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Mathematics and Informatics ICTAMI 2005 - Alba Iulia, Romania

SOME OBSERVATIONS ON A CLASS OF D-LINEAR CONNECTIONS ON THE TANGENT BUNDLE

Monica Purcaru and Mirela Tˆarnoveanu

Abstract.The present paper deals with some problems on the conformal structure on TM. The concepts of conformal structure and d-linear connection compatible with the conformal structure, corresponding to two 1-forms are introduced in the tangent bundle. The problem of determining the set of all d-linear connections compatible with the conformal structure, corresponding to two 1-forms is solved for an arbitrary nonlinear connection.Some important particular cases are considered.

2000 Subject Classification:53C05, 53C15, 53C21, 53C60, 53B05, 53B35, 53B40.

Keywords and phrases:tangent bundle, conformal structure, d-linear con-nection compatible with the conformal structure, corresponding to two 1-forms.

1. Preliminaries

The geometry of the tangent bundle (T M, π, M) has been studied by M. Matsumoto in [4], by R. Miron and M. Anastasiei in [5], [6], by R. Miron and M. Hashiguchi in [7], by V. Oproiu in [8], by Gh. Atanasiu and I. Ghinea in [1], by R. Bowman in [2], by K. Yano and S. Ishihara in [10],etc. Concerning the terminology and notations, we use those from [6].

Let M be a real n-dimensional C∞_{-differentiable manifold and (T M, π, M)}

its tangent bundle.

If (xi_{) is a local coordinates system on a domain U of a chart on M, the}

induced system of coordinates on π−1_{(U) is (x}i_{, y}i_),_(i_{= 1, ..., n).}

Let N be a nonlinear connection onT M, with the coefficientsNi

j(x, y),(i, j =

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2. The notion of d-linear connection compatible with a conformal structure

We consider on T M a metrical (almost symplectic) structure G defined by: G(x, y) = 1

2gij(x, y)dx

i

∧dxj +1

2g˜ij(x, y)δy

i

∧δyj, (1) where (dxi_{, δy}i_),_(i _{= 1, ..., n) is the dual basis of} δ

δxi,

∂ ∂yi

, and (gij(x, y),

˜

gij(x, y)) is a pair of given d-tensor fields on T M, of the type (0,2), each of

them nondegenerate and symmetric (alternate), in the last case it is necessary that n=2n’.

We asociate to the lift G the Obata’s operators:

(

Ωir

sj = 12(δ

i

sδjr−gsjgir), Ω∗sjir = 12(δ

i

sδjr+gsjgir),

˜ Ωir

sj = 12(δ

i

sδjr−˜gsjg˜ir), Ω˜∗sjir = 12(δ

i

sδjr+ ˜gsjg˜ir).

(2) Obata’s operators have the same properties as the ones associated with a Finsler space [7].

Let S2(T M) be the set of all symmetric d-tensor fields, of the type (0,2) on T M (A2(T M) be the set of all alternate d-tensor fields, of the type (0,2) on T M). As is easily shown, the relations on S2(T M) (A2(T M)) defined by (3):

 



(aij ∼bij)⇔

(∃)λ(x, y)∈ F(T M), aij(x, y) =e2λ(x,y)bij(x, y)

,

˜

aij ∼˜bij

⇔(∃)µ(x, y)∈ F(T M),a˜ij(x, y) = e2µ(x,y)˜bij(x, y)

, (3) is an equivalence relation on S2(T M) (A2(T M)).

Theorem 1. The equivalent class: Gˆ of S2(T M)/∼ (A2(T M)/∼) to which

the metrical (almost symplectic) tensor field G belongs, is called conformal structure on T M.

Thus:

ˆ

G={G′|G′_ij(x, y) = e2λ(x,y)_g

ij(x, y)andG˜′ij(x, y) = e

2µ(x,y)_˜_g

ij(x, y)}. (4) Definition 1. A d-linear connection, D, on T M, with local coefficients DΓ(N) = (Li

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T M:

ω =ωidxi+ ˙ωiδyi, ω˜ = ˜ωidxi+ ˙ω˜iδyi such that: (

gij|k = 2ωkgij, gij|k = 2 ˙ωkgij,

˜

gij|k = 2˜ωkg˜ij, ˜gij|k = 2 .

˜ ωkg˜ij,

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where and denote the h-and v-covariant derivatives with respect to D,is said to be compatible with the conformal structure G, corresponding to the 1-formsˆ ω,ω˜ and is denoted by: DΓ(N, ω,ω).˜

3. The set of all d- linear connections compatible with the

conformal structure ˆG, corresponding to two 1-forms

Let N0 and N be two nonlinear connections on TM, with the coefficients (

0 N i

(1) j,

0 N i

(2) j) and (N i

(1) j, N i

(2) j) respectively.

Let D0Γ(N0) = ( 0 Li

jk,

0 ˜ Li

jk,

0 ˜ Ci

jk,

0 Ci

jk) be the local coefficients of a fixed

d-linear connection D0 on T M. Then any d-linear connection, D, on T M, with local coefficients: DΓ(N) = (Li

jk,L˜ijk,C˜ijk, Cijk),can be expresed in the form: 

                   

                   

Ni j =

0 Ni

j −Aij,

Li jk =

0 Li

jk +Alk

0 ˜ Ci

jl −Bijk,

˜ Li

jk =

0 ˜ Li

jk +Alk

0 Ci

jl −B˜ijk,

˜ Ci

jk =

0 ˜ Ci

jk −D˜ijk,

Ci jk =

0 Ci

jk −Dijk,

Al j

0

|k

= 0,

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where (Ai

j, Bijk,B˜ijk,D˜ijk, Dijk) are components of the difference tensor fields

of DΓ(N) fromD0 Γ(N0), [4] and 0, 0

denotes the h-and v-covariant derivatives with respect to D0.

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Theorem 2.Let D0 be a given d-linear connection on T M, with local co-efficients DΓ(0 N0) = (

0 Li jk, 0 ˜ Li jk, 0 ˜ Ci jk, 0 Ci

jk). Then set of all d-linear connections

compatible with the conformal structureG, corresponding to the 1-formsˆ ω and ˜

ω, with local coefficients DΓ(N, ω,ω) = (L˜ i

jk,L˜ijk,C˜ijk, Cijk) is given by:                                              Ni j = 0 Ni

j −Xij,

Li jk = 0 Li jk + 0 ˜ Ci

jm Xmk+ 12g

is_(g sj

0

|k+gsj

0 |_m Xm

k)−δijωk+ ΩirhjXhrk,

˜ Li jk = 0 ˜ Li jk + 0 Ci

jm Xmk+ 12g˜

is_(˜_g sj

0

|k+ ˜gsj

0 |_m Xm

k)−δijω˜k+ ˜ΩirhjX˜hrk,

˜ Ci jk = 0 ˜ Ci

jk +1₂gisgsj

0 |_k −δi

jω˙k+ ΩirhjY˜hrk,

Ci jk =

0 Ci

jk +12g˜

is_g_˜ sj

0 |_k −δi

jω˙˜k+ ˜ΩirhjYhrk,

Xi j 0 |k = 0, (7)

where Xi

j, Xijk, X˜ijk, Y˜ijk, Yijk are arbitrary tensor fields on T M, ω =

ωidxi + ˙ωiδyi and respective ˜ω = ˜ωidxi + ˙˜ωiδyi are arbitrary 1-forms in T M

and 0, 0

denote the h-and respective v-covariant derivatives with respect toD0.

Particular cases

1. IfXi

j =Xijk = ˜Xijk = ˜Yijk =Yijk = 0 in Theorem 2, we have:

Theorem 3.Let D0 be a given d-linear connection onT M, with local coeffi-cients DΓ(0 N0) = (

jk). Then the following d-linear connection

D, with local coefficients DΓ(N , ω,0 ω) = (L˜ i

jk,L˜ijk,C˜ijk, Cijk) given by (8) is

compatible with the conformal structureG, corresponding to the 1-formsˆ ω and ˜

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                         Li jk = 0 Li

jk +12g

is_g sj

0

|k−δ i jωk,

˜ Li jk = 0 ˜ Li

jk +1₂˜gis˜g sj

0

|k−δ i jω˜k,

˜ Ci jk = 0 ˜ Ci

jk +12g

is_g sj

0 |_k −δi

jω˙k,

Ci jk =

0 Ci

jk +1₂g˜isg˜sj

0 |_k −δi

jω˙˜k,

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where 0, 0

denote the h-and respective v-covariant derivatives with respect to the given d-linear connection D0 and ω = ωidxi + ˙ωiδyi and respective

˜

ω = ˜ωidxi+ ˙˜ωiδyi are two given 1-forms in T M.

2. If we take a metrical (almost symplectic) d-linear connection as D0 in Theorem 3, then (8) becomes:

                     Li jk = 0 Li

jk −δjiωk,

˜ Li jk = 0 ˜ Li

jk −δjiω˜k,

˜ Ci jk = 0 ˜ Ci

jk −δjiω˙k,

Ci jk =

0 Ci

jk −δjiω˙˜k.

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3. If we consider a d-linear connection compatible with conformal structure ˆ

G, corresponding to the 1-forms ω and ˜ω, as D0 in Theorem 2, we have

Theorem 4.Let D0 be a given d-linear connection compatible with confor-mal structure G, corresponding to the 1-formsˆ ω and ω˜ on T M, with local coefficients: D0Γ(N , ω,0 ω) = (˜

jk). The set of all d-linear

con-nections compatible with conformal structure G, corresponding to the 1-formsˆ ω and ω˜ on T M, with local coefficients DΓ(N, ω,ω) = (L˜ i

jk,L˜ijk,C˜ijk, Cijk) is

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                    

                   

Ni j =

0 Ni

j −Xij,

Li jk =

0 Li

jk +(

0 ˜ Ci

jm +δijω˙m)Xmk+ +ΩirhjXhrk,

˜ Li

jk =

0 ˜ Li

jk +(

0 Ci

jm +δijω˙˜m)Xmk+ ˜ΩirhjX˜hrk,

˜ Ci

jk =

0 ˜ Ci

jk +ΩirhjY˜hrk,

Ci jk =

0 Ci

jk + ˜ΩirhjYhrk,

Xi j

0

|k

= 0,

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where Xi

j, Xijk, X˜ijk, Y˜ijk, Yijk are arbitrary tensor fields on T M, ω =

ωidxi + ˙ωiδyi and respective ˜ω = ˜ωidxi + ˙˜ωiδyi are two arbitrary 1-forms in

T M and0, 0

denote h-and respective v-covariant derivatives with respect toD0. The result obtained in this particular case support the findings of R.Miron and M.Hashiguchi for the Finsler connections in their paper [7].

4. If we take Xi

j = 0 in Theorem 3 we obtain a result given by R.Miron

and M.Anastasiei in [6]:

Theorem 5.Let D0 be a given d-linear connection compatible with confor-mal structure G, corresponding to the 1-formsˆ ω and ω˜ on T M, with local coefficients: D0Γ(N , ω,0 ω) = (˜

0 Li

jk,

0 ˜ Li

jk,

0 ˜ Ci

jk,

0 Ci

jk). The set of all d-linear

con-nections compatible with conformal structure G, which preserve the nonlinearˆ connection N0, corresponding to the 1-forms ω and ω˜ on T M, with local coef-ficients DΓ(N , ω,0 ω) = (L˜ i

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            

           

Li jk =

0 Li

jk +ΩirhjXhrk,

˜ Li

jk =

0 ˜ Li

jk + ˜ΩirhjX˜hrk,

˜ Ci

jk =

0 ˜ Ci

jk +ΩirhjY˜hrk,

Ci jk =

0 Ci

jk + ˜ΩirhjYhrk,

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where Xi

j, Xijk, X˜ijk, Y˜ijk, Yijk are arbitrary tensor fields on T M. References

[1.] Atanasiu, Gh., Ghinea, I., Connexions Finsleriennes G´en´erales Presque Symplectiques, An. St. Univ. “Al. I. Cuza”, Ia¸si, Sect. I a Mat. 25 (Supl.), 1979, 11-15.

[2.] Bowman, R.,Tangent Bundles of Higher Order, Tensor, N.S., Japonia, 47_{, 1988, 97-100.}

[3.] Cruceanu, V., Miron, R., Sur les connexions compatible `a une

Structure Mˆetrique ou Presque symplectique, Mathematica (Cluj),9_{(32), 1967,} 245-252.

[4.] Matsumoto, M.,The Theory of Finsler Connections, Publ. of the

Study Group of Geometry 5, Depart.Math., Okayama Univ., 1970, XV+220 pp.

[5.] Miron, R., Introduction to the Theory of Finsler Spaces, The Proc. of Nat. Sem. on Finsler Spaces, Bra¸sov, 1980, 131-183.

[6.] Miron, R., Anastasiei, M., The Geometry of Lagrange Spaces:

Theory and Applications,Kluwer Acad. Publ., FTPH, no.59, 1994.

[7.] Miron R., Hashiguchi, M., Conformal Finsler Connections, Rev.

Roumaine Math. Pures Appl., 26, 6(1981), 861-878.

[8.] Oproiu, V., On the Differential Geometry of the Tangent Bundle, Rev. Roum. Math. Pures Appl., 13_{, 1968, 847-855.}

[9.] Purcaru, M., Structuri geometrice remarcabile ˆın geometria La-grange de ordinul al doilea, Tez˘a de doctorat, Univ. “Babe¸s-Bolyai” Cluj-Napoca, 2002.

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M. Purcaru and M. Tˆarnoveanu Department of Algebra and Geometry ”Transilvania” University

Bra¸sov, Romania