Ninth International Conference on
Geometry, Integrability and Quantization
June 8–13, 2007, Varna, Bulgaria
Ivaïlo M. Mladenov, Editor
SOFTEX, Sofia 2008, pp 198–209
Geometry,
Integrability
and
IX
Quantization
THE SECTIONAL CURVATURE OF THE TANGENT BUNDLES
WITH GENERAL NATURAL LIFTED METRICS
SIMONA L. DRUŢĂ
Faculty of Mathematics, “Al. I. Cuza” University, 700506 Iaşi, Romania
Abstract. We study some properties of the tangent bundles with metrics
of general natural lifted type. We consider a Riemannian manifold (M, g)
and we find the conditions under which the Riemannian manifold (T M, G),
where T M is the tangent bundle of M and G is the general natural lifted
metric of g, has constant sectional curvature.
1. Introduction
In the geometry of the tangent bundle T M of a smooth n-dimensional Riemannian
manifold (M, g) one uses several Riemannian and pseudo-Riemannian metrics, induced by the Riemannian metric g on M . Among them, we may quote the Sasaki
metric, the Cheeger-Gromoll metric and the complete lift of the metric g. The
possibility to consider vertical, complete and horizontal lifts on the tangent bundle
T M (see [18]) leads to some interesting geometric structures, studied in the last
years (see [1–3, 8, 9, 17]), and to interesting relations with some problems in Lagrangian and Hamiltonian mechanics. On the other hand, the natural lifts of g to
T M (introduced in [5, 6]) induce some new Riemannian and pseudo-Riemannian
geometric structures with many nice geometric properties (see [4, 5]).
Oproiu [11–13] has studied some properties of a natural lift G, of diagonal type,
of the Riemannian metric g and a natural almost complex structure J of diagonal
type on T M (see also [15, 16]). In [10], the same author has presented a general
expression of the natural almost complex structures on T M . In the definition of
the natural almost complex structure J of general type there are involved eight
parameters (smooth functions of the density energy on T M ). However, from the
condition for J to define an almost complex structure, four of the above parameters
can be expressed as (rational) functions of the other four parameters. A Riemannian metric G which is a natural lift of general type of the metric g depends on other
198
The Sectional Curvature of Tangent Bundles . . .
199
six parameters. In [14] we have found the conditions under which the Kählerian
manifold (T M, G, J) has constant holomorphic sectional curvature.
In the present paper we study the sectional curvature of the tangent bundle of a
Riemannian manifold (M, g). Namely, we are interested in finding the conditions
under which the Riemannian manifold (T M, G), where G is the general natural
lifted metric of g, has constant sectional curvature. We obtain that the sectional
curvature of (T M, G) is zero and the base manifold must be flat.
2. Preliminary Results
Consider a smooth n-dimensional Riemannian manifold (M, g) and denote its
tangent bundle by τ : T M −→ M . Recall that T M has a structure of a 2ndimensional smooth manifold, induced from the smooth manifold structure of M .
This structure is obtained by using local charts on T M induced from usual local
charts on M . If (U, ϕ) = (U, x1 , . . . , xn ) is a local chart on M , then the corresponding induced local chart on T M is (τ −1 (U ), Φ) = (τ −1 (U ), x1 , . . . , xn ,
y 1 , . . . , y n ), where the local coordinates xi , y j , i, j = 1, . . . , n, are defined as follows. The first n local coordinates of a tangent vector y ∈ τ −1 (U ) are the local
coordinates in the local chart (U, ϕ) of its base point, i.e., xi = xi ◦ τ , by an abuse
of notation. The last n local coordinates y j , j = 1, . . . , n, of y ∈ τ −1 (U ) are the
vector space coordinates of y with respect to the natural basis in Tτ (y) M defined by
the local chart (U, ϕ). Due to this special structure of differentiable manifold for
T M , it is possible to introduce the concept of M -tensor field on it. The M -tensor
fields are defined by their components with respect to the induced local charts on
T M (hence they are defined locally), but they can be interpreted as some (partial)
usual tensor fields on T M . However, the essential quality of an M -tensor field on
T M is that the local coordinate change rule of its components with respect to the
change of induced local charts is the same as the local coordinate change rule of
the components of an usual tensor field on M with respect to the change of local
charts on M . More precisely, an M -tensor field of type (p, q) on T M is defined by
sets of np+q components (functions depending on xi and y i ), with p upper indices
and q lower indices, assigned to induced local charts (τ −1 (U ), Φ) on T M , such
that the local coordinate change rule of these components (with respect to induced
local charts on T M ) is that of the local coordinate components of a tensor field of
type (p, q) on the base manifold M (with respect to usual local charts on M ), when
a change of local charts on M (and hence on T M ) is performed (see [7] for further
details); e.g., the components y i , i = 1, . . . , n, corresponding to the last n local
coordinates of a tangent vector y, assigned to the induced local chart (τ −1 (U ), Φ)
define an M -tensor field of type (1, 0) on T M . An usual tensor field of type (p, q)
on M may be thought of as an M -tensor field of type (p, q) on T M . If the considered tensor field on M is covariant only, the corresponding M -tensor field on
200
Simona L. Druţă
T M may be identified with the induced (pullback by τ ) tensor field on T M . Some
useful M -tensor fields on T M may be obtained as follows. Let u : [0, ∞) −→ R
be a smooth function and let kyk2 = gτ (y) (y, y) be the square of the norm of the
tangent vector y ∈ τ −1 (U ). If δji are the Kronecker symbols (in fact, they are the
local coordinate components of the identity tensor field I on M ), then the components u(kyk2 )δji define an M -tensor field of type (1, 1) on T M . Similarly, if
gij (x) are the local coordinate components of the metric tensor field g on M in the
local chart (U, ϕ), then the components u(kyk2 )gij define a symmetric M -tensor
field of type (0, 2) on T M . The components g0i = y k gki define an M -tensor field
of type (0, 1) on T M .
˙ the Levi-Civita connection of the Riemannian metric g on M . Then
Denote by ∇
we have the direct sum decomposition
T T M = V T M ⊕ HT M
(1)
of the tangent bundle to T M into the vertical distribution V T M = Ker τ∗
˙ The set of vector fields
and the horizontal
distribution HT M defined by ∇.
∂
∂
−1
, . . . , ∂yn on τ (U ) defines a local frame field for V T M and for HT M
∂y 1
we have the local frame field
δ
, . . . , δxδn
δx1
δ
∂
∂
=
− Γh0i h ,
i
i
δx
∂x
∂y
, where
Γh0i = y k Γhki
and Γhki (x) are the Christoffel symbols of g.
The set ∂y∂ 1 , . . . , ∂y∂n , δxδ 1 , . . . , δxδn defines a local frame on T M , adapted to the
direct sum decomposition (1). Remark that
∂
=
∂y i
∂
∂xi
V
,
δ
=
δxi
∂
∂xi
H
where X V and X H denote the vertical and horizontal lift of the vector field X
on M respectively. We can use the vertical and horizontal lifts in order to obtain
invariant expressions for some results in this paper. However, we should prefer to
work in local coordinates since the formulas are obtained easier and, in a certain
sense, they are more natural.
We can easily obtain the following
Lemma 1. If n > 1 and u, v are smooth functions on T M such that
ugij + vg0i g0j = 0
on the domain of any induced local chart on T M , then u = 0, v = 0.
Remark. In a similar way we obtain from the condition
uδji + vg0j y i = 0
201
The Sectional Curvature of Tangent Bundles . . .
the relation u = v = 0.
Consider the energy density of the tangent vector y with respect to the Riemannian
metric g
1
1
1
t = kyk2 = gτ (y) (y, y) = gik (x)y i y k ,
2
2
2
Obviously, we have t ∈ [0, ∞) for all y ∈ T M .
y ∈ τ −1 (U ).
3. The Sectional Curvature of the Tangent Bundle with General
Natural Lifted Metric
Let G be the general natural lifted metric on T M , defined by
δ
δ
(1)
, j = c1 gij + d1 g0i g0j = Gij
i
δx δx
∂
∂
(2)
,
= c2 gij + d2 g0i g0j = Gij
G
∂y i ∂y j
∂
δ
δ
∂
(3)
G
, j =G
, j = c3 gij + d3 g0i g0j = Gij
i
i
∂y δx
δx ∂y
G
(2)
where c1 , c2 , c3 , d1 , d2 , d3 are six smooth functions of the density energy on T M .
The Levi-Civita connection ∇ of the Riemannian manifold (T M, G) is obtained
from the formula
2G(∇X Y, Z) = X(G(X, Z)) + Y (G(X, Z)) − Z(G(X, Y )) + G([X, Y ], Z)
− G([X, Z], Y ) − G([Y, Z], X)
for all X, Y, Z ∈ χ(M ) and is characterized by the conditions
∇G = 0,
T =0
where T is the torsion tensor of ∇.
In the case of the tangent bundle T M we can obtain the explicit expression of ∇.
The symmetric 2n × 2n matrix

 (1)
(3)
Gij Gij


(3)
Gij
associated to the metric G in the base
(2)
Gij
δ
, . . . , δxδn , ∂y∂ 1 , . . . , ∂y∂n
δx1
 ij

ij
H(1) H(3)


ij
ij
H(3)
H(2)
has the inverse
202
Simona L. Druţă
where the entries are the blocks
kl
H(1)
= p1 g kl + q1 y k y l
kl
H(2)
= p2 g kl + q2 y k y l
kl
H(3)
kl
(3)
k l
= p 3 g + q3 y y .
Here g kl are the components of the inverse of the matrix (gij ) and p1 , q1 , p2 , q2 ,
p3 , q3 : [0, ∞) → R, some real smooth functions. Their expressions are obtained
by solving the system
(1)
(3)
(1)
(3)
(3)
(2)
(3)
(2)
hk
hk
Gih H(1)
+ Gih H(3)
= δik
hk
hk
Gih H(3)
+ Gih H(2)
=0
hk
hk
Gih H(1)
+ Gih H(3)
=0
hk
hk
Gih H(3)
+ Gih H(2)
= δik
in which we substitute the relations (2) and (3). By using Lemma 1, we get p1 , p2 ,
p3 as functions of c1 , c2 , c3
c2
c1
c3
p1 =
,
p2 =
,
p3 = −
(4)
2
2
c1 c2 − c3
c1 c2 − c3
c1 c2 − c23
and q1 , q2 , q3 as functions of c1 , c2 , c3 , d1 , d2 , d3 , p1 , p2 , p3
c2 d1 p1 − c3 d3 p1 − c3 d2 p3 + c2 d3 p3 + 2d1 d2 p1 t − 2d23 p1 t
c1 c2 − c23 + 2c2 d1 t + 2c1 d2 t − 4c3 d3 t + 4d1 d2 t2 − 4d23 t2
d2 p2 + d3 p3
(5)
q2 = −
c2 + 2d2 t
(c3 + 2d3 t)[(d3 p1 + d2 p3 )(c1 + 2d1 t) − (d1 p1 + d3 p3 )(c3 + 2d3 t)]
+
(c2 + 2d2 t)[(c1 + 2d1 t)(c2 + 2d2 t) − (c3 + 2d3 t)2 ]
(d3 p1 + d2 p3 )(c1 + 2d1 t) − (d1 p1 + d3 p3 )(c3 + 2d3 t)
q3 = −
·
(c1 + 2d1 t)(c2 + 2d2 t) − (c3 + 2d3 t)2
q1 = −
In [14] we obtained the expression of the Levi-Civita connection of the Riemannian
metric G on T M .
Theorem 1. The Levi-Civita
connection ∇ of G has the following expression in
∂
the local adapted frame ∂y1 , . . . , ∂y∂n , δxδ 1 , . . . , δxδn
∂
∂
eh δ ,
= Qhij h + Q
ij
j
∂y
∂y
δxh
δ
δ
∂
∇ ∂
= Pijh h + Peijh h ,
j
δx
∂y
∂y i δx
∇
∂
∂y i
∂
∂
δ
= (Γhij + Pejih ) h + Pjih h
j
∂y
∂y
δx
δ
δ
h
h ∂
∇ δ
= (Γhij + Seij
) h + Sij
j
δx
∂y h
δxi δx
∇
δ
δxi
The Sectional Curvature of Tangent Bundles . . .
203
˙ and the M -tensor fields
where Γhij are the Christoffel symbols of the connection ∇
appearing as coefficients in the above expressions are given as
1
1
(2)
(2)
(2)
(3)
(3)
kh
kh
+ (∂i Gjk + ∂j Gik )H(3)
Qhij = (∂i Gjk + ∂j Gik − ∂k Gij )H(2)
2
2
e h = 1 (∂i G(2) + ∂j G(2) − ∂k G(2) )H kh + 1 (∂i G(3) + ∂j G(3) )H kh
Q
ij
(3)
(1)
ij
jk
ik
jk
ik
2
2
1
1
(3)
(3)
(1)
(2)
kh
l
kh
Pijh = (∂i Gjk − ∂k Gij )H(3)
+ (∂i Gjk + R0jk
Gli )H(1)
2
2
1
1
(3)
(3)
(1)
(2)
kh
l
kh
+ (∂i Gjk + R0jk
Gli )H(3)
Peijh = (∂i Gjk − ∂k Gij )H(2)
2
2
1
(2)
(2)
h
l
kh
kh
Sij
= − (∂k Gij + R0ij
Glk )H(2)
+ c3 Ri0jk H(3)
2
1
(1)
(2)
l
kh
kh
h
Glk )H(3)
+ c3 Ri0jk H(1)
Seij
= − (∂k Gij + R0ij
2
h are the components of the curvature tensor field of the Levi Civita
where Rkij
˙ of the base manifold (M, g).
connection ∇
Taking into account the expressions (2), (3) and by using the formulas (4), (5) we
h, P
eh , Q
e h , Seh .
can obtain the detailed expressions of Pijh , Qhij , Sij
ij
ij
ij
The curvature tensor field K of the connection ∇ is defined by the well known
formula
K(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,
X, Y, Z ∈ Γ(T M ).
By using the local adapted frame δxδ i , ∂y∂ j , i, j = 1, . . . , n, we obtained in [14],
after a standard straightforward computation
δ
δ
δ
δ
h
h ∂
K
, j
= XXXXkij
+ XXXYkij
i
k
h
δx δx δx
δx
∂y h
δ
δ
∂
δ
h
h ∂
K
, j
= XXY Xkij
+ XXY Ykij
i
k
h
δx δx ∂y
δx
∂y h
∂
∂
δ
δ
h
h ∂
K
, j
= Y Y XXkij
+ Y Y XYkij
i
k
h
∂y ∂y δx
δx
∂y h
∂
∂
∂
δ
h
h ∂
K
,
= Y Y Y Xkij
+ Y Y Y Ykij
∂y i ∂y j ∂y k
δxh
∂y h
∂
δ
δ
δ
h
h ∂
K
, j
= Y XXXkij
+ Y XXYkij
i
k
h
∂y δx δx
δx
∂y h
∂
δ
∂
δ
h
h ∂
K
, j
= Y XY Xkij
+ Y XY Ykij
i
k
h
∂y δx ∂y
δx
∂y h
where the M -tensor fields appearing as coefficients denote the horizontal and vertical components of the curvature tensor of the tangent bundle, and they are given
204
Simona L. Druţă
by
h
l
l
h el
l
h
l
h
XXXXkij
= Seilh Sejk
+ Plih Sjk
− Sejl
Sik − Pljh Sik
+ Rkij
+ R0ij
Plk
h
l
l
l
h
l
h l
XXXYkij
= Sejk
Silh + Pelih Sjk
− Seik
Sjl
− Peljh Sik
+ Pelk
R0ij
1˙ r
(2) (3)
˙
− ∇
i R0jk Grl Hhl + c3 ∇i Rj0kh
2
h
l
l eh
l
l eh
l eh
XXY Xkij
= Pekj
Plih + Pkj
Sil − Peki
Pljh − Pki
Sjl + R0ij
Qlk
h
l eh
l
l eh
l h
l
h
XXY Ykij
= Pekj
Pli + Pkj
Silh − Peki
Plj − Pki
Sjl + R0ij
Qhlk + Rkij
h
h
h
l eh
l
l eh
l
Y Y XXkij
= ∂i Pjk
− ∂j Pik
+ Pejk
Qil + Pjk
Pilh − Peik
Qjl − Pik
Pjlh
h
l
l eh
l
l eh
h
h
+ Pejk
Qhil + Pjk
Pil − Peik
Qhjl − Pik
Pjl
− ∂j Peik
Y Y XYkij
= ∂i Pejk
h
l eh
h
e h + Ql Q
eh
el
el h
e h − ∂j Q
Y Y Y Xkij
= ∂i Q
ik
jk il + Qjk Pil − Qik Qjl − Qik Pjl
jk
h
e l Pe h − Ql Qh − Q
e l Pe h
Y Y Y Ykij
= ∂i Qhjk − ∂j Qhik + Qljk Qhil + Q
jk il
ik jl
ik jl
h
h
l eh
l
l
l eh
˙ j Rr G(2) H (3)
Y XXXkij
= ∂i Sejk
+ Sjk
Qil + Sejk
Pilh − Peik
Pljh − Pik
Sjl − ∇
0ik rl
hl
h
h
l
l eh
l eh
l h
˙ j Rr G(2) H (1)
Y XXYkij
= ∂i Sjk
+ Sjk
Qhil + Sejk
Pil − Peik
Plj − Pik
Sjl − ∇
0ik rl
hl
h
h
l eh
l
e l Seh
Y XY Xkij
= ∂i Pkj
+ Pekj
Qil + Pkj
Pilh − Qlik Pljh − Q
ik jl
h
h
l
l eh
el Sh .
Y XY Ykij
= ∂i Pekj
+ Pekj
Qhil + Pkj
Pil − Qlik Peljh − Q
ik jl
We mention that we used the character X on a certain position to indicate that the
argument on that position was a horizontal vector field and, similarly, we used the
character Y for vertical vector fields.
We compute the partial derivatives with respect to the tangential coordinates y i of
(α)
jk
of Gjk and H(α)
, for α = 1, 2, 3
(α)
∂i Gjk = c0α g0i gjk + d0α g0i g0j g0k + dα gij g0k + dα g0i gjk
jk
∂i H(α)
= p0α g jk g0i + qα0 g0i y j y k + qα δij y k + qα y j δik
(α)
∂i ∂j Gkl = c00α g0i g0j gkl + c0α gij gkl + d00α g0j g0k g0l + d0α gij g0k g0l + d0α g0j gik g0l
+ d0α g0j g0k gil + d0α g0i gjk g0l + d0α g0i g0k gjl + dα gjk gil + dα gik gjl .
Next we get the first order partial derivatives with respect to the tangential coordih, P
eh , Q
e h , Seh
nates y i of the M -tensor fields Pijh , Qhij , Sij
ij
ij
ij
1
(2)
(2)
(2)
hl
(∂j Gkl + ∂k Gjl − ∂l Gjk )
∂i Qhjk = ∂i H(2)
2
1 hl
(2)
(2)
(2)
+ H(2)
(∂i ∂j Gkl + ∂i ∂k Gjl − ∂i ∂l Gjk )
2
The Sectional Curvature of Tangent Bundles . . .
eh
∂i Q
jk
h
∂i Pejk
h
∂i Pjk
h
∂i Sjk
205
1
1 hl
(3)
(3)
(3)
(3)
hl
+ ∂i H(3)
(∂j Gkl + ∂k Gjl ) + H(3)
(∂i ∂j Gkl + ∂i ∂k Gjl )
2
2
1
(2)
(2)
(2)
hl
= ∂i H(3)
(∂j Gkl + ∂k Gjl − ∂l Gjk )
2
1 hl
(2)
(2)
(2)
+ H(3)
(∂i ∂j Gkl + ∂i ∂k Gjl − ∂i ∂l Gjk )
2
1
1 hl
(3)
(3)
(3)
(3)
hl
+ ∂i H(1)
(∂j Gkl + ∂k Gjl ) + H(1)
(∂i ∂j Gkl + ∂i ∂k Gjl )
2
2
1
1 hl
(3)
(3)
(3)
(3)
hl
= ∂i H(2) (∂j Gkl − ∂l Gjk ) + H(2) (∂i ∂j Gkl − ∂i ∂l Gjk )
2
2
1
(1)
(2)
hl
r
+ ∂i H(3)
(∂j Gkl + R0kl
Grj )
2
1 hl
(1)
(2)
(2)
r
r
+ H(3) (∂i ∂j Gkl + Rikl
Grj + R0kl
∂i Grj )
2
1
1 hl
(3)
(3)
(3)
(3)
hl
= ∂i H(3)
(∂j Gkl − ∂l Gjk ) + H(3)
(∂i ∂j Gkl − ∂i ∂l Gjk )
2
2
1
(1)
(2)
hl
r
+ ∂i H(1)
(∂j Gkl + R0kl
Grj )
2
1 hl
(1)
(2)
(2)
r
r
+ H(1)
(∂i ∂j Gkl + Rikl
Grj + R0kl
∂i Grj )
2
1
(1)
(2)
(1)
(2)
l
rh
l
rh
= − [(∂i ∂r Gjk + Rijk
Glr )H(2)
+ (∂r Gjk + R0jk
Glr )∂i H(2)
]
2
rh
rh
rh
+c03 g0i Rj0kr H(3)
+ c3 (Rjikr H(3)
+ Rj0kr ∂i H(3)
)
1
(1)
(2)
(1)
(2)
l
rh
l
rh
h
Glr )H(3)
+ (∂r Gjk + R0jk
Glr )∂i H(3)
]
∂i Sejk
= − [(∂i ∂r Gjk + Rijk
2
rh
rh
rh
+c03 g0i Rj0kr H(1)
+ c3 (Rjikr H(1)
+ Rj0kr ∂i H(1)
).
It was not convenient to think c1 , c2 , c3 , d1 , d2 , d3 and p1 , p2 , p3 , q1 , q2 , q3
as functions of t since RICCI did not make some useful factorizations after the
command TensorSimplify. We decided to consider these functions as well as
their derivatives of first, second and third order, as constants, the tangent vector y
(1)
(2)
(3)
ij
ij
ij
as a first order tensor, the components Gij , Gij , Gij , H(1)
, H(2)
, H(3)
as second
order tensors and so on, on the Riemannian manifold M , the associated indices
being h, i, j, k, l, r, s.
The tensor field corresponding to the curvature tensor field of a Riemannian manifold (T M, G) having constant sectional curvature k, is given by the formula
K0 (X, Y )Z = k[G(Y, Z)X − G(X, Z)Y ].
206
Simona L. Druţă
After a straightforward computation we obtain
δ
δ
δ
, j
i
δx δx δxk
δ
δ
∂
K0
,
δxi δxj ∂y k
∂
δ
∂
,
K0
∂y i ∂y j δxk
∂
∂
∂
K0
, j
i
∂y ∂y ∂y k
∂
δ
δ
K0
, j
i
∂y δx δxk
∂
δ
∂
,
K0
i
j
∂y δx ∂y k
K0
δ
∂
+ XXXY 0hkij h
h
δx
∂y
δ
∂
= XXY X0hkij h + XXY Y 0hkij h
δx
∂y
δ
∂
= Y Y XX0hkij h + Y Y XY 0hkij h
δx
∂y
δ
∂
= Y Y Y X0hkij h + Y Y Y Y 0hkij h
δx
∂y
δ
∂
= Y XXX0hkij h + Y XXY 0hkij h
δx
∂y
δ
∂
= Y XY X0hkij h + Y XY Y 0hkij h
δx
∂y
= XXXX0hkij
where the M -tensor fields appearing as coefficients are the horizontal and vertical
components of the tensor K0 and they are given by
(1)
(1)
XXXY 0hkij = 0
(3)
(3)
XXY Y 0hkij = 0
XXXX0hkij = k[Gjk δih − Gik δjh ],
XXY X0hkij = k[Gjk δih − Gik δjh ],
(3)
(3)
(2)
(2)
Y Y XX0hkij = 0,
Y Y XY 0hkij = k[Gjk δih − Gik δjh ]
Y Y Y X0hkij = 0,
Y Y Y Y 0hkij = k[Gjk δih − Gik δjh ]
(3)
Y XXY 0hkij = kGjk δih
(2)
Y XY Y 0hkij = kGjk δih .
Y XXX0hkij = −kGik δjh ,
Y XY X0hkij = −kGik δjh ,
(1)
(3)
In order to get the conditions under which (T M, G) is a Riemannian manifold
of constant sectional curvature, we study the vanishing of the components of the
difference K − K0 . In this study it is useful the following generic result similar to
the Lemma 1.
Lemma 2. If α1 , . . . , α10 are smooth functions on T M such that
α1 δih gjk + α2 δjh gik + α3 δkh gij + α4 δkh g0i g0j + α5 δjh g0i g0k + α6 δih g0j g0k
+ α7 gjk g0i y h + α8 gik g0j y h + α9 gij g0k y h + α10 g0i g0j g0k y h = 0
then α1 = · · · = α10 = 0.
After a detailed analysis of several terms in the vanishing problem of the components of the above difference we can formulate the following proposition.
207
The Sectional Curvature of Tangent Bundles . . .
Proposition 1. Let (M, g) be a Riemannian manifold. If the tangent bundle T M
with the general natural lifted metric G has constant sectional curvature, then the
base manifold is flat.
h − XXY Y 0h reduces to Rh .
Proof: For y = 0 the difference XXY Ykij
kij
kij
If the sectional curvature of the tangent bundle is constant, this difference vanishes,
so the curvature of the base manifold must vanish too.
4. Tangent Bundles with Constant Sectional Curvature
Theorem 2. Let (M, g) be a Riemannian manifold. The tangent bundle T M with
the natural lifted metric G has constant sectional curvature if and only if the base
manifold is flat and the metric G has the associated matrix of the form


βgij + β 0 g0i g0j
cgij
βgij + β 0 g0i g0j αgij +

α0 β 2 +2α0 ββ 0 t−2αβ 02 t
g0i g0j
β2

where α and β are two real smooth function depending on the energy density and
c is an arbitrary constant. Moreover, in this case, T M is flat, i.e. k = 0.
Proof: In Proposition 1 we have proved that the base manifold of the tangent bundle with constant sectional curvature must be flat. By using the RICCI package
of the program Mathematica, we impose the vanishing condition for the curvature tensor of the base manifold in all the differences between the components of
the curvature tensors K and K0 of T M . After a long computation we find some
3 d1
g δ h in the
differences in which the third terms are of one of the forms: 2(c2c−c
c )) ij k
3
1 2
h − Y XXX0h and Y XY Y h − Y XY Y 0h ,
case of the differences Y XXXkij
kij
kij
kij
c1 d1
c2 d1
h for the difference Y XXY h − Y XXY 0h and
g
δ
g
δkh
ij
ij
2
2
k
kij
kij
2(c c −c )
2(c c −c )
1 2
1 2
3
3
h − Y XY X h .
for Y XY Xkij
kij
As all the coefficients which appear in these differences must vanish, we obtain
d1 = 0, because c1 and c3 , or c2 and c3 cannot vanish at the same time, the metric
g being non-degenerated.
h − XXXY 0h we obtain that this difference
If we impose d1 = 0 in XXXYkij
kij
0
0
contains the factors c1 c1 (c1 c3 − c1 c03 + c1 d3 ). Thus, for the annulation of this
c c0 −c d
difference, we have the cases c01 = 1 3c3 1 3 or c1 = const (c1 = 0 being a
particular case).
c c0 −c d
The first case, c01 = 1 3c3 1 3 is not a favorable one, because the difference
h − Y Y Y Y 0h contains two summands which cannot vanish
Y Y Y Ykij
kij
1
1
gjk δih − gik δjh .
2t
2t
208
Simona L. Druţă
In the case c1 = const we obtain
h
XXXXkij
− XXXX0hkij = −c1 k(gjk δih − gik δjh )
from which c1 = 0 or k = 0. If c1 = 0
h
XY Xkij
− XXY X0hkij = −k(c3 gjk δih − c3 gik δjh − d3 δjh g0i g0k + d3 δih g0j g0k ).
As we considered c1 = 0, we cannot have c3 = 0 because the metric g must be
non-degenerated, so the parenthesis cannot vanish and it remains k = 0. Now we
can conclude that the tangent bundle with general natural lifted metric cannot have
nonzero sectional curvature.
We continue the study of the general case c1 = const, since the case c1 = 0 is a
particular case only. Because the sectional curvature of the tangent bundle, k, is
h − XXY Y 0h vanishes if and only
null, we obtain that the difference XXY Ykij
kij
0
if d3 = c3 . This condition makes vanish all the differences that we study, except
h − Y Y Y X0h . From the annulation of this last difference, we obtain
Y Y Y Xkij
kij
d2 = c02 + 2t
c02 c3 c03 − c2 c02
3
·
c23
If we denote c1 by c, c2 by α and c3 by β, we obtain that the matrix associated to
the metric G has the form given in Theorem 2.
Therefore, Theorem 2 gives the unique form of the matrix associated to the metric
G.
Acknowledgements
Partially supported by the Grant AT No.191/2006, CNCSIS, Ministerul Educaţiei
şi Cercetării, România.
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