Dual-holomorphic Functions and Problems of Lifts

Arif SALIMOV Seher ASLANCI Fidan JABRAILZADE

Abstract The main purpose of this paper is to study the differential geometrical objects on tangent bundle corresponding to dual-holomorphic objects of dual-holomorphic manifold.As a result of this approach, the authors find a new class of lifts (deformed complete lifts)in the tangent bundle.

Keywords Dual numbers, Tangent bundle, Complete lift, Dual-holomorphic functions, Anti-K¨ahler manifold

1 Introduction

We can define the following classical numbers of order two: Dual numbers (or parabolic numbers): i.e., a+εb, a,b ∈R, ε2= 0; where R is the field of real numbers.Let Mnbe a differentiable manifold and T(Mn)be its tangent bundle.Two types of lift(extension)problems have been studied in the previous works(see for example[1–2,8]): a)The lift of various objects(functions,vector fields,forms,tensor fields,linear connections,etc.)from the base manifold to the tangent bundle.b)The lift on the total manifold T(Mn)by means of a specific geometric structure on T(Mn).In the present paper we continue such a study by considering the structure given by the dual numbers on the tangent bundle and defining new lifts of functions, vector fields, forms, tensor fields and linear connections.

1.1 We consider a two-dimensional dual algebra R(ε), ε2= 0 (ε is nilpotent)with a standard basis {e1,e2} = {1,ε} and structural constants, α,β,γ = 1,2,whereare components of the(1,2)-tensor C :R(ε)×R(ε)→R(ε).

Let Z = xαeαbe a variable in R(ε), where xα(α = 1,2)are real variables.Using realvalued C∞-functions fβ(x)= fβ(x1,x2), β = 1,2, we introduce a dual function F = fβ(x)eβof variable Z ∈R(ε).Let dZ = dxαeαand dF = dfαeαbe respectively the differentials of Z and F(Z).We shall say that the function F = F(Z)is a dual-holomorphic function if there exists a new dual function F′(Z)such that dF = F′(Z)dZ.The function F′(Z)is called the derivative of F(Z).It is well known that the dual function F = F(Z)is holomorphic if and only if the following Scheffers condition holds (see [3]):

From here follows that the dual-holomorphic function F =F(Z)has the following explicit form:

F(Z)=f(x1)+ε(x2f′(x1)+g(x1)),

where f(x1)=f1(x1), f′(x1)=and g =g(x1)is any real C∞-function.

By similar devices, we see that the dual-holomorphic multi-variable function F =F(Z1,··· ,Zn), Zi=xi+εxn+i, i=1,··· ,n, has the form:

where g =g(x1,··· ,xn)is any real multi-variable C∞-function, ?sf =.

A dual-holomorphic manifold (see [6])Xn(R(ε))of dimension n is a Hausdorffspace with a fixed atlas compatible with a group of R(ε)-holomorphic transformations of space Rn(ε),where Rn(ε)= R(ε)× · · · × R(ε)is the space of n-tupes of dual numbers (z1,z2,··· ,zn)with zi= xi+ εyi∈R(ε), xi,yi∈R, i = 1,··· ,n.We shall identify Rn(ε)with R2n,when necessary, by mapping (z1,z2,··· ,zn)∈Rn(ε)into (x1,··· ,xn,y1,··· ,yn)∈R2nand therefore the R(ε)-holomorphic manifold Xn(R(ε))is a real manifold M2nof dimension 2n.

1.2 Let now Mnbe a differentiable manifold and T(Mn)be its tangent bundle, and π be the projection T(Mn)→Mn.The tangent bundle T(Mn)consists of pair(x,v), where x ∈Mnand v ∈Tx(Mn)(Tx(Mn)is a tangent vector space at x ∈Mn).Let (U, x = (x1,··· ,xn))be a coordinate chart in Mn.Then it induces local coordinates (x1,··· ,xn,xn+1,··· ,x2n)in π?1(U), where xn+1,··· ,x2nrepresent the components of v ∈Tx(Mn)with respect to local frame {?i}.In the following we use the notation=i+n for all i=1,··· ,n.

If(U′, x′=(x1′,··· ,xn′))is another coordinate chart in Mn, then the induced coordinateswill be given by

The Jacobian of (1.3)is given by matrix

From here follows that there exists a tensor field of type (1,1),

with properties ?2= 0 and S? = ?S, i.e., the transformation S : {?α} →{?α′} preserving ? is an admissible dual transformation.Thus, the tangent bundle T(Mn)of a manifold Mncarries a natural dual structure ?,which is integrable(?k=0).Therefore,with each induced coordinates (xi,)in π?1(U)?T(Mn), we associate the local dual coordinates Xi= xi+ε2=0.Using (1.3)we see that the local dual coordinates Xi=xi+is transformed by

The equation(1.5)shows that the quantities Xi′are dual-holomorphic functions of Xi=xi+(see (1.2)with g(x1,··· ,xn)= 0).Thus the tangent bundle T(Mn)with a natural integrable ?-structure is a real image of dual-holomorphic manifold Xn(R(ε))(dim Xn(R(ε))= n)(see[5]).In such interpretation there exists a one-to-one correspondence between dual tensor fields on Xn(R(ε))and pure tensor fields with respect to ?-structure on T(Mn)(see [6]).A real C∞-tensor field t of type (1,q)or ω of type (0,q)on T(Mn)is called pure with respect to ?-structure if

or

ω(?X1,X2,··· ,Xq)=ω(X1,?X2,··· ,Xq)=···=ω(X1,··· ,?Xq).

In particular, vector and covector fields will be considered to be pure for convenience sake.

It is important that the dual tensor field on Xn(R(ε))corresponding to a pure C∞-tensor field is not necessarily dual-holomorphic.This tensor field is dual-holomorphic on Xn(R(ε))if and only if Φ-operator associated with ? and applied to a pure tensor field t of type (1,q)or ω of type (0,q)satisfies the following conditions (see [4, 7]),

where LYis the Lie derivation with respect to Y.

2 Deformed Complete Lifts of Functions

From (1.2)we immediately have

F =Vf +ε(Cf +Vg),

where g is any function on Mn,Vf = f ?π,Vg = g ?π are vertical lifts of f, g, respectively,andCf = xn+s?sf is complete lift of f from Mnto its tangent bundle T(Mn)(see [8]).We callDf =Cf +Vg the deformed complete lift of function f to tangent bundle T(Mn).

Thus we have the following theorem.

Theorem 2.1LetT(Mn)be a tangent bundle ofMn, which is a real image of dualholomorphic manifoldXn(R(ε)).Then the vertical and the deformed complete lifts toT(Mn)of any function onMnare a real and dual part of corresponding dual-holomorphic function onXn(R(ε)), respectively.

3 Deformed Complete Lifts of Vector Fields

In a tangent bundle T(Mn)with dual structure ?, a vector fieldis called a dual-holomorphic vector field if? = 0 (see [3]).Such vector field is a real image of corresponding dual-holomorphic vector field V = (Vi)on Xn(R(ε)), whereThe condition of dual-holomorphy of a vector fieldon T(Mn)may be now locally written as follows:

By (1.4), we have:

a)The case where α=i, β =j, (3.1)reduces to

from which follows

b)The cases where α=i, β =j and α=i, β =j, (3.1)reduces to 0=0.

from which follows

and after integrating, we find

where w

i=wi(x1,··· ,xn)are any real multi-variable C∞-functions.

Remark 3.1Using (1.3), (3.2)–(3.3)andwe easily see that v =(vi(x1,··· ,xn))and w =(wi(x1,··· ,xn))are vector fields on Mn.

Thus a real dual-holomorphic vector fieldon tangent bundle can be written in the form

whereCv andVw are the complete and vertical lifts of vector fields v =(vi)and w =(wi)from Mnto tangent bundle T(Mn), respectively (see [8]).Thus we have the following theorem.

Theorem 3.1LetT(Mn)be a tangent bundle ofMn, which is a real image of dualholomorphic manifoldXn(R(ε)).Then a real image of corresponding dual-holomorphic vectorfieldis a deformed complete lift in the formDV =Cv+Vw, whereCvandVware the complete and vertical lifts of vector fieldsv = (vi)andw = (wi)fromMntoT(Mn), respectively.

4 Deformed Complete Lifts of Tensor Fields of Type (1,1)

for any vector fields X on T(Mn).From here we see that, the condition of pure tensor fields may be expressed in terms of the local induced coordinates as follows:

Using (1.4), from the last conditon we have

We note that, such tensor field is a real image of corresponding dual-holomorphic tensor field from Xn(R(ε))(see[3]).Sometimes the tensor Φ?t of type(1,2)is called the Nijenhuis-Shirokov tensor field.It is clear that, if ?=then Φ?t is the Nijenhuis tensor N?, i.e., Φ??=N?.

The condition of dual-holomorphy of a pure tensor fieldon T(Mn)may be now locally written as follows:

By virtue of (1.4)and (4.1), (4.2)after some calculations reduces to

From here follows

Remark 4.1Using (1.3), (4.3)and, we easily see that(x1,··· ,xn)and gij(x1,··· ,xn)are components of any tensor fields t and g of type (1,1)on Mn.

Thus a dual-holomorphic tensor fieldt on tangent bundle can be written in the form

whereCt andVg are the complete and vertical lifts of (1,1)-tensor fields t and g from Mnto tangent bundle T(Mn), respectively (see [8]).Thus we have the following theorem.

Theorem 4.1LetT(Mn)be a tangent bundle ofMn, which is a real image of dualholomorphic manifoldXn(R(ε)).Then a real image of corresponding dual-holomorphic tensor fieldTof type(1,1)fromXn(R(ε))is a deformed complete lift in the formDt=Ct+Vg,whereCtandVgare the complete and vertical lifts of(1,1)-tensor fieldstandgfromMntoT(Mn),respectively.

Let (M4n,F,G,H)be an almost quaternion manifold, i.e.,

Then for three fields F, G and H of type(1,1),we now consider the following deformed complete lifts:

From here, we find

Similarly

Thus we have the following theorem.

Theorem 4.2Let(M4n,F,G,H)be an almost quaternion manifold.Then the deformed complete lifts of each structureF, GandHare almost complex structures on the tangent bundle.

5 Deformed Complete Lifts of 1-Forms

Such 1-form is a real image of corresponding dual-holomorphic 1-form from Xn(R(ε))(see [3]).The tensor field Φ?ω of type (0,2)has components

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

By virtue of (1.4), (Φ?)αβ=0 reduces to

From here we have

Remark 5.1Using (1.3), (5.1)andwe easily see that ωj(x1,··· ,xn)and θj(x1,··· ,xn)are components of any 1-forms ω and θ on Mn, respectively.

Thus a real dual-holomorphic 1-formon tangent bundle can be rewritten in the form

whereCω andVθ are the complete and vertical lifts of 1-forms ω =(ωj)and θ =(θj)from Mnto tangent bundle T(Mn), respectively (see [8]).Thus we have the following theorem.

Theorem 5.1LetT(Mn)be a tangent bundle ofMn, which is a real image of dualholomorphic manifoldXn(R(ε)).Then a real image of corresponding dual-holomorphic1-form fromXn(R(ε))is a deformed complete lift in the formDω =Cω+Vθ,whereCωandVθare the complete and vertical lifts of1-formsω =(ωj)andθ =(θj)fromMntoT(Mn), respectively.

6 Deformed Complete Lifts of Riemannian Metrics

for any vector fields X and Y on T(Mn).From here we see that, the condition of purity ofg may be expressed in terms of the local induced coordinates as follows:

Using (1.4), from the last conditon we have

Such tensor field is a real image of corresponding dual-holomorphic tensor field of type (0,2)from Xn(R(ε)).It is well known that, ifg is a Riemannian metric andis its Levi-Civita connection, then the condition Φ?g = 0 is equivalent to the condition? = 0 (see [3]), i.e.,the triple (T(Mn),g,?)is a dual anti-K¨ahler (or K¨ahler-Norden)manifold.

The tensor field Φ?g of type (0,3)has components

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

By virtue of (1.4), after some calculations, (Φ?g)αβγ=0 reduces to

from which we have

Remark 6.1Using (1.3), (6.1)and, we easily see that gjk(x1,··· ,xn)and hjk(x1,··· ,xn)are components of any tensor fields g and h of type (0,2)on Mn, respectively.

Thus a real dual-holomorphic tensor fieldg of type(0,2)on tangent bundle can be rewritten in the form

whereCg andVh are the complete and vertical lifts of tesor fields g = (gjk)and h = (hjk)of type (0,2)from Mnto tangent bundle T(Mn), respectively (see [8]).Therefore we have the following theorem.

Theorem 6.1LetT(Mn)be a tangent bundle ofMn, which is a real image of dualholomorphic manifoldXn(R(ε)).Then a real image of corresponding dual-holomorphic tensor field of type(0,2)fromXn(R(ε))is a deformed complete lift in the formDg =Cg+Vh,whereCgandVhare the complete and vertical lifts ofg = (gjk)andh = (hjk)fromMntoT(Mn),respectively.

Remark 6.2Now let g be a Riemannian metric, and h be any symmetric(0,2)-tensor field on Mn.It is clear that in such case the tensorDg =Cg+Vh is a Riemannian metric on T(Mn).We note that lifts of this kind have been also studied under the names: The metric I+II (see[8])if g =h and the synectic lift (see [5]).

7 Deformed Complete Lifts of Connections

Using (1.4), from the purity conditon we have

It is well known that, such connection is a real image of corresponding dual-holomorphic connection from Xn(R(ε)).

From here, by virtue of (1.4)and (7.1), we have

Remark 7.1Using (1.3), (7.1)–(7.2)and

after straightforward calculations we see that(x1,··· ,xn)and(x1,··· ,xn)are components of any connection ?and tensor field H of type (1,2)on Mn, respectively.

Taking account of the definition of the complete liftC?of connection ?(see [8]), we see that a real dual-holomorphic connectionon tangent bundle can be rewritten in the form

whereVH is the vertical lift of tensor field H =()of type (1,2)from Mnto tangent bundle T(Mn).Thus we have following theorem.

Theorem 7.1LetT(Mn)be a tangent bundle ofMn, which is a real image of dualholomorphic manifoldXn(R(ε)).Then a real image of corresponding dual-holomorphic connection fromXn(R(ε))is a deformed complete lift in the formD?=C?+VH,whereC?andVHare the complete and vertical lifts of?= ()andH = ()fromMntoT(Mn),respectively.

Example 7.1Let(M,g)be a Riemannian manifold, and(T(Mn),?)be its tangent bundle with natural dual ?-structure:

The complete and vertical lifts of vector and tensor fields from Mnto T(Mn)have the following properties

for any function f on Mn(see [8]).Using these formulas, we find

and

From here we see that the triple(T(Mn),Dg,?)is a dual anti-K¨ahler manifold(?Dg?=0)(see Section 6).In such manifolds, the Levi-Civita connection ?DgofDg also is dual-holomorphic(see [3]).Thus the Levi-Civita connection ?Dgis a simplest example of deformed complete lift of connection.

AcknowledgementThe authors wish to thank the anonymous referees for a careful reading and his/her helpful suggestions about the paper.