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 (( ∂
∂x1)π(y), ...,(∂x∂n)π(y)) i.e.y=yi( ∂
∂xi)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 np+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=g
π(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
components of the metric tensor field g on M, then the components a(k y k2)g ij
define a symmetricM-tensor field of type (0,2) onT M. The componentsg0i=yjgji
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 ( ∂
∂y1, ...,
∂
∂yn) define a local frame
field forV T M and for the horizontal distributionHT M we have the local frame field ( δ
δx1, ...,
δ
δxn), where
δ δxi =
∂ ∂xi −Γ
h i0
∂ ∂yh; Γ
h
i0= Γhikyk
and Γh
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=yjgji, 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
ugjkδih−ugijδkh+vg0ig0jδkh−vg0jg0kδhi = 0, g0i=yjgji, y∈π−1(U)
on the domain of any induced local chart onT M, thenu=v= 0.
Remark.From the relation
ugjkyiyh−ugikyjyh= 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 kyk
2=1
2gπ(y)(y, y) = 1 2gij(x)y
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])
The expression ofGin local adapted frames is defined by the followingM-tensor fields
G1ij =G(δi, ∂j) =a1gij+b1g0ig0j,
G2ij =G(δi, δj) =a2cij+b2g0ig0j.
The associated matrix ofGwith respect to the adapted local frame is
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
H1ij = 1
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+Pejih)∂h+Pjihδh,
∇∂iδj = P
h
ijδh+Peijh∂h, ∇δiδj = (Γ
h
ij+Seijh)δh+Sijh∂h,
where theM-tensor fieldsQh
ij,Pijh,Peijh,Sijh,Seijh are given by:
Qhij = 1 2H
hk
1 (∂iG1jk+∂jG1ik),
Pijh = 1 2H
hk
1 (∂iG1jk−∂kG1ij),
e Pijh =1
2H
hk
1 ∂iG2jk−
1 2H
rl
1(∂iG1jl−∂lG1ij)G2rkH1kh,
Sijh =a2
2 ( ˙∇icjk+ ˙∇jcki−∇k˙ cij)H
kh 1 − −a1R0ijkH1kh+
1 2H
sl
1(∂lG2ij)G2skH1kh,
e Sijh =−1
2H
hk 1 ∂kG2ij,
Rlijk denoting the local coordinate components of the Riemann-Christoffel tensor of
the Levi-Civita connection∇˙ onM andR0ijk=ylRlijk
Remark.Replacing the expressions ofG1ij,G2ij,H1ij,∂iG1jk,∂iG2jk 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 Ykijh ∂h,
K(∂i, ∂j)δk=Y Y XYkijh ∂h+Y Y XXkijh δh,
K(∂i, δj)∂k=Y XY Ykijh ∂h+Y XY Xkijh δh,
K(∂i, δj)δk=Y XXYkijh ∂h+Y XXXkijh δh,
K(δi, δj)∂k=XXY Ykijh ∂h+XXY Xkijh δh,
K(δi, δj)δk =XXXYkijh ∂h+XXXXkijh δh,
Y Y Y Yh
kij=∂iQhjk+QljkQilh−∂jQhik−QlikQhjl
Y Y XYkijh =∂iPejkh +PeilhPjkl +Pejkl Qhil−∂jPeikh−PejlhPikl −Peikl Qhjl
Y Y XXkijh =∂iPjkh +Pjkl Pilh−∂jPikh −Pikl Pjlh
Y XY Yh
kij =∂iPekjh +Pekjl Qhil+PeilhPkjl −PeljhQlik
Y XY Xh
kij=∂iPkjh +Pkjl Pilh−PljhQlik
Y XXYh
kij=∂iSjkh +Sjkl Qhil+Sejkl Peilh−SjlhPikl −Peikl Peljh−∇j˙ Peikh
Y XXXh
kij=∂iSejkh +Sejkl Pilh−SejlhPikl −Peikl Pljh
XXY Yh
kij= ˙∇iPekjh +Pekjl Pelih+Pkjl Silh−∇j˙ Pekih −PekilPeljh−Pkil Sjlh+
+Rh
kij+Rl0ijQhlk
XXY Xkijh =Pekjl Plih+Pkjl Seilh−Pekil Pljh−Pkil Sejlh
XXXYh
kij = ˙∇iSjkh +SilhSejkl +Sjkl Pelih−∇j˙ Sikh −SjlhSelik−SlikPeljh+R0ijl Pelkh
XXXXh
kij= ˙∇iSejkh +Sejkl Seilh+Sjkl Plih−∇j˙ Seikh −Seikl Sejlh−Sikl Pljh+
+Rh
kij+Rl0ijPlkh
Remark.Note that, as a first step, the formulae for the local expression ofKalso contain some other terms involving the Christoffel symbolsΓh
ij. However, all of these
terms are involved in the derivative∇˙. For example
˙
∇iPejkh =δiPejkh −ΓlijPelkh −ΓlikPejlh+ ΓhilPejkl ,
but using the expression ofPeh
ij and taking account of relations (2.2) we obtain after
a straightforward computation that
˙
∇iPejkh = a ′
2
2 H
hl
1 g0j∇i˙ ckl−
a2
2H
sl
1(∂jG1kl−∂lG1jk)( ˙∇icsr)H1rh
Remark also that the terms∇i˙ Qh
jk and∇i˙ Pjkh 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 fieldsQh
ij, Pijh, Peijh,Shij,
e Sh
ij 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 Ykijh =Y Y Y Ykijh −Y Y Y Y0kijh . The explicit expression ofDif f Y Y Y Yh
kij is
Dif f Y Y Y Ykijh = a ′
1−b1
2(a1+ 2tb1)
+ 1
Using again Lemma 2 and taking account that a1 6= 0 it follows that
Dif f Y Y XYkijh = 0 if and only ifk= 0. Under the conditionsb1=a
Now we may consider the following cases:
(i)a′2= 0, b2= 0 so the pseudo-Riemannian metricGis given by
the remaining differences we have
˙
∇i∇k˙ chj −∇j˙ ∇k˙ chi + ˙∇j∇˙hc
ik−∇i˙ ∇˙hcjk= 0.
Observing that the first bracket of the expression of Dif f XXXYh
kij 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˙ ∇l˙ cjk−∇i˙ ∇k˙ clj+ ˙∇j∇k˙ cli−∇j˙ ∇l˙ cik= 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)
G(∂i, ∂j) = 0,
G(δi, ∂j) =a1gij+a
′
1g0ig0j,
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)
G(∂i, ∂j) = 0,
G(δi, ∂j) =a1gij+a
′
1g0ig0j,
G(δi, δj) =b2g0ig0j,
wherea1, b2: [0,∞)→Rare smooth functions, b2(0) = 0. In this case we have
Dif f XXY Ykijh =ugjkyiyh−ugikyjyh,
whereu= b2(a1b2−2a′1b2t+2a1b
′
2t)
4a1(a1+2a′
1t)2
. FromDif f XXY Yh
kij= 0 we have, using the remark
b2(a1b2−2a
′
1b2t+ 2a1b
′
2t) = 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
a2 1
¢′ =a1b
2
2+ 2ta1b2b
′
2−2a
′
1b22t
a3 1
= 0,∀t≥0
It follows that tb22a−12 is a constant function, but since b2(0) = 0, we must have
tb22(t)a−12(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.
References
[1] C.L.Bejan, V. Oproiu Tangent bundles of quasi-constant holomorphic sectional curvatures,Balkan Journal of Geometry and Its Applications 11 (1) (2006), 11-22.
[2] J.M.Lee,A Mathematica package for doing tensor calculation in differential ge-ometry. User’s Manual, 2000.
[3] K.P.Mok, E.M. Patterson, Y.C. Wong, Structure of symmetric tensors of type (0,2) and tensors of type (1,1) on the tangent bundle, Trans. Amer. Math. Soc., 234 (1977), 253-278.
[4] C. Oniciuc, On the harmonic sections of tangent bundles, An. Univ. Buc., 1 (1998), 67-72.
[5] V. Oproiu,Some classes of natural almost Hermitian structures on the tangent bundles.Publ. Math. Debrecen 62 (2003), 561-576.
[6] V. Oproiu,On the harmonic sections of cotangent bundles, Rend. Sem. Fac. Sci. Univ. Cagliari 59 (2), (1989), 177-184.
[7] V. Oproiu, N. Papaghiuc, Some classes of almost anti-hermitian structures on the tangent bundle, Mediterr. J. Math. 1 (2004), 269-282.
[8] D.D. Poro¸sniuc, A class of locally symmetric K¨ahler Einstein structures on the nonzero cotangent bundle of a space form,Balkan Journal of Geometry and Its Applications, 9 (2), (2004), 68-81.
[9] K. Yano, S. Ishihara, Tangent and Cotangent Bundles, M. Dekker, New York 1973.
Cristian Eni
