On Deformed Riemannian Extensions Associated with Twin Norden Metrics?

Arif SALIMOV Rabia CAKAN

1 Introduction

LetMnbe ann-dimensionalC∞-manifold with torsion-free connection?,CT(Mn)be its cotangent bundle,andπbe the natural projectionCT(Mn)→Mn.A system of local coordinates(U,xi),i=1,···,ninMninduces onCT(Mn)a system of local coordinates=n+i=n+1,···,2n,wherexi=piare components of covectorspin each cotangent spaceCTx(Mn),x∈Uwith respect to the natural coframe{dxi}.

We denote bythe module overofC∞tensor fields of type(r,s),whereis the ring of real-valuedC∞functions onMn(CT(Mn)).

LetX=be the local expressions inU?Mnof vector and covector(1-form)fieldsandrespectively.Then the complete and horizontal liftsCX,ofXand the vertical liftofare given,respectively,by

with respect to the natural framewhereare components of the torsion-free connection?onMn.

A new(pseudo)Riemannian metricis defined by the equation(see[7,p.268])

for anyX,Ywhereis a function inwith a local expressionWe call?gthe Riemannian extension of the symmetric connectionThe Riemannian extension?ghas components of the form

with respect to the natural framewhereis the Kronecker delta.

On the other hand,the vector fieldsHXandspan the moduleHence the tensor field?gis also determined by its action ofHXandVξ.From(1.2)–(1.4),we have

for anyX,YThus?gis completely determined by the conditions(1.5)–(1.7).

It is well known thatCT(Mn)has a canonical symplectic structureω=dp,wherepis a basic 1-form inThe symplectic 2-form has components of the form

with respect to the natural frameandI,J=1,···,2n.

2 The Metric?g as a Pullback ofCg

Let nowMnbe a Riemannian manifold with metricgand?=?gbe the Levi-Civita connection ofg.We denote byT(Mn)the tangent bundle overMnwith local coordinateswhereyx=?x∈Mn.LetCgbe a complete lift of a Riemannian metricgtoT(Mn)with components

A very important feature of any Riemannian metricgis that it provides a musical(natural)isomorphismg?:CT(Mn)→T(Mn)between the cotangent and tangent bundles.The musicalisomorphismg?is expressed bywith respect to the local coordinates.The Jacobian matrix ofg?is given by

Using(2.1)–(2.2)we see that the pullback ofCgbyg?is the(0,2)-tensor fieldonCT(Mn)and has components

Thus,from(1.4)and(2.3),we havei.e.,the Riemannian extension?g∈should be considered as a pullback of the complete lift

3 The Deformed Metrics in the Cotangent Bundle

It is well known that the deformed complete lift of Riemannian metricgto the tangent bundle is defined by

whereais a symmetric tensor field onMn,andhas components

with respect to the natural frame inLifts of this kind have also been studied under the name:The synectic lift of metrics(see[1,4,p.88,5,6,p.165]).Also we note that,if(Mn,g)is flat,thenis not necessarily flat,butis necessarily flat.Ifa=g,then we haveThe metricCg+Vgis called a metric of type I+II.The metric I+II was used by Yano and Ishihara[7]to study the geometry of tangent bundles.

The pullback ofVahas components

Using(1.8),(2.3)and(3.1)we have

where

is a tensor field of type(2,0)in(see[7,p.230]).The tensor field

is non-singular and can be regarded as a new metric on the cotangent bundleCT(Mn).

4 The Deformed Metrics of Type ?g+

Let(Mn,J)be a 2n-dimensional almost complex manifold,whereJdenotes its almost complex structure.A semi-Riemannian metricgof the natural signature(n,n)is a Norden(anti-Hermitian)metric if(see[3])

for anyX,YAn almost complex manifold(Mn,J)with a Norden metric is referred to as an almost Norden manifold.Structures of this kind have also been studied under the name:Almost complex structures with pure(or B-)metrics.A Khler-Norden(anti-Khler)manifold can be defined as a triple(Mn,g,J)which consists of a smooth manifoldMnendowed with an almost complex structureJand a Norden metricgsuch that?J=0,where?is the Levi-Civita connection ofg.It is well known that the condition?J=0 is equivalent to C-holomorphicity(analyticity)of the Norden metricg(see[4]),i.e.,ΦJg=0,where(ΦJg)(X,Y,Z)=?LXG)(Y,Z)andG(Y,Z)=(g?J)(Y,Z)=g(JY,Z)is the twin Norden metric.It is a remarkable fact that(Mn,g,J)is K?hler-Norden if and only if the twin Norden structure(Mn,G,J)is K?hler-Norden.This is of special significance for Khler-Norden metrics since in such casegandGshare the same Levi-Civita connection(?g=?G=0).Since in dimension 2,a Khler-Norden manifold is flat,we assume in the sequel that dimM≥4.

Let now(Mn,g,J)be an almost Norden manifold.Ifa=G(see Section 3),whereGis a twin Norden metric,then we have a metric

The metrichas components

with respect to the induced coordinates

The main purpose of the next sections is to study the metricin the cotangent bundle and also the metric connection with respect to this metric.

The line element of(4.1)is given by

whereFrom here,we have the following theorem.

Theorem 4.1Let(Mn,g,J)be an almost Norden manifold.Then the fibre represented bydxi=0is a null submanifold inCT(Mn)with a deformed Riemann extension metricbut the horizontal distribution defined by=0is not null.

LetC?be the Levi-Civita connection determined by=0(C?is called the complete lift ofThe components ofare given by

with respect to induced coordinates inIf(Mn,g,J)is Khler-Norden(?g=?G=0),then using the expression

from(1.1)and(4.2)we find=0 for anyXThen we have

On the other hand,C?is torsion-free,so we have the following theorem.

Theorem 4.2Let(Mn,J,g)be a K?hler-Norden manifold.Then the Levi-Civita connection of?g coincides with the Levi-Civita connection of

Using(1.1)and(4.1),we easily see that the inner product of the complete liftsCXandCYof vector fieldsX,Y∈Mnwith respect to the metricis given by

From this equation,we have the following theorem.

Theorem 4.3Let(Mn,J,g)be a Khler-Norden manifold.Then the complete liftsCX,CY of two vector fields X,Y towith a metricare orthogonal if X,Y are orthogonal with respect to G and are parallel.

5 The Metric Connection with Respect to ?g+

Let?gbe the Levi-Civita connection onMn.InU?Mn,we put

Then from(1.2)–(1.3),we see thathave respectively local expressions of the form

We call the setthe frame adapted to the connection?g.The indicesα,β,γ,···=1,···,2nindicate the indices with respect to the adapted frame.

From equations(1.2)–(1.3)and(5.1)–(5.2),we see that the liftsHXandVωhave respectively components

with respect to the adapted frameandandωiare local components ofXandω,respectively.Also from(1.5)–(1.7),we see that

i.e.,?ghas components

with respect to the adapted frame

Using(5.1)–(5.2),we now consider local vector fieldsand 1-formsdefined by

where

We easily see that the setis the coframe dual to the adapted frame=

Using(3.1)and(5.6),we see thathas components

with respect to the adapted framehas components

with respect to the adapted frame

Since the adapted frameis non-holonomic,we put

from which we have

According to(4.2),(5.1)and(5.6)–(5.7),the components of the non-holonomic objectare given by

all the others being zero,whereare local components of the curvature tensorRof?.

Let now(Mn,J,g)be a K?hler-Norden manifold,and letC?be the Levi-Civita connection determined by the Riemannian extension?gor by the deformed Riemannian extension?g+(see Theorem 4.2).We put

Fromwe have

The equation=0 has the form

with respect to the adapted frameWe have from(5.10)–(5.11)that

whereand

Taking account of(5.3)–(5.5)and(5.9),we obtain(see[2])

with respect to the adapted frame

Untill now,we have given the metricto the cotangent bundleand considered the Levi-Civita connectionCThis is the unique connection which satisfies=0,and has no torsion.But there exists another connectione?which satisfies=0,and has the non-trivial torsion tensor.We call this connection the metric connection of

The horizontal liftH?of the torsion-free connection?to the cotangent bundleCT(Mn)is defined by

for anyX,Y∈andω,

We now putwhereThen taking account of=and writingfor the different indices,from(5.12)we have

LetTbe the torsion tensor of the horizontal lift.ThenTis the skew-symmetric tensor field of type(1,2)inCT(Mn)determined by[7,p.287]

whereRis the curvature tensor of?andγR(X,Y)=Thus the connectionH?has non-trivial torsion even for Levi-Civita connection?=?gdetermined byg,unlessgis locally flat.

Since?g=?G=0,by virtue of(1.5)–(1.7)and(5.8),we have

for anyX,Y,Zandω,θ,εi.e.,the horizontal liftis a metric connection ofThus we have the following theorem.

Theorem 5.1Let(Mn,J,g)be a Khler-Norden manifold,and let?gbe the Levi-Civita connection of g.Then the horizontal liftis a metric connection of the deformed Riemannian extension

Let nowHRbe a curvature tensor field ofThe curvature tensorHRof the metric connectionH(?g)has components

with respect to the adapted frame.Using(5.3)–(5.4),(5.9),(5.13)–(5.14)and computing components of the contracted curvature tensor field(the Ricci tensor field)we obtain

whereis the Ricci tensor field of?ginMn.

By virtue of(5.15),for the scalar curvatureofCT(Mn)with the metric connectionH(),we have

where

Thus we have the following theorem.

Theorem 5.2Let(Mn,J,g)be a Khler-Norden manifold.Then the cotangent bundle CT(Mn)with the metric connectionH(?g)has vanishing scalar curvature with respect to the deformed Riemannian extension

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