Habil FATTAYEV Arif SALIMOV

Abstract In this paper the authors consider the bundle of affinor frames over a smooth manifold,define the Sasaki metric on this bundle,and investigate the Levi-Civita connection of Sasaki metric.Also the authors determine the horizontal lifts of symmetric linear connection from a manifold to the bundle of affinor frames and study the geodesic curves corresponding to the horizontal lift of the linear connection.

Keywords Bundle of affinor frames,Riemannian manifold,Sasaki metric,Horizontal lift,Geodesic curve

1 Introduction

Fiber bundles play an important role in all major areas of modern differential geometry.Prime examples of fiber bundles are tangent,cotangent and tensor bundles over differentiable manifolds.The geometry of the tangent bundle was first investigated in the fundamental paper[16]of Sasaki.He used the metricg,given on the differentiable manifoldMn,to construct the Riemannian metricGon the tangent bundleT(Mn) ofMn.This metric is called the Sasaki metric (or the diagonal lift ofgtoT(Mn)).The curvature properties of the Sasaki metric onT(Mn) are studied by Kowalski in [6].Interesting relationships between the geometric properties of the base manifold (Mn,g) and its tangent bundle (T(Mn),G) with the Sasaki metric are investigated by Aso[1],Musso and Tricerri[11].The Sasaki metric on the cotangent bundle was studied by Mok [9],Salimov and Agca [13].In [3,12,14–15]the Sasaki metric was investigated on the tensor bundles.The Sasaki metrics on the linear frame and linear coframe bundles were studied by Mok[10],Kowalski and Sekizawa[7],Fattayev and Salimov[5].In the study of fiber bundles,a special place is also occupied by lifts of linear connections and their geodesic curves (see,for example,[2,8–9,17–18]).

In this paper,we consider a bundle of affinor frames over a smooth manifold,and using the diagonal lift of the Riemannian metric,we study some questions of the differential geometry of this bundle,also we define the horizontal lifts of the symmetric linear connection and investigate their geodesic curves.

2 Preliminaries

In this section,we summarize all the basic definitions and results that are needed later.In the following all manifolds,maps,tensor fields,connections and metrics in question are supposed to be differentiable of classC∞.

LetMnbe ann-dimensional differentiable manifold andL11(Mn) the bundle of an affinor frames ofMn(see [4]).The bundle of an affinor framesL11(Mn) consists of all pairs (x,Ax),wherexis a point ofMnandAxis a basis (an affinor frame) for the linear spaceT11(x) of all affinors at pointx.We denote byπ:L11(Mn) →Mnthe projection map,defined byπ(x,Ax)=x.For the coordinate system (U,xi) inMn,we putL11(U)=π−1(U) and an affinorXαβof the frameAxcan be uniquely expressed in the formso thatis a coordinate system inL11(Mn) (see [4]).Indicesi,j,k,···,α,β,γ,···have range in{1,2,···,n},while indicesA,B,C,···have range in{1,···,n,n+1,···,n+n4}and indicesiαβ,lγδ,kστ,···have range in {n+ 1,···,n+n4}.For simplicity,we give the following notation:Summation over repeated indices is always implied.

We denote by ℑrs(Mn) the set of all differentiable tensor fields of type (r,s) onMn.Let ∇be a linear connection,V∈ℑ10(Mn)a vector field andB∈ℑ11(Mn)an affinor field onMnwith local components Γkij,ViandBji,respectively.Then there is exactly one vector fieldHVonL11(Mn),called the horizontal lift ofV,and exactly one vector fieldVαβBonL11(Mn) for each pairα,β=1,2,···,n,called theαβth-vertical lift ofB,that are known to be defined inL11(U)(see [4]) by with respect to the natural framewhereδγαis the Kronecker delta.

Iffis a differentiable function onMn,Vf=f◦πdenotes its canonical vertical lift toL11(Mn).

Let (U,xi) be a coordinate system inMn.InU⊂Mn,we put

From (2.1)–(2.2),we have

with respect to the natural frame {∂i,∂iαβ} inL11(Mn).Thesen+n4vector fields are linearly independent and generate,respectively,the horizontal distribution of linear connection ∇and the vertical distribution ofL11(Mn).We call the set {HX(i),VαβΛji} the frame adapted to the linear connection ∇onπ−1(U)⊂L11(Mn).Putting

we write the adapted frame as {DI}={Di,Diαβ}.From(2.3)–(2.4)we see thatHVandVαβBhave respectively,components

with respect to the adapted frame {DI},Viandare the local components ofVandBonMn,respectively.

LetB∈ℑ11(Mn),which is locally represented byThe vector fieldsγBandare defined by

with respect to the natural frame {∂i,∂iαβ} inL11(Mn).

The brackets of vertical and horizontal lifts are expressed by the following formulae:

for allX,Y∈ℑ10(Mn) andB,C∈ℑ11(Mn),whereR(X,Y) = [∇X,∇Y]-∇[X,Y]andγ-γ:B→ℑ10(L11(Mn)) equals

for anyB∈ℑ11(Mn).

Remark 2.1Using equality (2.6),it is easy to establish that a vertical vector field(R(X,Y))∈ℑ11(L11(Mn)) can be represented as

3 Sasaki Metric on L11(Mn)

Let (Mn,g) be a Riemannian manifold.For eachx∈Mn,the extension of scalar productg(denoted byG) is defined on the linear spaceL11(x)=π−1(x) by

for allB,C∈ℑ11(Mn).

Definition 3.1The Sasaki metricSg(or diagonal lift ofg) is defined onL11(Mn) by the following three relations:

for allX,Y∈ℑ10(Mn) andB,C∈ℑ11(Mn).

We recall that any elementt∈ℑ02(L11(Mn)) is completely determined by its action on vector fields of typeHXandVαβB.From this it follows thatSgis completely determined by(3.1)–(3.3).

From (3.1)–(3.3),we see that the Sasaki metricSghas the components

with respect to the adapted frame {DI},wheregijandgijare the local covariant and contravariant components ofgonMn.For cotangent bundleCT(Mn),linear frame bundleF(Mn),linear coframe bundleF∗(Mn) and (1,1)-tensor bundleT11(Mn),see [5–6,13–14],respectively.

Now we consider local 1-forminπ−1(U),defined by

where

The matrix (3.6) is the inverse of the matrix

of the transformationDL=AJL∂J(see (2.3)–(2.4)).We easily see that the setis the coframe dual to the adapted frame

Since the adapted frame {DI} is non-holonomic,we put

and then we have

According to (2.3)–(2.4) and (3.6)–(3.7),the components of non-holonomic objectare given by

with all the others being zero,whereare the local components of the curvature tensorRof the Riemannian metricgonMn.

LetS∇be the Levi-Civita connection of the Sasaki metricSg.from the equation

we have

with respect to the adapted frame {DI},whereare the components of the Levi-Civita connectionS∇.

with respect to the adapted frame {DK}.Thus,we have from (3.9)–(3.10)

Taking account of (3.4)–(3.5),(3.8) and (3.11),we immediately get the following theorem.

Theorem 3.1Let(Mn,g)be a Riemannian manifold andS∇be the Levi-Civita connection of the bundle of affinor framesL11(Mn)equipped with the Sasaki metricSg.The particular values offor different indices,are then found to be

with respect to the adapted frame {DK},where

It is well-known that the Levi-Civita connection ∇of a Riemannian metricgis given by Koszul formula

for all vector fieldsX,Y,Z∈ℑ10(Mn).

The following theorem holds.

Theorem 3.2LetMnbe a Riemannian manifold with the metricgandL11(Mn) be the bundle of affinor frames ofMnequipped with the Sasaki metricSg.Then the corresponding Levi-Civita connectionS∇satisfies the following relations:

Proof(i) By the help of Koszul formula (3.12),(2.5)–(2.7),(2.9) and (3.1)–(3.3),we have

and

By combining of (3.13) and (3.14),we obtain

(ii) The statement is obtained as follows:

Using

we have

On the other hand,

Therefore,

(iii) By calculations analogy to those in (ii),we obtain

and

Thus,

(iv) By using the Koszul formula (3.12),(2.7) and (2.9),we yield

and

Therefore,we have

and Theorem 3.2 is proved.

4 Horizontal Lifts of Linear Connections

Let ∇be the symmetric linear connection onMnand Γkijits components.

Definition 4.1A horizontal lift of the symmetric linear connection ∇onMnto the bundle of affinor framesL11(Mn) is the linear connectionH∇defined by

for allX,Y∈ℑ10(Mn) andA,B∈ℑ11(M).

We note that the horizontal lifts of linear connections to tangent,cotangent,and linear frame bundles were defined in [2,8–9,17,19].

The components of the horizontal liftH∇of ∇onMnwith components Γkijin the natural frameare defined in the adapted frame {DI} by decomposition

From (4.1)–(4.2),by using (2.5)–(2.6),we obtain

(1)H∇VαγAVβσB=0,

consequently,

(2)H∇VαγAHY=0,

from which we find

(3)H∇HXHY=H(∇XY),H∇XiDi(YkDk)=(∇XY)pDp,

consequently,

(4)H∇HXVβσB=Vβσ(∇XB),

from which we obtain that

Thus,the following theorem holds.

Theorem 4.1The horizontal liftH∇of the symmetric linear connection ∇given onMn,to the bundle of affinor framesL11(Mn) has the components

with respect to the adapted frame {DI}.

Now consider the following transformation of frames on

i.e.,

We denote the components of the linear connectionH∇with respect to the natural frame{∂I} byi.e.,

Then

Using (4.3) and (3.6)–(3.7),from (4.5) we obtain

Thus,we have the following theorem.

Theorem 4.2The horizontal liftH∇of the symmetric linear connection ∇given onMn,to the bundle of affinor framesL11(Mn) has the components

with respect to the natural frame

5 Geodesics of the Horizontal Lift of Linear Connections

Geodesics of horizontal lifts of connections in tangent,cotangent and tensor bundles were investigated in[2]and [18,pp.114–117,pp.297–299].In the present section we study geodesics of the horizontal lifts of connections in the bundle of affinor frames.

is of the form

By using formulas (4.6) forfrom (5.1) we obtain

and

Let us consider the covariant differentiation of

Now applying

and taking into account the symmetry of the linear connection ∇given onMn,from (5.3) we get

Comparing expression (5.4) and the second equality of (5.2),we obtain the following theorem.

Theorem 5.1A geodesic curve on the bundle of affinor framesL11(Mn) with respect to the horizontal liftH∇of the symmetric linear connection ∇onMnhas in induced coordinatesthe equations of the form:

From Theorem 5.1 we get the following result.

Theorem 5.2A curveon the bundle of affinor framesL11(Mn)is a geodesic of horizontal liftH∇of the symmetric linear connection ∇if the projectiononMnis a geodesic of∇onMnand the second covariant differentiation of each affinorβ,σ= 1,2,···,nof the affinor frameAalongCvanishes,whereπ:L11(Mn) →Mnis the natural projection.