the vertical lift Vg of g. The Riemanian g-natural metric G is a particular subclass of all Riemannian g−natural metrics onT M in [1]. In [4], M.T.K. Abbassi and M. Sarih proved that the Riemanian g−natural metric G is as rigid as the Sasaki metric in the following sense: if (T M, G) is a space of constant scalar curvature, then (M, g) is at. Our aim is to study some properties of the Riemanian g−natural metric G in terms of adapted frame which allows the tensor calculus to be eciently done in T M.
Some results on Riemanniang−natural metrics generated by classical lifts 19 Throughout this paper, all manifolds, tensor elds and connections are always assumed to be dierentiable of class C∞. Also, we denote by=pq(M)the set of all tensor elds of type (p, q) onM, and by =pq(T M) the corresponding set on the tangent bundleT M.
2 Preliminaries
LetM be ann-dimensional Riemannian manifold and denote byπ:T M →M its tangent bundle which bres the tangent spaces toM. ThenT M is a2n−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) in M induces on T M a system of local coordinates
π−1(U), xi, xi =yi , i = n +i = n + 1, ...,2n, where (yi) is the cartesian coordinates in each tangent space TPM at P ∈ M with respect to the natural base ∂
∂xi |P , P being an arbitrary point in U whose coordinates are(xi). Summation over repeated indices is always assumed.
Let X =Xi ∂∂xi be the local expression in U of a vector eldX onM. Then the vertical lift
VX and the horizontal liftHX of X are given, with respect to the induced coordinates, by
VX =Xi∂i, (2.1)
HX =Xi∂i−ysΓiskXk∂i, (2.2) where ∂i = ∂x∂i, ∂i = ∂y∂i and Γisk are the coecients of the Levi-Civita connection ∇ of g.
LetSbe a(p, q)−tensor eld onM, q >1. We then consider the tensor eldγS ∈ =pq−1(T M) onπ−1(U)dened by
γS = (ysSsij12...j...ipq)∂j1 ⊗...⊗∂jp⊗dxi2 ⊗...⊗dxiq
with respect to the induced coordinates(xi, yi)([20], p. 12). The tensor eld γS dened on each π−1(U) determines global tensor eld on T M. For any C ∈ =11(M), we easily see that γC has components, with respect to the induced coordinates (xi, yi),
(γC) =ysCis∂i. (2.3)
Also, note that(γC)(Vf) = 0,f ∈ =00(M), i.e. γC is a vertical vector eld on T M.
With the connection ∇g of g onM, we can introduce on each induced coordinate neighbour- hood π−1(U) of T M a frame eld which is very useful in our computation. The adapted frame onπ−1(U)consists of the following 2n linearly independent vector elds:
Ej = ∂j −ysΓhsj∂h, Ej = ∂j.
We write the adapted frame as{Eβ}=
Ej, Ej . Straightforward calculations give the following lemma.
Lemma 2.1. [19, 20] The Lie brackets of the adapted frame of T M satisfy the following identi- ties:
[Ej, Ei] =ybRijbaEa, [Ej, Ei] = ΓajiEa, Ej, Ei
= 0,
where Rijba denote the components of the curvature tensor of M.
20 L. Bilen, A. Gezer
Using (2.1), (2.2) and (2.3), we have
HX =XjEj, (2.4)
VX =XjEj, (2.5)
and
γC =ysCjsEj (2.6)
with respect to the adapted frame {Eβ} (for details, see [20]).
3 On locally conformally at tangent bundles with Riemannian g− natural metrics of the form G = a
Sg + b
Hg + c
Vg
The Riemannian g−natural metric G=aSg +bHg+cVg on the tangent bundleT M over the Riemannian manifold(M, g) is dened by the following three equations:
G(HX,HY) = (a+c)V(g(X, Y)), (3.1) G(VX,HY) =G(HX,V Y) =bV(g(X, Y)), (3.2) G(VX,V Y) = aV(g(X, Y)) (3.3) for all X, Y ∈ =10(M), where the inequalities a >0 and a(a+c)−b2 >0hold [4].
If g =gijdxidxj is the expression of the Riemannian metric g on M, then from (3.1)-(3.3), the Riemannian g−natural metric G is expressed in the adapted local frame by
G= (Gαβ) =
(a+c)gij bgij bgij agij
. (3.4)
For the Levi-Civita connection of the Riemanniang−natural metricG, we have the following.
Proposition 3.1. [4] Let (M, g) be a Riemannian manifold, ∇ its Levi-Civita connection and R its curvature tensor. Then the corresponding Levi-Civita connection ∇e of T M with the Rie- mannian g-natural metric of the form G = aSg +bHg +cVg is characterized by the following equalities
i) ∇eHX H
Y = H(∇XY) + ab 2α
H[R(y, X)Y +R(y, Y)X] (3.5) +b2
α
V(R(X, y)Y − a(a+c) 2α
V(R(X, Y)y), ii) ∇eHX VY = −a2
2α
H(R(Y, y)X) + V(∇XY) + ab 2α
V(R(Y, y)X), iii) ∇eVX HY = −a2
2α
H(R(X, y)Y + ab 2α
V(R(X, y)Y), iv) ∇eVX VY = 0
for all X, Y ∈ =10(M), where α=a(a+c)−b2 and the inequalities a >0 and a(a+c)−b2 >0 hold.
Some results on Riemanniang−natural metrics generated by classical lifts 21 For the tangent bundle (T M, G), the conformal curvature tensor Ce is given by:
Ceαγβσ = Reαγβσ − 1
2(n−1)(GγσReαβ−GασReγβ+Gαβ Reγσ− Gγβ Reασ)
+ Se
2(n−1)(2n−1)(GαβGγσ−GασGγβ).
A Riemannian manifold (M, g) (dimM ≥ 4) is called locally conformally at if the conformal curvature tensor Ce= 0. In this section, we shall prove the following theorem.
Theorem 3.1. Let M be an n−dimensional Riemannian manifold and T M its tangent bundle with the Riemanniang−natural metricG=aSg+bHg+cVg, such thata >0and a(a+c)−b2 >
0. The tangent bundle T M is locally conformally at if and only if M is locally at and the Riemannian g−natural metric G is of the form G=a(Sg+H g) +cVg.
Now, let us consider the formula ∇eEαEβ = eΓγαβEγ with respect to the adapted frame {Eβ}, where eΓγαβ denote the components of the Levi-Civita connection ∇e of G. On taking account of (3.5) for the cases VX =Ei,HX =Ei and VY =Ej, HY =Ej, we have the following.
Lemma 3.1. The Levi-Civita connection ∇e of (T M, G) is characterized by the following equal- ities
∇eEiEj ={Γhij − ab
2αys(Risjh+Rjsih)}Eh+{b2
αysRisjh− a(a+c)
2α ysRijsh}Eh,
∇eEiEj ={−a2
2αysRjsih}Eh+{Γhij+ ab
2αysRjsih}Eh,
∇eE
iEj ={−a2
2αysRisjh}Eh+{ab
2αysRisjh}Eh,
∇eEiEj = 0,
(3.6)
with respect to the adapted frame{Eβ}, whereΓhij, and Rhji s respectively, denote the components of the Levi-Civita connection ∇ and the curvature tensor eld R of g on M with respect to the natural frame {∂i}.
The Riemannian curvature tensor Re of (T M, G) is obtained from the well-known formula
Re X,e Ye
Ze=∇e
Xe∇e
YeZe−∇e
Ye∇e
XeZe−∇e[X,eeY]Ze
for allX,e Y ,e Ze ∈ =10(T M). For pairsXe =Em, EmandYe =Ei, Ei andZe=Ej, Ej, by Lemma 2.1 and 3.1, we get the following lemma.
22 L. Bilen, A. Gezer
Lemma 3.2. The components of the curvature tensor Re of (T M, G) are as follows:
(i) R(Ee m, Ei)Ej
=
Rmijk+ ab 2αys
∇i Rmsjk+Rjsmk
− ∇m Risjk+Rjsik +a2
4αypys
RhsmkRijph−RhsikRmjph−2RmishRhpjk +a2b2
4α2ypys
Rmshk Ripjh+Rjpih
−Rishk Rmpjh+Rjpmh
Ek +
b2 αys
∇mRisjk− ∇iRmsjk
+ a(a+c) 2α ys
∇iRmjsk− ∇mRijsk +ab3
4α2ypys
Rsmhk Ripjh+Rjpih
+Rishk Rmpjh+Rjpmh +ab
4αypys[Rmhsk Ripjh+Rjpih
−Rihsk Rmpjh+Rjpmh +RhsikRmjph−RhsmkRijph+ 2RmishRhpjk] Ek
(ii) R(Ee m, Ei)Ej
= a2
2αys ∇iRjsmk− ∇mRjsik + a3b
4α2ypys
RjpihRmshk−RjpmhRishk
Ek +
Rmijk+ ab
2αys ∇mRjsik− ∇iRjsmk
+ a2b2 4α2ypys
RjpihRsmhk+RjpmhRishk + a2
4αypys
RmhskRjpih −RihskRjpmh
Ek (iii) R(Ee m, Ei)Ej
= ab
2α(Rmijk+Rjimk)− a2
2αys∇mRisjk + a3b
4α2ypys
RipjhRmshk−Rishk Rmpjh+Rjpmh
Ek +
−b2
α Rmijk+ a(a+c)
2α Rmjik+ ab
2αys∇mRisjk+ a2
4αypysRmhskRipjh + a2b2
4α2ypys
Rishk Rmpjh+Rjpmh
+RipjhRsmhk
Ek (iv) R(Ee m, Ei)Ej
=
−ab
2α (Rimjk+Rjmik) + a2
2αys∇iRmsjk +a3b
4α2ypys
Rmphk Rjsih+Risjh
−RishkRmpjh
Ek +
b2
αRimjk− a(a+c)
2α Rijmk− ab
2αys∇iRmsjk
− a2b2 4α2ypys
Rmshk Rjpih+Ripjh
+RmpjhRsihk
−a2
4αypysRishkRmpjh
Ek
Some results on Riemanniang−natural metrics generated by classical lifts 23
(v) R(Ee m, Ei)Ej = a2
2αRjimk− a4
4α2ypysRishkRjpmh
Ek +
−ab
2α Rjimk+ a3b
4α2ypysRishkRjpmh
Ek
(vi) R(Ee m, Ei)Ej =
−a2
2αRjmik+ a4
4α2ypysRmshkRjpih
Ek +
ab
2αRjmik− a3b
4α2ypysRmshkRjpih
Ek
(vii) R(Ee m, Ei)Ej = a2
αRmijk+ a4 4α2ypys
RmshkRipjh−RishkRmpjh
Ek +
ab
αRimjk− a3b 4α2ypys
RmshkRipjh−RishkRmpjh
Ek (viii) R(Ee m, Ei)Ej = 0
with respect to the adapted frame {Eβ}, where α =a(a+c)−b2 (for invariant forms of Re, see [4]).
Let Reαβ = Reσαβ σ denote the components of Ricci tensor of (T M, G). From Lemma 3.2 we have
Lemma 3.3. The components Reαβ of the Ricci tensor of (T M, G) are as follows:
(i) Reij = − a4
4α2ypysRishmRjpmh (ii) Reij = −ab
2αRji+ a2
2αys(∇sRji− ∇jRsi)
−a3b
4α2ypysRpjmhRsihm, (iii) Reij = −ab
2αRij + a2
2αys(∇sRij − ∇iRsj)
−a3b
4α2ypysRpimhRsjhm, (iv) Reij = (1−b2
α)Rij + ab
2αys(2∇sRij − ∇iRsj− ∇jRis) +a2
4αypys(RshimRmjph+RhismRmpjh−2RmishRhpjm)
−a2b2
4α2ypys(RishmRmpjh+RishmRjpmh) with respect to the adapted frame {Eβ}.
The scalar curvature Se of (T M, G) is dened by Se= GαβReαβ, where Gαβ denote the com- ponents of the inverse matrix of (Gαβ) in (3.4) which has the following local expression:
G−1 = (Gβγ) = a
αgjk −αbgjk
−αbgjk (a+c)α gjk
. In view of Lemma 3.3, the following result is obtained.
24 L. Bilen, A. Gezer
Lemma 3.4. The scalar curvature Se of (T M, G) is given by
Se= a
αS− a3
2α4ypysRpijmRsijm, where S denote the scalar curvature of (M, g).
Proof of Theorem 3.1. The tangent bundle(T M, G)is locally conformally at if and only if the components of the curvature tensor Re of (T M, G)satisfy the following equations:
Reαγβσ = 1
2(n−1)(GγσReαβ−GασReγβ+GαβReγσ− Gγβ Reασ) (3.7)
− Se
2(2n−1)(n−1)(GαβGγσ−GασGγβ), where Reαγβσ =GσεReαγβε.
In the cases of α=m, γ =i, β =j, σ =l and α =m, γ =i, β =j, σ =l in (3.7), we get Remijl = 1
2(n−1)(bgilRemj−bgmlReij+agmjReil−agijReml) (3.8)
− Se
2(2n−1)(n−1)(abgmjgil−abgmlgij) and
Remijl = 1
2(n−1)(agilRemj−agmlReij+agmjReil−agijReml) (3.9)
− Se
2(2n−1)(n−1)(a2gmjgil−a2gmlgij).
By using (viii) of Lemma 3.2 and (3.4) we have Remijk = 0 and Remijk = 0. Thus the equations (3.8) and (3.9) reduce to the followings:
Se
2(2n−1)(n−1)(abgmjgil−abgmlgij) (3.10)
= 1
2(n−1)(bgilRemj−bgmlReij+agmjReil−agijReml) and
Se
2(2n−1)(n−1)(a2gmjgil−a2gmlgij) (3.11)
= 1
2(n−1)(agilRemj−agmlReij+agmjReil−agijReml).
Comparing (3.10) with (3.11), we obtain a = b and Reil = Reil. From here, by means of (i) and (iii)of Lemma 3.3, we get
Rij = 0 and
Reil =− a4
4α2ypysRpihmRslmh. (3.12)
Some results on Riemanniang−natural metrics generated by classical lifts 25 Transvecting (3.11) by gil and then bygmj respectively, we have
an
2n−1Se= 2gilReil. (3.13)
By (3.12), we calculate
gilReil = − a4
4α2ypysgilRpihmRslmh (3.14)
= a4
4α2ypysRpihjRsihj
= −aα2 2 S.e Substituting (3.14) into (3.13), we obtain
( an
2n−1 +aα2)Se= 0
from which Se= 0, i.e. RpihjRsihj = 0 which gives Rpihj = 0. This completes the proof.
4 Metric connection with torsion of Riemannian g -natural metrics of the form G = a
Sg + b
Hg + c
Vg
It is well-known that a linear connection∇on a Riemannian manifold(M, g)is metric connection with repect to g if ∇g = 0. The (unique) metric connection ∇ which is torsion-free is called the Levi-Civita connection of g. But there exist other metric connections whose torsion tensor is non-zero on a Riemannian manifold (M, g). In this section we consider a metric connection on(T M, G) with a non-zero torsion tensor.
The horizontal lift H∇ of a linear connection on M to T M is the unique linear connection dened on T M by the following conditions
H∇HXHY = H(∇XY), H∇HXVY = V(∇XY)
H∇VXHY = 0, H∇VXVY = 0
for all X, Y ∈ =10(M).The torsion tensor Teof H∇ satises the conditions Te(VX,V Y) = 0, Te(VX,HY) =V (T(X, Y)), Te(HX,HY) = H(T(X, Y))−γR(X, Y)
where T and R are respectively the torsion and curvature tensor elds of the linear connection
∇ on M (for details, see, [20]). From the last identities above, we say that the connection H∇ has non-zero torsion tensor even if ∇ is selected as the Levi-Civita connection ∇g of g on the Riemannian manifold (M, g). By using the denition of the horizontal lift H∇ and (3.1)-(3.3), on calculating
(H∇XeG)(Y ,e Ze) =X(G(e Y ,e Z))e −G(H∇XeY ,e Ze)−G(Y ,e H∇XeZ)e for all X,e Y ,e Ze∈ =10(T M), we get
(H∇VXG)(VY,V Z) = 0, (H∇HXG)(VY,V Z) =a V((∇Xg)(Y, Z)), (H∇VXG)(VY,HZ) = 0, (H∇HXG)(VY,HZ) = b V((∇Xg)(Y, Z)), (H∇VXG)(HY,V Z) = 0, (H∇HXG)(HY,V Z) = b V((∇Xg)(Y, Z)), (H∇VXG)(HY,HZ) = 0, (H∇HXG)(HY,HZ) = (a+c) V((∇Xg)(Y, Z)).
26 L. Bilen, A. Gezer
If ∇ = ∇g, where ∇g is the Levi-Civita connection of g on the Riemannian manifold (M, g), then H∇G= 0. This gives the following result.
Theorem 4.1. The horizontal lift H(∇g) of the Levi-Civita connection ∇g of a Riemannian manifold (M, g) is a metric connection with a non-zero torsion on T M with respect to the Riemannian g−natural metric G=aSg+bHg+cVg, such that a >0 and a(a+c)−b2 >0.
The metric connection H(∇g) is given by
∇EiEj = ΓhijEh,
∇eEiEj = ΓhijEh,
∇eE
iEj = 0, ∇eE
iEj = 0,
with respect to the adapted frame {Eβ}. Also we can say that the metric connection H(∇g) and the Levi-Civita connection ∇e of G coincide if and only if (M, g) is at. On computing the contracted curvature tensor (Ricci tensor) of H(∇g), we have the components HRαβ =H Rγαβγ such that
HRij =Rij, HRij = 0, HRij = 0, HRij = 0
with respect to the adapted frame, where Rij is the Ricci tensor of ∇g onM ([20], p.154). For the scalar curvature of H(∇g)with respect to G, we get
HS = Gαβ HRαβ = a αgijRij
= a αS.
From the last identity, we can state the following theorem.
Theorem 4.2. Let (M, g) be a Riemannian manifold and T M its tangent bundle with the Riemannian g−natural metricG=aSg+bHg+cVg, such that a >0and a(a+c)−b2 >0. The scalar curvatureHS of the tangent bundleT M with the metric connection H(∇g)with respect to G is zero if and only if the scalar curvature S of ∇g on (M, g) is zero.
5 Ane Killing and Killing Vectors with respect to Riemannian g - natural metrics of the form G = a
Sg + b
Hg + c
Vg
LetA is a (1,1)−tensor eld on M with the components(Aij), then ∗A dened by
∗A={Aisyf}Ei,
with respect to the adapted frame {Eβ} is a smooth vector eld onT M [17].
LetLXe be the Lie derivation with respect to the vector eldXe. We shall rst state following lemma which are needed later on.
Lemma 5.1. (see [8]) The Lie derivations of the adapted frame and its dual basis with respect to Xe =vhEh+vhEh are given as follows:
(1) L
XeEh =−(Ehva)Ea+n
ybvcRhcba−vbΓab h−(Ehva)o Ea (2) LXeEh =−(Ehva)Ea+
vbΓab h−(Ehva) Ea
(3) L
Xedxh = (Eavh)dxa+ (E¯avh)δya (4) LXeδyh =
n
ycvbRbach+vbΓhb a+ (Eavh) o
dxa
−n
vbΓhb a−(Eavh)o δya.
