1 Necessary facts about the tangent bundle T M

of a Riemannian manifold

Cristian Eni

Abstract. On the tangent bundle of a Riemannian manifold (M, g) we consider a pseudo-Riemannian metric defined by a symmetric tensor fieldc onM and four real valued smooth functions defined on [0,∞). We study the conditions under which the above pseudo-Riemannian manifold has constant sectional curvature.

M.S.C. 2000: Primary 53C55, 53C15, 53C05.

Key words: tangent bundle, pseudo-Riemannian metric, curvature.

1 Necessary facts about the tangent bundle T M

Let (M, g) be a smoothn-dimensional Riemannian manifold and let π: T M →M be its tangent bundle. ThenT M has a structure of a 2n-dimensional smooth mani- fold induced from the structure of smoothn-dimensional manifold of M as follows:

every local chart (U, φ) = (U, x¹, ..., xⁿ) on M induced a local chart (π⁻¹(U),Φ) = (π⁻¹(U), x¹, ..., xⁿ, y¹, ..., yⁿ) onT M, where we made an abuse of notation,identifying xⁱ with π^∗xⁱ =xⁱ◦π andyⁱ being the vector space coordinates ofy ∈π⁻¹(U) with respect to the natural local frame ((_∂x^∂1)_π(y), ...,(_∂x^∂n)_π(y)) i.e.y=yⁱ(_∂x^∂i)_π(y)

This special structure ofT M allows us to introduce the notion ofM-tensor fields on it (see [3]). An M-tensor field of type (p, q) on T M is defined by sets of n^p+q functions depending onxⁱ and yⁱ, assigned to induced local charts (π⁻¹(U),Φ) on T M, thus the change rule is that of the components of a tensor field of type (p, q) on M, when a change of local charts on the base manifold is performed. Remark that the componentsyⁱ define anM-tensor field of type (1,0) onT M. It is also obvious that a usual tensor field of type (p, q) onM may be thought as anM-tensor field of type (p, q) onT M. In the case of a covariant tensor field, the correspondingM-tensor field on the tangent bundleT Mis nothing else but the pullback of the initial tensor field by the submersionπ:T M →M. Other usefulM-tensor fields onT M may be obtained as follows. Leta: [0,∞)→R be a smooth function and letkyk²=gπ(y)(y, y) be the square of the norm of the tangent vectory. Then the components a(ky k²)δⁱ_j define a M-tensor field of type (1,1) on T M. Similarly, if gij(x) are the local coordinate

Balkan Journal of Geometry and Its Applications, Vol.13, No.2, 2008, pp. 35-42.∗

c

°Balkan Society of Geometers, Geometry Balkan Press 2008.

components of the metric tensor field g on M, then the components a(k y k²)gij

define a symmetricM-tensor field of type (0,2) onT M. The componentsg0i=y^jgji

define anM-tensor field of type (0,1) on T M.

Recall that the Levi-Civita connection ˙∇of the Riemannian metric gdefines the direct sum decomposition

T T M =V T M⊕HT M

of the tangent bundle toT M into the vertical distribution V T M =ker π∗ and the horizontal distribution HT M. The vector fields (_∂y^∂1, ...,_∂y^∂n) define a local frame field forV T M and for the horizontal distributionHT M we have the local frame field (_δx^δ1, ...,_δx^δn), where

δ δxⁱ = ∂

∂xⁱ −Γ^h_i0 ∂

∂y^h; Γ^h_i0= Γ^h_iky^k

and Γ^h_ij are the Christoffel symbols defined by the Riemannian metric g. In [5] the author proves the following

Lemma 1. Ifn >1 andu, v are smooth function on T M such that ugij+vg0ig0j= 0, g0i=y^jgji, y∈π⁻¹(U) on the domain of any induced local chart onT M, thenu=v= 0.

In a similar way we can obtain

Lemma 2. Ifn >1 andu, v are smooth function on T M such that ugjkδ_i^h−ugijδ^h_k+vg0ig0jδ_k^h−vg0jg0kδ^h_i = 0, g0i=y^jgji, y∈π⁻¹(U) on the domain of any induced local chart onT M, thenu=v= 0.

Remark.From the relation

ugjkyiy^h−ugikyjy^h= 0, y∈π⁻¹(U), we obtainu= 0.

Since we work in a fixed local chart (U, φ) onM and in the corresponding induced local chart (π⁻¹(U),Φ) on T M, we shall use the following simpler notations

∂

∂yⁱ =∂i, δ δxⁱ =δi

We also denote by t=1

2 kyk²=1

2g_π(y)(y, y) =1

2gij(x)yⁱy^j, y∈π⁻¹(U).

2 A pseudo-Riemannian metric on T M

Letcbe a symmetric tensor field of type (0,2) onM, and leta1, b1, a2, b2: [0,∞)→R be smooth functions. Consider the following symmetric tensor field of type (0,2) on T M (see [6],[7],[4])

(2.1)













Gy(X^V, Y^V) = 0,

Gy(X^H, Y^V) =a1(t)g_π(y)(X, Y) +b1(t)g_π(y)(y, X)g_π(y)(y, Y), Gy(X^H, Y^H) =a2(t)cπ(y)(X, Y) +b2(t)gπ(y)(y, X)gπ(y)(y, Y).

The expression ofGin local adapted frames is defined by the followingM-tensor fields

G¹_ij =G(δi, ∂j) =a1gij+b1g0ig0j, G²_ij =G(δi, δj) =a2cij+b2g0ig0j.

The associated matrix ofGwith respect to the adapted local frame is



 0 G¹_ij G¹_ij G²_ij





The conditions forGto be nondegenerate are ensured if a1(a1+ 2tb1)6= 0.

Under these conditions the matrix (G¹_ij) has the inverse with the entries H₁^ij = 1

a1g^ij+ b1

a1+ 2tb1yⁱy^j We shall denote by

∂hG¹_ij =∂G¹_ij

∂y^h , ∂hG²_ij = ∂G²_ij

∂y^h, δhG¹_ij =δG¹_ij

δx^h , δhG²_ij = δG²_ij δx^h .

The following formulae can be easily checked and will be useful in our next compu- tation:

(2.2)































∇˙iG¹_jk=δiG¹_jk−Γ^h_ijG¹_hk−Γ^h_ikG¹_jh= 0,

∇˙iG²_jk=δiG²_jk−Γ^h_ijG²_hk−Γ^h_ikG²_jh=a2∇˙icjk,

∇˙iH₁^jk=δiH₁^jk+ Γ^j_ihH₁^hk+ Γ^k_ihH₁^jh= 0,

∇˙i∂jG¹_kl=δi∂jG¹_kl−Γ^h_ij∂hG¹_kl−Γ^h_ik∂jG¹_hl−Γ^h_il∂jG¹_kh= 0,

∇˙i∂jG²_kl=δi∂jG²_kl−Γ^h_ij∂hG²_kl−Γ^h_ik∂jG²_hl−Γ^h_il∂jG²_kh=a⁰₂g0j∇˙ickl.

Proposition 3.The Levi-Civita connection∇of the pseudo-Riemannian manifold (T M, G)has the following expression in the local adapted frame (∂1, ..., ∂n,

δ1, ..., δn)

∇∂i∂j = Q^h_ij∂h, ∇δi∂j = (Γ^h_ij+Pe_ji^h)∂h+P_ji^hδh,

∇∂iδj = P_ij^hδh+Pe_ij^h∂h, ∇δiδj = (Γ^h_ij+Se_ij^h)δh+S_ij^h∂h, where theM-tensor fieldsQ^h_ij,P_ij^h,Pe_ij^h,S_ij^h,Se_ij^h are given by:

Q^h_ij = 1

2H₁^hk(∂iG¹_jk+∂jG¹_ik), P_ij^h = 1

2H₁^hk(∂iG¹_jk−∂kG¹_ij), Pe_ij^h =1

2H₁^hk∂iG²_jk−1

2H₁^rl(∂iG¹_jl−∂lG¹_ij)G²_rkH₁^kh, S_ij^h =a2

2 ( ˙∇icjk+ ˙∇jcki−∇˙kcij)H₁^kh−

−a1R0ijkH₁^kh+1

2H₁^sl(∂lG²_ij)G²_skH₁^kh, Se_ij^h =−1

2H₁^hk∂kG²_ij,

Rlijk denoting the local coordinate components of the Riemann-Christoffel tensor of the Levi-Civita connection∇˙ onM andR_0ijk=y^lR_lijk

Remark.Replacing the expressions ofG¹_ij,G²_ij,H₁^ij,∂iG¹_jk,∂iG²_jk by their local coordinate components we obtain some quite complicated expressions.

The curvature tensor field K of the connection ∇ is defined by the well-known formula

K(X, Y)Z=∇X∇YZ− ∇Y∇XZ− ∇_[X,Y]Z, X, Y, Z∈Γ(T M)

Proposition 4. The local coordinate expression of the curvature tensor field in the adapted local frame(∂1, ..., ∂n, δ1, ..., δn)is given by

K(∂i, ∂j)∂k =Y Y Y Y_kij^h ∂h,

K(∂i, ∂j)δk=Y Y XY_kij^h ∂h+Y Y XX_kij^h δh, K(∂i, δj)∂k=Y XY Y_kij^h ∂h+Y XY X_kij^h δh, K(∂i, δj)δk=Y XXY_kij^h ∂h+Y XXX_kij^h δh, K(δi, δj)∂k=XXY Y_kij^h ∂h+XXY X_kij^h δh, K(δi, δj)δk =XXXY_kij^h ∂h+XXXX_kij^h δh, where we have denoted

Y Y Y Y_kij^h =∂iQ^h_jk+Q^l_jkQ^h_il−∂jQ^h_ik−Q^l_ikQ^h_jl

Y Y XY_kij^h =∂iPe_jk^h +Pe_il^hP_jk^l +Pe_jk^l Q^h_il−∂jPe_ik^h−Pe_jl^hP_ik^l −Pe_ik^l Q^h_jl Y Y XX_kij^h =∂iP_jk^h +P_jk^l P_il^h−∂jP_ik^h −P_ik^l P_jl^h

Y XY Y_kij^h =∂iPe_kj^h +Pe_kj^l Q^h_il+Pe_il^hP_kj^l −Pe_lj^hQ^l_ik Y XY X_kij^h =∂iP_kj^h +P_kj^l P_il^h−P_lj^hQ^l_ik

Y XXY_kij^h =∂iS_jk^h +S_jk^l Q^h_il+Se_jk^l Pe_il^h−S_jl^hP_ik^l −Pe_ik^l Pe_lj^h−∇˙jPe_ik^h Y XXX_kij^h =∂iSe_jk^h +Se_jk^l P_il^h−Se_jl^hP_ik^l −Pe_ik^l P_lj^h

XXY Y_kij^h = ˙∇iPe_kj^h +Pe_kj^l Pe_li^h+P_kj^l S_il^h−∇˙jPe_ki^h −Pe_ki^lPe_lj^h−P_ki^l S_jl^h+ +R_kij^h +R^l_0ijQ^h_lk

XXY X_kij^h =Pe_kj^l P_li^h+P_kj^l Se_il^h−Pe_ki^l P_lj^h−P_ki^l Se_jl^h

XXXY_kij^h = ˙∇iS_jk^h +S_il^hSe_jk^l +S_jk^l Pe_li^h−∇˙jS_ik^h −S_jl^hSe^l_ik−S^l_ikPe_lj^h+R_0ij^l Pe_lk^h XXXX_kij^h = ˙∇iSe_jk^h +Se_jk^l Se_il^h+S_jk^l P_li^h−∇˙jSe_ik^h −Se_ik^l Se_jl^h−S_ik^l P_lj^h+

+R_kij^h +R^l_0ijP_lk^h

Remark.Note that, as a first step, the formulae for the local expression ofKalso contain some other terms involving the Christoffel symbolsΓ^h_ij. However, all of these terms are involved in the derivative∇. For example˙

∇˙_iPe_jk^h =δ_iPe_jk^h −Γ^l_ijPe_lk^h −Γ^l_ikPe_jl^h+ Γ^h_ilPe_jk^l ,

but using the expression of Pe_ij^h and taking account of relations (2.2) we obtain after a straightforward computation that

∇˙iPe_jk^h = a⁰₂

2 H₁^hlg0j∇˙ickl−a2

2H₁^sl(∂jG¹_kl−∂lG¹_jk)( ˙∇icsr)H₁^rh

Remark also that the terms∇˙iQ^h_jk and∇˙iP_jk^h do not appear because they are zero as follows from the formulae (2.2).

Now, we have to replace the expression of the M tensor fieldsQ^h_ij, P_ij^h, Pe_ij^h,S^h_ij, Se_ij^h in order to obtain the explicit expression of the components ofK. However, the final expressions are quite complicated, but they may be obtained after some long and hard computation made by using the Mathematica package RICCI.

Recall that the pseudo-Riemannian manifold (T M, G) has constant sectional cur- vaturekif its curvature tensor fieldKis given by

K(X, Y)Z=K0(X, Y)Z =k(G(Y, Z)X−G(X, Z)Y),∀X, Y, Z∈Γ(T M).

In order to find under which conditions (T M, G) has constant sectional curvature we shall consider the differences between the components of the tensor fieldsK andK0

and we shall denote them byDif f. For example

Dif f Y Y Y Y_kij^h =Y Y Y Y_kij^h −Y Y Y Y_0kij^h . The explicit expression ofDif f Y Y Y Y_kij^h is

Dif f Y Y Y Y_kij^h = a⁰₁−b1

2(a1+ 2tb1)(gjkδ^h_i −gijδ^h_k)+

+ 1

4a²₁(a1+ 2tb1)(3a1a⁰₁²−2a²₁a⁰⁰₁−3a1b²₁+ 2a²₁b⁰₁+ 2a⁰₁²b1t−4a1a⁰⁰₁b1t−

−2b³₁t+ 4a1a⁰₁b⁰₁t)(δ_k^hg0ig0j−δ^h_ig0jg0k).

From Lemma 2 it follows thatDif f Y Y Y Y_kij^h = 0 if and only ifb1=a⁰₁. By replacing b1=a⁰₁ in the expression ofDif f Y Y XY_kij^h we obtain

Dif f Y Y XY_kij^h =−ka1gjkδ^h_i +ka1gikδ_j^h+ka⁰₁δ^h_jg0ig0k−ka⁰₁δ_i^hg0jg0k. Using again Lemma 2 and taking account that a1 6= 0 it follows that Dif f Y Y XY_kij^h = 0 if and only ifk= 0. Under the conditionsb1=a⁰₁ andk= 0 we have

Dif f Y Y XX_kij^h =Dif f Y XY X_kij^h =Dif f XXY X_kij^h = 0.

ComputingDif f XXXX_kij^h and takingy= 0 it follows thatR= 0, so (M, g) is flat.

Takingy= 0 in the formulaeDif f Y XXX_kij^h = 0 we obtain





na⁰₂(0)cjk=−2b2(0)gjk

a⁰₂(0)cjk=−(n+ 1)b2(0)gjk

from which we have

(n²+n−2)b2(0)gjk= 0

Assuming thatn >1 it follows thatb2(0) = 0, soa⁰₂(0)cjk= 0.

Now we may consider the following cases:

(i)a⁰₂= 0, b2= 0 so the pseudo-Riemannian metricGis given by

(2.3)







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



G(∂i, ∂j) = 0,

G(δi, ∂j) =a1gij+a⁰₁g0ig0j, G(δi, δj) =a2cij,

wherea₁: [0,∞)→Ris a smooth function anda₂is a nonzero constant. Computing the remaining differences we have

Dif f Y XY Y_kij^h =Dif f Y XXY_kij^h =Dif f Y XXX_kij^h =

=Dif f XXY Y_kij^h =Dif f XXXX_kij^h = 0, and

Dif f XXXY_kij^h = a2

2a1

( ˙∇i∇˙kc^h_j −∇˙j∇˙kc^h_i + ˙∇j∇˙^hcik−∇˙i∇˙^hcjk)+

+ a⁰₁a2

2a1(a1+ 2ta⁰₁)( ˙∇i∇˙lcjk−∇˙i∇˙kclj+ ˙∇j∇˙kcli−∇˙j∇˙lcik)y^hy^l. Takingy= 0 inDif f XXXY_kij^h = 0 it follows that

∇˙i∇˙kc^h_j −∇˙j∇˙kc^h_i + ˙∇j∇˙^hcik−∇˙i∇˙^hcjk= 0.

Observing that the first bracket of the expression of Dif f XXXY_kij^h is zero if and only if the second bracket of it is zero we may state:

Theorem 5. If the tangent bundle (T M, G) has constant sectional curvature, where G has the entries given by (2.3), then it must be flat. Moreover, (T M, G) is flat if and only if(M, g)is flat and the tensor field c satisfies the condition

(2.4) ∇˙i∇˙lcjk−∇˙i∇˙kclj+ ˙∇j∇˙kcli−∇˙j∇˙lcik= 0.

A symmetric tensor fieldc of type (0,2) onM is Codazzi tensor field if ( ˙∇Xc)(Y, Z) = ( ˙∇Yc)(X, Z), X, Y, Z ∈Γ(M)

Note that the condition (2.4) is fulfilled if c is parallel with respect to ∇ or it is a Codazzi tensor field onM.

(ii)a2= 0,b2= 0 so the pseudo-Riemannian metricGis given by

(2.5)



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

G(∂i, ∂j) = 0,

G(δi, ∂j) =a1gij+a⁰₁g0ig0j, G(δi, δj) = 0,

wherea1: [0,∞)→Ris a smooth function. In this case all the differencesDif f are zero, so we have the following

Theorem 6. If the tangent bundle (T M, G) has constant sectional curvature, where G has the entries given by (2.5), then it must be flat. Moreover, (T M, G) is flat if and only if(M, g)is flat.

(iii)a2= 0, so the pseudo-Riemannian metricGis given by

(2.6)



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G(∂i, ∂j) = 0,

G(δi, ∂j) =a1gij+a⁰₁g0ig0j, G(δi, δj) =b2g0ig0j,

wherea1, b2: [0,∞)→Rare smooth functions, b2(0) = 0. In this case we have Dif f XXY Y_kij^h =ugjkyiy^h−ugikyjy^h,

whereu= ^b2(a1b2−2a

0

1b2t+2a1b⁰₂t)

4a1(a1+2a⁰₁t)² . FromDif f XXY Y_kij^h = 0 we have, using the remark made in the first section,

b2(a1b2−2a⁰₁b2t+ 2a1b⁰₂t) = 0.

Remark thatb2= 0 is a solution of this equation. Next we shall prove thatb2= 0 is the unique solution of this equation. First of all let us observe that

¡tb²₂ a²₁

¢⁰

=a1b²₂+ 2ta1b2b⁰₂−2a⁰₁b²₂t

a³₁ = 0,∀t≥0

It follows that tb²₂a⁻²₁ is a constant function, but since b2(0) = 0, we must have tb²₂(t)a⁻²₁ (t) = 0,∀t ≥0, sob₂(t) = 0, for all t ≥0. As a consequence of Theorem 6 we obtain:

Theorem 7. If the tangent bundle (T M, G) has constant sectional curvature, where G has the entries given by (2.6), then it must be flat. Moreover, (T M, G) is flat if and only if(M, g)is flat and b₂= 0.

Acknowledgement.The author is indebted to Professor V. Oproiu for suggesting the subject and for the advice during the preparation of this paper.

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Cristian Eni

University of Galat¸i, Department of Mathematics, 48 Domneasc˘a Str., RO-800008, Galat¸i, Romania.

E-mail: [email protected]