Rabia CAKANKursat AKBULUTArif SALIMOV

1 Introduction

Let(M,g)be a smooth pseudo-Riemannian manifold of dimension n.We denote byM the tangent and cotangent bundles over M with local coordinatesrespectively,whereand px=pidxi∈M,?x∈M.

A very important feature of any pseudo-Riemannian metric g is that it provides musical isomorphisms g?:TM → T?M and g?:T?M → TM between the tangent and cotangent bundles.Some properties of geometric structures on cotangent bundles with respect to the musical isomorphisms are proved in[1–5].

The musical isomorphisms g?and g?are expressed by

and

with respect to the local coordinates,respectively.The Jacobian matrices of g?and g?are given by

and

respectively,where δ is the Kronecker delta.

We denote by(M)the set of all diff erentiable tensor fields of type(p,q)on M.LetCXT∈(TM),C?T∈TM)andCST∈(TM)be complete lifts of tensor fields X∈(M),?∈M)and S∈(M)to the tangent bundle TM.

The aim of this paper is to study the lift properties of cotangent bundles of Riemannian manifolds.The results are significant for a better understanding of the geometry of the cotangent bundle of a Riemannian manifold.In this paper,we transfer via the differentialthe complete liftsandfrom the tangent bundle TM to the cotangent bundle T?M.The transferred liftsare compared with the complete liftsin the cotangent bundle and we show that(a)if and only if the vector field X is a Killing vector field,if and only if the triple(M,g,?),?2=?IdMis an anti-Khler manifold,if and only if the metric g satisfies the Yano-Ako equations.Also we give a new interpretation of the Riemannian extension?g should be considered as the pullback:?g=(g?)?Cg,whereCg is the complete lift of g to the tangent bundle TM.

2 Transfer of Complete Lifts of Vector Fields

Let X=Xi?ibe the local expression in U ? M of a vector field X ∈(M).Then the complete liftCXTof X to the tangent bundle TM is given by

with respect to the natural frame{?i,?i}.

Using(1.1)and(2.1),we have

where LXis the Lie derivation of g with respect to the vector field X:

In a manifold(M,g),a vector field X is called a Killing vector field if LXg=0.It is well known that the complete liftCXT? of X to the cotangent bundle T?M is given by

From(2.2)wefind

where γ(LXg)is defined by

Thus we have the following theorem.

Theorem 2.1 Let(M,g)be a pseudo-Riemannian manifold,and letCXTandCXT?be complete lifts of a vector field X to the tangent and cotangent bundles,respectively.Then the differential(pushforward)ofCXTby g?coincides withCXT?,i.e.,

if and only if X is a Killing vector field.

Let X and Y be Killing vector fields on M.Then we have

i.e.,[X,Y]is a Killing vector field.SinceC[X,Y]T=[CXT,CYT]andC[X,Y]T? =[CXT?,CYT?],from Theorem 2.1 we have the following result.

Corollary 2.1 If X and Y are Killing vector fields on M,then

whereis a differential(pushforward)of the musical isomorphism g?.

3 Transfer of Complete Lifts of Almost Complex Structures

Let(M,?)be a 2n-dimensional,almost complex manifold,where ? (?2= ?I)denotes its almost complex structure.A semi-Riemannian metric g of the neutral signature(n,n)is an anti-Hermitian(also known as a Norden)metric if

for any X,Y ∈(M).An almost complex manifold(M,?)with an anti-Hermitian metric is referred to as an almost anti-Hermitian manifold.Structures of this kind have also been studied under the name:Almost complex structures with pure(or B-)metric.An anti-K¨ahler(K¨ahler-Norden)manifold can be defined as a triple(M,g,?)which consists of a smooth manifold M endowed with an almost complex structure ? and an anti-Hermitian metric g such that??=0,where? is the Levi-Civita connection of g.It is well known that the condition??=0 is equivalent to C-holomorphicity(analyticity)of the anti-Hermitian metric g(see[6]),i.e.,

for any X,Y,Z ∈(M),where Φ?g ∈(M)and G(Y,Z)=(g? ?)(Y,Z)=g(?Y,Z)is the twin anti-Hermitian metric.It is a remarkable fact that(M,g,?)is anti-K¨ahler if and only if the twin anti-Hermitian structure(M,G,?)is anti-K¨ahler.This is of special significance for anti-K¨ahler metrics since in such case g and G share the same Levi-Civita connection.

Let ? =?i? dxjbe the local expression in U ? M of an almost complex strucure ?.Then the complete liftC?Tof ? to the tangent bundle TM is given by(see[8,p.21])

with respect to the induced coordinates(xi,xi)=(xi,yi)in TM.It is well known thatC?Tdefines an almost complex structure on TM,if and only if so does ? on M.

Using(1.1)–(1.2)and(3.1),we have

Since g=(gij)and g?1=(gij)are pure tensor fields with respect to ?,wefind

and

where

Substituting(3.3)–(3.4)into(3.2),we obtain

It is well known that the complete liftC?T? of ? ∈ ?10(M)to the cotangent bundle is given by(see[8,p.242])

with respect to the induced coordinates in T?M.Thus we obtain

where

From here,we have the following theorem.

Theorem 3.1 Let(M,g,?)be an almost anti-Hermitian manifold,and letC?TandC?T?be complete lifts of an almost complex structure ? to the tangent and cotangent bundles,respectively.Then the differential ofC?Tby g?coincides withC?T?,i.e.,g??C?T=C?T?if and only if(M,g,?)is an anti-K¨ahler(Φ?g=0)manifold.

4 Transfer of Complete Lifts of the Vector-Valued 2-Form

Let S be a vector-valued 2-form on M.A semi-Riemannian metric g is called pure with respect to S if

for any X1,X2,Y∈M),where SYdenotes a tensor field of type(1,1)such that

for any Y,Z∈(M).The condition of purity of g may be expressed in terms of the local components as follows:

We now define the Yano-Ako operator

where(g?S)(X,Y1,Y2)=g(S(X,Y1),Y2).The Yano-Ako operator has the following components with respect to the natural coordinate system:

The non-zero components of the complete liftCSTof S to the tangent bundle TM are given by(see[8,p.22])

Using(1.1)and(1.2),we can easily verify that

and I,J,···=1,···,2n has non-zero components of the form

i.e.,the transfer g??CSTcoincides with the complete liftCST?of the vector-valued 2-form S∈∧2(M)to the cotangent bundle if and only if

Thus we have the following theorem.

Theorem 4.1 Let g be a pure pseudo-Riemanian metric with respect to the vector-valued 2-form S ∈ ∧2(M),and letCSTandCST?be complete lifts of S to the tangent and cotangent bundles,respectively.Then

if and only if g satisfies the following Yano-Ako equation:

①論證范圍內(nèi)地下水資源評(píng)價(jià)，根據(jù)論證范圍內(nèi)的地下水補(bǔ)、徑、排條件，計(jì)算各項(xiàng)補(bǔ)給量和排泄量，并進(jìn)行均衡分析，分析補(bǔ)給量計(jì)算的可靠性。

where ΦSg is the operator defined by(4.1).

5 Transfer of Complete Lifts of Metrics

LetCg be a complete lift of a pseudo-Riemannian metric g to TM with components

Using(1.2)and(5.1)we see that the pullback ofCg by g?is the(0,2)-tensor field(g?)?Cg on T?M and has components

On the other hand,a new pseudo-Riemannian metric?g∈(T?M)on T?M is defined by the equation(see[8,p.268])

for any X,Y ∈(M),where γ(?XY+ ?YX)is a function in π?1(U) ? T?M with a local expression

and is called a Riemannian extension of the Levi-Civita connection?g to T?M.The Riemannian extension?g has components of the form

with respect to the natural frame{?i,??}.Thus,from(5.2)and(5.3)we obtain(g?)?Cg=?g,i.e.,we have the following theorem.

Theorem 5.1 The Riemannian extension?g∈(T?M)is a pullback of the complete liftCg∈TM).

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