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Tangent sphere bundles of natural diagonal lift type
S. L. Druţă and V. Oproiu
Dedicated to the 70-th anniversary
of Professor Constantin Udriste
Abstract. We show that the tangent sphere bundles endowed with a
Riemannian metric induced from the natural diagonal lifted metric of the
tangent bundle are never space forms. Next we study the conditions under
which these tangent sphere bundles are Einstein.
M.S.C. 2000: 53C05, 53C15, 53C55.
Key words: natural lift; tangent sphere bundle; sectional curvature; Einstein manifold.
1
Introduction
The tangent sphere bundle Tr M consisting of spheres of constant radius r centered
in the null tangent vectors are hyper-surfaces of the tangent bundle T M , obtained by
considering only the tangent vectors having the norm equal to r.
In the most part of the papers, like [3]-[6], [17], [25], [27], [28], the metric considered
on the tangent bundle T M was the Sasaki metric (see [26]), but E. Boeckx noticed
that the unit tangent bundle equipped with the induced Cheeger-Gromoll metric is
isometric to the tangent sphere bundle T √1 M , of radius √12 endowed with the metric
2
induced by the Sasaki metric. This suggested to O. Kowalski and M. Sekizawa the
idea that the tangent sphere bundles with different constant radii and with the metrics
induced from the Sasaki metric might possess different geometrical properties, and in
the paper [11] from 2000 they showed how the geometry of the tangent sphere bundles
depends on the radius.
Some other metrics on the tangent sphere bundles may be constructed by using a
few lifts to the tangent bundle, which have been considered in [21]-[24], [29]-[31], and
studied in some very recent papers, such as [2], [7] and [14].
In the last years some interesting results were obtained by endowing the tangent
sphere bundles with Riemannian metrics induced by the natural lifted metrics from
T M , which are not Sasakian (see [1], [16], [18] – [20]).
Roughly speaking, a natural operator (in the sense of [8] – [10], [13]) is a fibred manifold mapping, which is invariant with respect to the group of local diffeomorphisms
of the base manifold.
∗
Balkan
Journal of Geometry and Its Applications, Vol.15, No.1, 2010, pp. 53-67.
c Balkan Society of Geometers, Geometry Balkan Press 2010.
°
54
S. L. Druţă and V. Oproiu
In the present paper we consider on the tangent bundle T M of a Riemannian
e obtained by the second author in [24] as the diagonal
manifold M a natural metric G,
lift of the Riemannian metric g from the base manifold. We prove that the tangent
e may
sphere bundle Tr M endowed with the Riemannian metric G induced from G
never have constant sectional curvature, then we find the conditions under which
(Tr M, G) is an Einstein manifold.
The manifolds, tensor fields and other geometric objects considered in this paper
are assumed to be differentiable of class C ∞ (i.e. smooth). The Einstein summation
convention is used throughout this paper, the range of the indices h, i, j, k, l, m, r,
being always {1, . . . , n}.
2
Preliminary results
Let (M, g) be a smooth n-dimensional Riemannian manifold and denote its tangent
bundle by τ : T M → M . The total space T M has a structure of a 2n-dimensional
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 (see [15]). The vertical lift to T M of a vector field X defined on
M will be denoted by X V .
˙ the Levi Civita connection of the Riemannian metric g on M . Then
Denote by ∇
we have the direct sum decomposition
(2.1)
T T M = V T M ⊕ HT M
of the tangent bundle to T M into the vertical distribution V T M = Ker τ∗ and
˙ (see [32]). The set of vector fields
the horizontal distribution HT M defined by ∇
{ ∂y∂ 1 , . . . , ∂y∂n } on τ −1 (U ) defines a local frame field for V T M and for HT M we have
∂
h ∂
the local frame field { δxδ 1 , . . . , δxδn }, where δxδ i = ∂x
Γh0i = y k Γhki , and
i − Γ0i ∂y h ,
Γhki (x) are the Christoffel symbols of g.
δ
∂
The set { ∂y
i , δxj }i,j=1,n , denoted also by {∂i , δj }i,j=1,n , defines a local frame on
T M , adapted to the direct sum decomposition (2.1). The horizontal lift to T M of a
∂ H
and
vector field X defined on M will be denoted by X H . Remark that δi = ( ∂x
i)
∂ V
∂i = ( ∂xi ) .
The second author considered in [24] a Riemannian metric, which is obtained as
a natural diagonal lift of the Riemannian metric g from the base manifold M to the
tangent bundle T M , the coefficients being functions of the energy density defined by
the tangent vector y as follows
t=
1
1
1
kyk2 = gτ (y) (y, y) = gik (x)y i y k ,
2
2
2
y ∈ τ −1 (U ).
Tangent sphere bundles of natural diagonal lift type
55
Obviously, we have t ∈ [0, ∞) for all y ∈ T M .
e the natural diagonal Riemannian metric on T M , defined by:
Let us denote by G
(2.2)
e H , Y H ) = c1 (t)gτ (y) (X, Y ) + d1 (t)gτ (y) (X, y)gτ (y) (Y, y),
G(X
y
y
e V , Y V ) = c2 (t)gτ (y) (X, Y ) + d2 (t)gτ (y) (X, y)gτ (y) (Y, y),
G(X
y
y
e yV , YyH ) = G(X
e yH , XyV ) = 0,
G(X
∀X, Y ∈ T01 (T M ), ∀y ∈ T M, where c1 , c2 , d1 , d2 are smooth functions of the energy
e to be a Riemannian metric on T M (i.e. to be
density on T M . The conditions for G
positive definite) are c1 > 0, c2 > 0, c1 + 2td1 > 0, c2 + 2td2 > 0 for every t ≥ 0.
The symmetric matrix of type 2n × 2n
! µ
Ã
¶
e (1)
G
0
c1 (t)gij + d1 (t)g0i g0j
0
ij
=
,
e (2)
0
c2 (t)gij + d2 (t)g0i g0j
0
G
ij
e in the adapted frame {δj , ∂i }
associated to the metric G
i,j=1,n , has the inverse
! µ
Ã
¶
e kl
H
0
p1 (t)g kl + q1 (t)y k y l
0
(1)
=
,
e kl
0
p2 (t)g kl + q2 (t)y k y l
0
H
(2)
where g kl are the entries of the inverse matrix of (gij )i,j=1,n , and p1 , q1 , p2 , q2 ,
are some real smooth functions of the energy density. More precisely, they may be
expressed as rational functions of c1 , d1 , c2 , d2 :
(2.3)
p1 =
1
1
d1
d2
, p2 = , q 1 = −
, q2 = −
.
c1
c2
c1 (c1 + 2td1 )
c2 (c2 + 2td2 )
e associated to the Riemannian metric
Proposition 2.1. The Levi-Civita connection ∇
e from the tangent bundle T M has the form
G
V
V
V
H
V
V
V
˙
e
e
∇X V Y = Q(X , Y ), ∇X H Y = (∇X Y ) + P (Y , X )
, ∀X, Y ∈ T01 (M ),
e
˙ X Y )H + S(X H , Y H ),
e X H Y H = (∇
∇X V Y H = P (X V , Y H ), ∇
where the M -tensor fields Q, P, S, have the following components with respect to the
adapted frame {∂i , δj }i,j=1,n :
Qhij
(2.4)
Pijh
h
Sij
e (2) + ∂j G
e (2) − ∂k G
e (2) )H
e kh ,
= 21 (∂i G
ij
jk
ik
(2)
(1)
(2)
e + Rl G
e
e kh
= 21 (∂i G
0jk li )H(1) ,
jk
e (2) + Rl G
e (2) e kh
= − 21 (∂k G
0ij lk )H(2) ,
ij
h
Rkij
being the components of the curvature tensor field of the Levi Civita connection
h
h
˙
∇ from the base manifold (M, g) and R0ij
= y k Rkij
. The vector fields Q(X V , Y V )
and S(X H , Y H ) are vertically valued, while the vector field P (Y V , X H ) is horizontally
valued.
56
S. L. Druţă and V. Oproiu
Using the relations (2.4), we may easily prove that the M -tensor fields Q, P, S,
have invariant expressions of the forms
Q(X V , Y V )
=
c′2
V
2c2 [g(y, X)Y
+ g(y, Y )X V ] −
c′2 −2d2
2(c2 +2td2 ) g(X, Y
)y V
c d′ −2c′ d
2 2
g(y, X)g(y, Y )y V ,
+ 2c22 (c2 2 +2td
2)
P (X V , Y H ) =
+
c′1
H
2c1 g(y, X)Y
+
d1
2c1 g(y, Y
c1 d′1 −c′1 d1 −d21
2c1 (c1 +2td1 ) g(y, X)g(y, Y
)X H +
)y H −
d1
2(c1 +2td1 ) g(X, Y
)y H
c2
H
2c1 (R(X, y)Y )
d1
− 2c1 (cc12+2td
g(X, R(Y, y)y)y H ,
1)
S(X H , Y H )
d1
[g(y, X) Y V + g(y, Y ) X V ] −
= − 2c
2
c′1
2(c2 +2td2 ) g(X, Y
) yV
c d′ −2d d
1 2
g(y, X)g(y, Y )y V − 21 (R(X, Y )y)V ,
− 2c22 (c1 2 +2td
2)
for every vector fields X, Y ∈ T01 (M ) and every tangent vector y ∈ T M .
Since in the following sections we shall work on the subset Tr M of T M consisting
of spheres of constant radius r, we shall consider only the tangent vectors y for which
2
the energy density t is equal to r2 , and the coefficients from the definition (2.2) of the
e become constant. So we may consider them constant from the beginning.
metric G
Then the M -tensor fields involved in the expression of the Levi-Civita connection
become simpler:
Q(X V , Y V )
=
d2
g(X, Y
c2 +r 2 d2
)y V ,
P (X V , Y H )
=
d1
g(Y, y)X H
2c1
+
d1
[c g(X, Y
2c1 (c1 +r 2 d1 ) 1
c2
− 2c
(R(X, y)Y )H −
1
(2.5)
S(X H , Y H )
) − d1 g(X, y)g(Y, y)]y H
c2 d1
g(X, R(Y, y)y)y H ,
2c1 (c1 +r 2 d1 )
d1
[g(X, y)Y V + g(Y, y)X V ]
= − 2c
2
d2
g(y, X)g(y, Y )y V − 21 (R(X, Y )y)V .
+ c2 (cd21+2td
2)
3
Tangent sphere bundles of constant radius r
Let Tr M = {y ∈ T M : gτ (y) (y, y) = r2 }, with r ∈ (0, ∞), and the projection
τ : Tr M → M , τ = τ ◦ i, where i is the inclusion map.
The horizontal lift of any vector field on M is tangent to Tr M , but the vertical
lift is not always tangent to Tr M . The tangential lift of a vector X to (p, y) ∈ Tr M ,
used in some recent papers like [4], [11], [12], [18], [25], is defined by
XyT = XyV −
1
gτ (y) (X, y)yyV .
r2
The tangent bundle to Tr M is spanned by δi and ∂jT = ∂j − r12 g0j y k ∂k , i, j, k = 1, n.
Remark that the vector fields {∂jT }j=1,n are not independent; there is the relation
y j ∂jT = 0, which can be directly checked. In any point of Tr M the vectors ∂iT ; i =
1, . . . n span an (n − 1)-dimensional subspace of T Tr M . The vector field
N = y i ∂i
57
Tangent sphere bundles of natural diagonal lift type
is normal to Tr M in T M .
e defined on T M . If
Denote by G the metric on Tr M induced from the metric G
we consider only the tangent vectors y ∈ Tr M , the energy density t defined by them
2
e become
is equal to r2 , and the coefficients from the definition (2.2) of the metric G
constants. Thus, the metric G from Tr M has the form
G(XyH , YyH ) = c1 gτ (y) (X, Y ) + d1 gτ (y) (X, y)gτ (y) (Y, y),
G(XyT , YyT ) = c2 [gτ (y) (X, Y ) − r12 gτ (y) (X, y)gτ (y) (Y, y)],
(3.1)
G(XyH , YyT ) = G(YyT , XyH ) = 0,
for every vector fields X, Y on M and every tangent vector y, where c1 , d1 , c2 are
constants. The conditions for G to be positive are c1 > 0, c2 > 0, c1 + r2 d1 > 0.
Proposition 3.1. The Levi-Civita connection ∇, associated to the Riemannian metric G on the tangent sphere bundle Tr M of constant radius r has the expression
h
∇∂ T ∂jT = Ahij ∂hT , ∇δi ∂jT = Γhij ∂hT + Bji
δh ,
i
∇ T δ = B h δ , ∇ δ = Γh δ + C h ∂ T ,
δi j
∂i j
ij h
ij h
ij h
where the M -tensor fields involved as coefficients have the expressions:
Ahij =
h
=
(3.2) Bij
− r12 g0j δih ,
Pijh −
d1
h
h
Cij
= − 2c
(δjh g0i − δih g0j ) − 21 R0ij
,
2
1
h
r 2 gi0 P0j
c2
h
− 2c
Rjik
yk −
1
=
d1 h
2c1 δi g0j
+
d1
2(c1 +r 2 d1 ) (gij
−
2c1 +r 2 d1
g0i g0j )y h
r 2 c1
c2 d1
h k l
2c1 (c1 +r 2 d1 ) Rikjl y y y ,
h
where g0j = gj0 = y i gij and P0j
= y i Pijh .
We mention that A and C have these quite simple expressions, since they are
the coefficients of the tangential part of the Levi-Civita connection. All the terms
containing y h ∂hT = 0 have been canceled.
The invariant form of the Levi-Civita connection from Tr M is
∇X T Y T = − r12 g(Y, y)X T ,
d1
d1
H
g(Y, y) X H + 2(c1 +d
∇X T Y H = 2c
2 g(X, Y )y
1
1r )
2
d1 (2c1 +r d1 )
c2
c2 d1
H
H
H
− 2r2 c1 (c1 +r2 d1 ) g(X, y)g(Y, y)y − 2c1 (R(X, y)Y ) − 2c1 (c1 +r2 d1 ) g(X, R(Y, y)y)y ,
d1
H
˙ X Y )T + d1 g(X, y) Y H +
∇X H Y T = ( ∇
2c1
2(c1 +r 2 d1 ) g(X, Y )y
2
c2
c2 d1
1 +r d1 )
H
H
H
− 2rd12 (2c
c1 (c1 +r 2 d1 ) g(X, y)g(Y, y)y − 2c1 (R(Y, y)X) − 2c1 (c1 +r 2 d1 ) g(Y, R(X, y)y)y ,
˙ X Y )H − d1 [g(X, y) Y T − g(Y, y) X T ] − 1 (R(X, Y )y)T .
∇X H Y H = ( ∇
2c2
2
˙ k Qh , ∇
˙ kP h , ∇
˙ k S h of the M -tensor fields Q, P, S are
The covariant derivatives ∇
ij
ij
ij
defined by some formulas similar to those from the classical cases. The covariant
derivative of P is given by
˙ k Pijh = δk Pijh + Γhkl Pijl − Γlki Pljh − Γlkj Pilh .
∇
58
S. L. Druţă and V. Oproiu
Remark that instead of the partial derivative ∂x∂ k we use the operator δk = δxδ k =
∂
˙ k S h . It can be
˙ k Qh and for ∇
− Γhk0 ∂y∂ k . Similar expressions are obtained for ∇
ij
ij
∂xk
˙ k S h are M -tensors
˙ kP h , ∇
˙ k Qh , ∇
proved by a straightforward computation that ∇
ij
ij
ij
of type (1, 3). The expression of the covariant derivative of an arbitrary M -tensor can
˙ k y i = 0, ∇
˙ k gij = 0. In the case where Q, P, S
be obtained easily. It follows that ∇
are given by (2.5), their covariant derivatives are:
h l
˙ k Qh = 0, ∇
˙ kSh = − 1 ∇
˙
∇
ij
ij
2 k Rlij y
h l
˙ k P h = − c2 ∇
˙
∇
ij
2c1 k Rjil y −
c2 d1
l r h
˙
2c1 (c1 +r 2 d1 ) ∇k Riljr y y y ,
˙ k Rh , ∇
˙ k Rhkij are the usual local coordinate expressions of the covariant
where ∇
kij
derivatives of the curvature tensor field and the Riemann-Christoffel tensor field of
˙ on the base manifold M . If the base manifold (M, g) is
the Levi Civita connection ∇
˙ k Qh = 0, ∇
˙ k P h = 0, ∇
˙ k S h = 0.
locally symmetric then, obviously, ∇
ij
ij
ij
In a similar way we obtain
h l
˙
˙ k Ah = 0, ∇
˙ kCh = − 1 ∇
∇
ij
ij
2 k Rlij y ,
h l
˙ k B h = − c2 ∇
˙
∇
ij
2c1 k Rjil y −
c2 d1
l r h
˙
2c1 (c1 +r 2 d1 ) ∇k Riljr y y y ,
The curvature tensor field K of the connection ∇ from Tr M is defined by the formula
K(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,
X, Y, Z ∈ T01 (Tr M ).
We obtain by a standard straightforward computation the horizontal and tangential
components of the curvature tensor field K
h
h
K(δi , δj )δk = HHHHkij
δh + HHHTkij
∂hT ,
h
h
K(δi , δj )∂kT = HHT Hkij
δh + HHT Tkij
∂hT ,
h
h
K(∂iT , ∂jT )δk = T T HHkij
δh , K(∂iT , ∂jT )∂kT = T T T Tkij
∂hT ,
h
h
h
K(∂iT , δj )δk = T HHHkij
δh + T HHTkij
∂hT , K(∂iT , δj )∂kT = T HT Hkij
δh ,
where the non-zero M -tensor fields which appear as coefficients are given by
h
h
l
h l
h l
HHHHkij
= Rkij
+ Blih Cjk
− Blj
Cik + Blk
R0ij ,
h
˙ j Rh y l − ∇
˙ i Rh y l ),
HHHTkij
= 12 (∇
lik
ljk
h
HHT Hkij
=
c2 ˙
h
2c1 (∇j Rikl
˙ i Rh )y l +
−∇
jkl
c2 d1
˙
2c1 (c1 +r 2 d1 ) (∇j Rilkm
˙ i Rjlkm )y l y m y h ,
−∇
h
h
l
h l
l
)∂hT ,
HHT Tkij
∂hT = (Rkij
+ Cilh Bkj
− Cjl
Bki + Ahlk R0ij
h
h
h
l
h l
T T HHkij
= ∂iT Bjk
− ∂jT Bik
+ Bilh Bjk
− Bjl
Bik −
h
T T T Tkij
=
1
h
r 2 (gjk δi
h
T HHHkij
=
− gik δjh ) +
c2 ˙
h
l
2c1 ∇j Rkil y
+
1
h
r 4 (δj g0i g0k
1
h
r 2 (gi0 Bjk
h
− gj0 Bik
),
− δih g0j g0k ),
c2 d1
l r h
˙
2c1 (c1 +r 2 d1 ) ∇j Rilkr y y y ,
h
l
h l
h
˙ j Rh y l )∂ T ,
+ Ahil Cjk
− Cjl
Bik − ∇
T HHTkij
∂hT = (∂iT Cjk
lik
h
h
h
h
h
l
˙ j Ah = 0.
, T HT Tkij
= −∇
T HT Hkij
= ∂iT Bkj
+ Bkj
Bilh − Alik Blj
ik
Tangent sphere bundles of natural diagonal lift type
59
h
h
We mention that from the final values of HHT Tkij
∂hT and T HHTkij
∂hT , obtained by
h
h
replacing the expressions (3.2) of Ahij , Bij
, Cij
, we shall eliminate the terms containing
h T
y ∂h , because they vanish.
We want to find the manifolds (Tr M, G) of constant sectional curvature k, i.e the
manifolds for which the curvature tensor field K satisfies the relation
K(X, Y )Z − k[G(Y , Z)X − G(X, Z)Y ] = 0, ∀X, Y , Z ∈ T01 (Tr M ).
(3.3)
Let us consider X, Y , Z ∈ T01 (Tr M ) as being the tangential lifts of X, Y, Z ∈
We have to study the vanishing conditions for the difference
T01 (M ).
K(X T , Y T )Z T − k[G(Y T , Z T )X T − G(X T , Z T )Y T ] = 0, ∀X, Y, Z ∈ T01 (M ),
which has the detailed expression
(3.4)
− g(X, Z)Y T ] + r14 g(y, Z)[g(y, X)Y T − g(y, Y )X T ]
−k[G(Y T , Z T )X T − G(X T , Z T )Y T ] = 0,
1
T
r 2 [g(Y, Z)X
for all the vector fields X, Y, Z and for every vector y tangent to M .
The local coordinate form of (3.4) is
¤
kr2 c2 − 1 £
1
gjk δih − gik δjh − 2 (δjh g0i g0k − δih g0j g0k ) = 0,
2
r
r
from which we obtain a first necessary condition for the manifold (Tr M, G) to have
constant sectional curvature k:
(3.5)
c2 =
1
.
kr2
If in the relation (3.3) we take instead of X, Y , Z ∈ T01 (Tr M ) the horizontal lifts of
X, Y, Z ∈ T01 (M ), we obtain that the following vanishing condition must be satisfied:
(3.6)
K(X H , Y H )Z H − k[G(Y H , Z H )X H − G(X H , Z H )Y H ] = 0,
for every tensor fields X, Y, Z ∈ T01 (M ).
Since
d2
K(X H , Y H )Z H = (R(X, Y )Z)H − 4c11c2 [g(y, X)Y H − g(y, Y )X H ]g(y, Z)
2
d1
+ 4c2 (cd1
[g(Y, Z)g(y, X) − g(X, Z)g(y, Y )]y H + 4c
[(R(y, Y )Z)H g(y, X)
2
1 +r d1 )
1
3d1
H
H
−(R(y, X)Z) g(y, Y ) + (R(X, Y )y) g(y, Z)] + 4(c1 +r2 d1 ) g(X, R(Z, y)Y )y H
d2
H
1
− 4c1 (c1 +r
2 d ) [g(Y, R(Z, y)y)g(X, y) − g(X, R(Z, y)y)g(Y, y)]y
1
c2
c2
[R(R(Y, Z)y, y)X − R(R(X, Z)y, y)Y ]H − 2c
[R(R(X, Y )y, y)Z]H
+ 4c
1
1
c2 d1
+ 4c1 (c1 +r2 d1 ) [g(X, R(R(Y, Z)y, y)y) − g(X, R(R(Y, y)y, y)Z)
˙ Z R(X, Y )y)T ,
−2g(X, R(R(Z, y)y, y)Y )]y H + 21 (∇
the coefficient of the horizontal part of the difference (3.6) has the following local
60
S. L. Druţă and V. Oproiu
coordinate expression:
h
HHHHkij
− k[(c1 gjk + d1 g0j g0k )δih − (c1 gik + d1 g0i g0k )δjh ] =
h
= Rkij
+ c1 k(gik δjh − gjk δih ) +
d2
(3.7)
d1 (4c1 c2 k−d1 ) h
(δj g0i g0k
4c1 c2
h
1
+ 4c2 (c1 +r
+
2 d ) (gjk g0i − gik g0j )y
1
h
1
+ 4(c13d
+r 2 d1 ) Rijk0 y −
d1
h
4c1 (Rk0j g0i
d21
4c1 (c1 +r 2 d1 ) (Rj0k0 g0i
− δih g0j g0k )
h
h
− Rk0i
g0j + R0ij
g0k )
− Ri0k0 g0j )y h
c2
l
h
l
h
l
h
+ 4c
(R0jk
Ril0
− R0ik
Rjl0
− 2R0ij
Rkl0
)
1
d1
l
l
l
h
+ 4c1 (cc12+r
2 d ) (R0i0 Rjkl0 − R0ik Rj0l0 − 2R0ij Rk0l0 )y .
1
h
For the indices h, i, j, k fixed, the condition for HHHHkij
− k[(c1 gjk + d1 g0j g0k )δih −
h
(c1 gik + d1 g0i g0k )δj ] to be zero leads to an equation of the type A + Al1 l2 y l1 y l2 +
Al1 l2 l3 l4 y l1 y l2 y l3 y l4 = 0, in the tangential coordinates y i ; i = 1, n, where the coefficients A’s are obtained from the above expression by full symmetrization. Differentiating the obtained expression four times, it follows that Al1 l2 l3 l4 = 0. Then differentiating the remaining expression A + Al1 l2 y l1 y l2 two times, it follows Al1 l2 = 0. We
conclude that the constant term A is zero, whence the base manifold must be a space
form, with the curvature of the form:
h
Rkij
= c1 k(gjk δih − gik δjh ).
(3.8)
Replacing this expression and the value (3.5) of c2 into (3.7), we obtain the following
vanishing condition:
i
k(3c1 − r2 d1 ) h c1 + r2 d1 h
h
h
= 0,
(δ
g
−
δ
g
)g
−
(g
g
−
g
g
)y
0i
0j
0k
jk
0i
ik
0j
j
i
4r2
c1
which is satisfied if and only if c1 has the form
(3.9)
c1 =
r 2 d1
.
3
h
The same values (3.5) of c2 and (3.9) of c1 lead to the identity T T HHkij
= 0, which
is another necessary condition for (Tr M, G) to be a space form. More precisely, after
imposing the expression (3.8) for the curvature of the base manifold, and the value
h
(3.5) of c2 into the expression of T T HHkij
, this becomes
h
T T HHkij
=
3c1 − r2 d1
[gjk δih − gik δjh + (δjh g0i − δih g0j )g0k − (gjk g0i − gik g0j )y h ],
4c1 r2
which vanishes if and only if c1 takes the value (3.9).
Due to the condition for the base manifold to be a space form, the components
h
h
h
HHHTkij
, HHT Hkij
, T HHHkij
of the curvature tensor field of (Tr , G) and the corh
responding components of the difference from (3.3) become zero. Since HHT Tkij
is
also vanishing when we substitute the expression of the curvature (3.8) and the values
(3.9) for c1 and (3.5) for c2 , we have immediately that
K(X H , Y H )Z T = k[G(Y H , Z T )X H − G(X H , Z T )Y H ], ∀X, Y, Z ∈ T01 (M ).
Tangent sphere bundles of natural diagonal lift type
61
The identity
K(X T , Y H )Z H = k[G(Y H , Z H )X T − G(X T , Z H )Y H ], ∀X, Y, Z ∈ T01 (M ),
reduces, after imposing the conditions (3.8), (3.9), (3.5), to the relation
2
T
T
+ g(X, Y )Z T
− d1 kr
3 [g(Y, Z)X + g(X, Z)Y
−g(y, X)g(y, Y )Z T − g(y, X)g(y, Z)Y T − g(y, Y )g(y, Z)X T ] = 0,
which is true for every vector fields X, Y, Z ∈ T01 (M ), and for every tangent vector
y ∈ T M , if and only if d1 = 0 or k = 0 (i.e the manifold (Tr , G) is flat).
Finally, we have that the manifold (Tr M, G) may never be a space form, since the
relation
K(X T , Y H )Z T = k[G(Y H , Z T )X T − G(X T , Z T )Y H ], ∀X, Y, Z ∈ T01 (M ),
which has the final form
1
T
r 2 [g(Y, Z)X
(3.10)
+ g(X, Z)Y T + g(X, Y )Z T ]
− r14 {g(y, X)g(y, Y )Z T + g(y, X)g(y, Z)Y T + g(y, Y )g(y, Z)X T
+[g(Y, Z)g(y, X) + g(X, Z)g(y, Y ) + g(X, Y )g(y, Z)
− r32 g(y, X)g(y, Y )g(y, Z)]y h } = 0,
may never be satisfied. Hence we may state
Theorem 3.2. The tangent sphere bundle Tr M , with the Riemannian metric G
e of diagonal lift type on the tangent bundle T M , has never
induced from the metric G
constant sectional curvature.
Corollary 3.3. The tangent sphere bundle Tr M , endowed with the metric induced
by the Sasaki metric g s from the tangent bundle T M is never a space form.
In fact, the Sasaki metric can be obtained as a particular case of our natural
diagonal lifted metric, with the coefficients c1 = c2 = 1, d1 = 0.
4
Einstein tangent sphere bundles
We shall find the conditions under which the manifold (Tr M, G), with G given by
the relations (3.1), is Einstein. To this aim, we shall compute the Ricci tensor of the
manifold (Tr M, G).
First, let us remark some facts concerning the obtaining of the Ricci tensor field
for the tangent bundle T M . We have the well known formula
Ric(Y, Z) = trace(X → K(X, Y )Z),
where X, Y, Z are vector fields on T M . Then we get easily the components of the
Ricci tensor field on T M
h
h
g
g
RicHH
jk = Ric(δj , δk ) = HHHHkhj + V HHVkhj ,
g Vjk = Ric(∂
g j , ∂k ) = V V V V h − V HV H h ,
RicV
khj
kjh
62
S. L. Druţă and V. Oproiu
h
h
h
where the components V V V Vkij
, V HV Hkij
, V HHVkij
, are obtained from the curh
h
vature tensor field on T M in a similar way as the components T T T Tkij
, T HT Hkij
,
h
T HHTkij are obtained from the curvature tensor field on Tr M . In the expression
g
g
g
g
of RicHV
jk = Ric(δj , ∂k ) = RicV Hjk = Ric(∂j , δk ) there are involved the covariant derivatives of the curvature tensor field R. If (M, g) is locally symmetric then
RicHVjk = RicV Hjk = 0. In particular we have RicHVjk = RicV Hjk = 0 in the
case where (M, g) has constant sectional curvature.
Equivalently, we may use an orthonormal frame (E1 , . . . , E2n ) on T M and we may
use the formula
2n
X
g
Ric(Y, Z) =
G(Ei , K(Ei , Y )Z).
i=1
We may choose the orthonormal frame (E1 , . . . , E2n ) such that the first n vectors
E1 , . . . , En are the vectors of a (orthonormal) frame in HT M and the last n vectors
En+1 , . . . E2n are the vectors of a (orthonormal) frame in V T M . Moreover, we may
assume that the last vector E2n is the unitary vector of the normal vector N = y i ∂i
to Tr M .
The components of the Ricci tensor field of Tr M can be obtained in a similar
way by using the above traces. However the vector fields ∂1T , . . . , ∂nT are not independent. On the open set from Tr M , where y n 6= 0 we can consider the basis
T
for T Tr M . The last vector ∂nT is expressed as
δ1 , ..., δn , ∂1T , ..., ∂n−1
∂nT = −
n−1
1 X i T
y ∂i .
y n i=1
T
Remark that the basis δ1 , ..., δn , ∂1T , ..., ∂n−1
can be completed with the normal vector
V
h
N = y = y ∂h .
h
h
For the components HHHHkij
and HT T Hkij
the traces can pe computed easily,
just like in the case of T M . The components for which we must compute more
h
h
carefully the traces are T T T Tkij
, and T HHTkij
. We have:
h
T HHTkij
∂hT
=
=
Pn−1
l=1
Pn−1
l=1
l
n
T HHTkij
∂lT + T HHTkij
∂nT
l
n 1
T HHTkij
∂lT − T HHTkij
yn
h
n 1 h T
= T HHTkij
∂hT − T HHTkij
y n y ∂h .
Pn−1
l=1
y l ∂lT
Thus the trace involved in the definition of RicHH on Tr M is
h
T HHTkhj
−
1 h
n
y T HHTkhj
.
yn
h
A short computation made by using the above expression of of T HHTkij
gives
h
y i T HHTkij
=
thus we get
d1 (4c1 + r2 d1 )
g0j g0k y h ,
4c1 c2 r2
1 i
d1 (4c1 + r2 d1 )
n
y
T
HHT
=
g0j g0k .
kij
yn
4c1 c2 r2
63
Tangent sphere bundles of natural diagonal lift type
It follows that
h
h
−
+ T HHTkhj
RicHHjk = HHHHkhj
d1 (4c1 + r2 d1 )
g0j g0k .
4c1 c2 r2
In a similar way we get that the trace involved in the definition of RicT T on Tr M is
h
T T T Tkhj
−
Then
n
y h T T T Tkhj
=
1 h
n
y T T T Tkhj
.
yn
1
1
gjk y n − 4 g0j g0k y n .
2
r
r
Hence
h
h
RicT Tjk = Ric(∂jT , ∂kT ) = HT T Hkhj
+ T T T Tkhj
−
1
1
gjk + 4 g0j g0k .
2
r
r
After the computations we obtain the detailed expressions of RicHHjk and RicT Tjk :
RicHHjk =
Ricjk −
d1 (2c1 +r 2 d1 )
2c2 (c1 +r 2 d1 ) gjk
c2
− 2c
Rh0kl Rl jh0 +
1
RicT Tjk =
+
d1 (2c1 +r 2 d1 )[c1 n+r 2 d1 (n−1)]
g0j g0k
2r 2 c1 c2 (c1 +r 2 d1 )
c2 d1
h
2c1 (c1 +r 2 d1 ) R 0k0 Rh0j0 ,
r 4 d21 −2c1 (c1 +r 2 d1 )(n−2) 1
( r2 g0j g0k
2c1 r 2 (c1 +d1 r 2 )
− gjk ) +
c22
R
Rhij0
4c21 hik0
c2 d
− 2c2 (c12+r1 2 d1 ) Rh0j0 Rh0k0 .
1
The components Rh0kl , Rl jh0 , Rh0k0 , Rh0j0 , etc. are obtained as usual from the com˙ by transvecting with y’s. E.g. Rh =
ponents of the curvature tensor field R of ∇
0kl
h
i
l
l
i
h
h
R ikl y , R jh0 = R jhi y , R 0k0 = R ikj y i y j , Rh0j0 = Rhijk y i y k . Recall that
˙
Rhijk = ghl Rl ijk are the components of the Riemann-Christoffel tensor field of ∇.
Taking the above relations into account, we obtain that the differences between the
components of the Ricci tensor and the corresponding components of the metric G
multiplied by a constant ρ have the following forms:
(1)
RicHHjk − ρGjk = Ricjk −
+ d1 (2c1 +r
2
d1 (2c1 +r 2 d1 )+2ρc1 c2 (c1 +r 2 d1 )
gjk
2c2 (c1 +r 2 d1 )
d1 )[c1 n+r 2 d1 (n−1)]−2ρr 2 c1 c2 d1 (c1 +r 2 d1 )
g0j g0k
2c1 c2 r 2 (c1 +r 2 d1 )
c2
l
h
l
h
(Rkh0
R0jl
+ Rjh0
R0kl
)+
+ 4c
1
(2)
RicT Tjk − ρGjk =
+
c22
R
Rhi
4c21 hik0 j0
c2 d1
h
2c1 (c1 +r 2 d1 ) Rh0j0 R0k0 ,
−
c22 d1
R
Rh
2c21 (c1 +r 2 d1 ) h0j0 0k0
r 4 d21 +2c1 (c1 +r 2 d1 )(n−2)+ρ2c1 c2 r 2 ) 1
( r2 g0j g0k
2r 2 c1 (c1 +r 2 d1 )
− gjk ).
When the base manifold (M, g) has constant sectional curvature c, i.e. when
(4.1)
h
Rkij
= c(gjk δih − gik δjh ),
64
S. L. Druţă and V. Oproiu
the above differences take the forms
(1)
RicHHjk − ρGjk =
+
cc2 [2(n−1)(c1 +r 2 d1 )−cc2 r 2 ]−d1 (2c1 +r 2 d1 )−2ρc1 c2 (c1 +r 2 d1 )
gjk
2c2 (c1 +d1 r 2 )
d1 [2c21 n+r 4 (d21 −c2 c22 )(n−1)]+r 2 c1 [d21 (3n−2)−c2 c22 (n−2)]−2ρr 2 c1 c2 d1 (c1 +r 2 d1 )
g0j g0k ,
2r 2 c1 c2 (c1 +d1 r 2 )
(2)
RicT Tjk − ρGjk =
r 4 (d21 −c2 c22 )+2c1 (c1 +r 2 d1 )(2−n+ρc2 r 2 ) ¡ 1
2r 2 c1 (c1 +r 2 d1 )
r 2 g0j g0k
− gjk
(2)
¢
The difference RicT Tjk − ρGjk vanishes if and only if the constant ρ has the value
ρ=
2c21 (n−2)+r 2 [2c1 d1 (n−2)+r 2 (c2 c22 −d21 )]
.
2r 2 c1 c2 (c1 +r 2 d1 )
(1)
After replacing this expression, the difference RicHHjk − ρGjk becomes
(1)
(4.2)
RicHHjk − ρGjk =
2(cc2 r 2 −c1 )
{c1 (n
r2
− 2) − r2 [cc2 − d1 (n − 1)]}gjk
+{4c21 d1 + r2 c1 [d21 (n + 2) − c2 c22 (n − 2)] + r4 d1 (d21 − c2 c22 )n}g0j g0k .
By doing a detailed analysis of all the situations in which the difference expressed by
(4.2) vanishes, we may prove the following theorem.
Theorem 4.1. The tangent sphere bundle Tr M of an n-dimensional Riemannian
(M, g) of constant sectional curvature c is Einstein with respect to the metric G ine defined on T M , i.e. it exists a real
duced from the natural diagonal lifted metric G
constant ρ such that
Ric(X, Y ) = ρG(X, Y ), ∀X, Y ∈ T01 (Tr M ),
if and only if
c1 =
r 2 d1 n
d1 n
c(n − 1)2 (n − 2)
, c2 =
, ρ=
.
n−2
c(n − 2)
r 2 d1 n 2
Proof. The difference expressed by (4.2) vanishes if and only if the both coefficients
are equal to zero. The vanishing condition for the coefficient of gjk leads to two cases
which must be studied:
I)
c1 = cc2 r2 , II) c1 =
r2 [cc2 − d1 (n − 1)]
.
n−2
In the case I, the coefficient of g0j g0k from (4.2) becomes
−(cc2 + d1 )2 [cc2 (n − 2) − d1 n]r4 = 0,
so we have two possible values for c2 :
(4.3)
c2 = −
d1 n
d1
, or c2 =
.
c
c(n − 2)
Replacing the first one into the expression of c1 , we obtain that c1 + r2 d1 = 0, thus
the constant ρ, some components of K as well as the Levi Civita connection are not
defined. So this situation should be excluded.
65
Tangent sphere bundles of natural diagonal lift type
When c2 takes the second value from (4.3), c1 and ρ have the forms presented in
the case I from the theorem.
If c1 has the expression from case II the coefficient of g0j g0k given in (4.2) decomposes as
(4.4)
−
r4 (cc2 − d1 ){c2 c22 (n − 2)2 + d1 [d1 n2 + 2cc2 (n2 − 4n + 2)]}
= 0.
(n − 2)2
The vanishing condition for the first factor and the value of c1 from case II lead us
again to the situation where c1 + r2 d1 = 0, so it should be excluded.
The second factor which appears in (4.4) is a second degree function of c2 . The
discriminant of the attached equation is
∆ = 16c2 d21 (1 − n)(n2 − 3n + 1).
Since the dimension n of the base manifold is a natural number bigger then one, ∆ is
positive only for n = 2, value which makes vanish the denominator of the expression
(4.4), so the case must be treated separately, starting with the expressions of the
(1)
(2)
differences RicHHjk − ρGjk and RicT Tjk − ρGjk , which become of the forms
(1)
RicHHjk − ρGjk =
cc2 [2(c1 +r 2 d1 )−cc2 r 2 ]−d1 (2c1 +r 2 d1 )−2ρc1 c2 (c1 +r 2 d1 )
gjk
2c2 (c1 +d1 r 2 )
+
(2)
RicT Tjk − ρGjk =
d1 [4c21 +r 4 (d21 −c2 c22 )]+4r 2 c1 d21 −2ρr 2 c1 c2 d1 (c1 +r 2 d1 )
g0j g0k ,
2r 2 c1 c2 (c1 +d1 r 2 )
r4 (d21 − c2 c22 ) + 2ρr2 c1 c2 (c1 + r2 d1 )
2r2 c1 (c1 + r2 d1 )
µ
¶
1
g
g
−
g
0j 0k
jk .
r2
From these relations we get
ρ=
r2 (c2 c22 − d21 )
,
2c1 c2 (c1 + r2 d1 )
and then
(1)
RicHHjk − ρGjk = 2(cc2 r2 − c1 )(d1 − cc2 )gjk
+2d1 [2c1 (c1 + r2 d1 ) + r4 (d21 − c2 c22 )]g0j g0k .
This last difference vanishes if and only if c1 = r2 cc2 and d1 = −cc2 , or d1 = cc2 and
c1 = −r2 cc2 , but the both cases must be excluded, since they lead to c1 + r2 d1 = 0.
Thus the theorem is proved.
¤
Corollary 4.2. The tangent sphere bundle Tr M of an n-dimensional Riemannian
space form (M, g) may never be Einstein with respect to the metric induced by the
Sasaki metric g s from T M .
In fact, in this case, the coefficients should be c1 = c2 = 1, d1 = 0, and they do
not satisfy the condition from theorem 4.1.
66
S. L. Druţă and V. Oproiu
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Authors’ addresses:
Simona-Luiza Druţă
Faculty of Mathematics, ”AL. I. Cuza” University of Iaşi,
Bd. Carol I, No. 11, RO-700506 Iaşi, Romania.
E-mail: [email protected]
Vasile Oproiu
Faculty of Mathematics, ”AL. I. Cuza” University of Iaşi,
Bd. Carol I, No. 11, RO-700506 Iaşi, Romania.
Institute of Mathematics ”O. Mayer”,
Romanian Academy, Iaşi Branch.
E-mail: [email protected]
S. L. Druţă and V. Oproiu
Dedicated to the 70-th anniversary
of Professor Constantin Udriste
Abstract. We show that the tangent sphere bundles endowed with a
Riemannian metric induced from the natural diagonal lifted metric of the
tangent bundle are never space forms. Next we study the conditions under
which these tangent sphere bundles are Einstein.
M.S.C. 2000: 53C05, 53C15, 53C55.
Key words: natural lift; tangent sphere bundle; sectional curvature; Einstein manifold.
1
Introduction
The tangent sphere bundle Tr M consisting of spheres of constant radius r centered
in the null tangent vectors are hyper-surfaces of the tangent bundle T M , obtained by
considering only the tangent vectors having the norm equal to r.
In the most part of the papers, like [3]-[6], [17], [25], [27], [28], the metric considered
on the tangent bundle T M was the Sasaki metric (see [26]), but E. Boeckx noticed
that the unit tangent bundle equipped with the induced Cheeger-Gromoll metric is
isometric to the tangent sphere bundle T √1 M , of radius √12 endowed with the metric
2
induced by the Sasaki metric. This suggested to O. Kowalski and M. Sekizawa the
idea that the tangent sphere bundles with different constant radii and with the metrics
induced from the Sasaki metric might possess different geometrical properties, and in
the paper [11] from 2000 they showed how the geometry of the tangent sphere bundles
depends on the radius.
Some other metrics on the tangent sphere bundles may be constructed by using a
few lifts to the tangent bundle, which have been considered in [21]-[24], [29]-[31], and
studied in some very recent papers, such as [2], [7] and [14].
In the last years some interesting results were obtained by endowing the tangent
sphere bundles with Riemannian metrics induced by the natural lifted metrics from
T M , which are not Sasakian (see [1], [16], [18] – [20]).
Roughly speaking, a natural operator (in the sense of [8] – [10], [13]) is a fibred manifold mapping, which is invariant with respect to the group of local diffeomorphisms
of the base manifold.
∗
Balkan
Journal of Geometry and Its Applications, Vol.15, No.1, 2010, pp. 53-67.
c Balkan Society of Geometers, Geometry Balkan Press 2010.
°
54
S. L. Druţă and V. Oproiu
In the present paper we consider on the tangent bundle T M of a Riemannian
e obtained by the second author in [24] as the diagonal
manifold M a natural metric G,
lift of the Riemannian metric g from the base manifold. We prove that the tangent
e may
sphere bundle Tr M endowed with the Riemannian metric G induced from G
never have constant sectional curvature, then we find the conditions under which
(Tr M, G) is an Einstein manifold.
The manifolds, tensor fields and other geometric objects considered in this paper
are assumed to be differentiable of class C ∞ (i.e. smooth). The Einstein summation
convention is used throughout this paper, the range of the indices h, i, j, k, l, m, r,
being always {1, . . . , n}.
2
Preliminary results
Let (M, g) be a smooth n-dimensional Riemannian manifold and denote its tangent
bundle by τ : T M → M . The total space T M has a structure of a 2n-dimensional
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 (see [15]). The vertical lift to T M of a vector field X defined on
M will be denoted by X V .
˙ the Levi Civita connection of the Riemannian metric g on M . Then
Denote by ∇
we have the direct sum decomposition
(2.1)
T T M = V T M ⊕ HT M
of the tangent bundle to T M into the vertical distribution V T M = Ker τ∗ and
˙ (see [32]). The set of vector fields
the horizontal distribution HT M defined by ∇
{ ∂y∂ 1 , . . . , ∂y∂n } on τ −1 (U ) defines a local frame field for V T M and for HT M we have
∂
h ∂
the local frame field { δxδ 1 , . . . , δxδn }, where δxδ i = ∂x
Γh0i = y k Γhki , and
i − Γ0i ∂y h ,
Γhki (x) are the Christoffel symbols of g.
δ
∂
The set { ∂y
i , δxj }i,j=1,n , denoted also by {∂i , δj }i,j=1,n , defines a local frame on
T M , adapted to the direct sum decomposition (2.1). The horizontal lift to T M of a
∂ H
and
vector field X defined on M will be denoted by X H . Remark that δi = ( ∂x
i)
∂ V
∂i = ( ∂xi ) .
The second author considered in [24] a Riemannian metric, which is obtained as
a natural diagonal lift of the Riemannian metric g from the base manifold M to the
tangent bundle T M , the coefficients being functions of the energy density defined by
the tangent vector y as follows
t=
1
1
1
kyk2 = gτ (y) (y, y) = gik (x)y i y k ,
2
2
2
y ∈ τ −1 (U ).
Tangent sphere bundles of natural diagonal lift type
55
Obviously, we have t ∈ [0, ∞) for all y ∈ T M .
e the natural diagonal Riemannian metric on T M , defined by:
Let us denote by G
(2.2)
e H , Y H ) = c1 (t)gτ (y) (X, Y ) + d1 (t)gτ (y) (X, y)gτ (y) (Y, y),
G(X
y
y
e V , Y V ) = c2 (t)gτ (y) (X, Y ) + d2 (t)gτ (y) (X, y)gτ (y) (Y, y),
G(X
y
y
e yV , YyH ) = G(X
e yH , XyV ) = 0,
G(X
∀X, Y ∈ T01 (T M ), ∀y ∈ T M, where c1 , c2 , d1 , d2 are smooth functions of the energy
e to be a Riemannian metric on T M (i.e. to be
density on T M . The conditions for G
positive definite) are c1 > 0, c2 > 0, c1 + 2td1 > 0, c2 + 2td2 > 0 for every t ≥ 0.
The symmetric matrix of type 2n × 2n
! µ
Ã
¶
e (1)
G
0
c1 (t)gij + d1 (t)g0i g0j
0
ij
=
,
e (2)
0
c2 (t)gij + d2 (t)g0i g0j
0
G
ij
e in the adapted frame {δj , ∂i }
associated to the metric G
i,j=1,n , has the inverse
! µ
Ã
¶
e kl
H
0
p1 (t)g kl + q1 (t)y k y l
0
(1)
=
,
e kl
0
p2 (t)g kl + q2 (t)y k y l
0
H
(2)
where g kl are the entries of the inverse matrix of (gij )i,j=1,n , and p1 , q1 , p2 , q2 ,
are some real smooth functions of the energy density. More precisely, they may be
expressed as rational functions of c1 , d1 , c2 , d2 :
(2.3)
p1 =
1
1
d1
d2
, p2 = , q 1 = −
, q2 = −
.
c1
c2
c1 (c1 + 2td1 )
c2 (c2 + 2td2 )
e associated to the Riemannian metric
Proposition 2.1. The Levi-Civita connection ∇
e from the tangent bundle T M has the form
G
V
V
V
H
V
V
V
˙
e
e
∇X V Y = Q(X , Y ), ∇X H Y = (∇X Y ) + P (Y , X )
, ∀X, Y ∈ T01 (M ),
e
˙ X Y )H + S(X H , Y H ),
e X H Y H = (∇
∇X V Y H = P (X V , Y H ), ∇
where the M -tensor fields Q, P, S, have the following components with respect to the
adapted frame {∂i , δj }i,j=1,n :
Qhij
(2.4)
Pijh
h
Sij
e (2) + ∂j G
e (2) − ∂k G
e (2) )H
e kh ,
= 21 (∂i G
ij
jk
ik
(2)
(1)
(2)
e + Rl G
e
e kh
= 21 (∂i G
0jk li )H(1) ,
jk
e (2) + Rl G
e (2) e kh
= − 21 (∂k G
0ij lk )H(2) ,
ij
h
Rkij
being the components of the curvature tensor field of the Levi Civita connection
h
h
˙
∇ from the base manifold (M, g) and R0ij
= y k Rkij
. The vector fields Q(X V , Y V )
and S(X H , Y H ) are vertically valued, while the vector field P (Y V , X H ) is horizontally
valued.
56
S. L. Druţă and V. Oproiu
Using the relations (2.4), we may easily prove that the M -tensor fields Q, P, S,
have invariant expressions of the forms
Q(X V , Y V )
=
c′2
V
2c2 [g(y, X)Y
+ g(y, Y )X V ] −
c′2 −2d2
2(c2 +2td2 ) g(X, Y
)y V
c d′ −2c′ d
2 2
g(y, X)g(y, Y )y V ,
+ 2c22 (c2 2 +2td
2)
P (X V , Y H ) =
+
c′1
H
2c1 g(y, X)Y
+
d1
2c1 g(y, Y
c1 d′1 −c′1 d1 −d21
2c1 (c1 +2td1 ) g(y, X)g(y, Y
)X H +
)y H −
d1
2(c1 +2td1 ) g(X, Y
)y H
c2
H
2c1 (R(X, y)Y )
d1
− 2c1 (cc12+2td
g(X, R(Y, y)y)y H ,
1)
S(X H , Y H )
d1
[g(y, X) Y V + g(y, Y ) X V ] −
= − 2c
2
c′1
2(c2 +2td2 ) g(X, Y
) yV
c d′ −2d d
1 2
g(y, X)g(y, Y )y V − 21 (R(X, Y )y)V ,
− 2c22 (c1 2 +2td
2)
for every vector fields X, Y ∈ T01 (M ) and every tangent vector y ∈ T M .
Since in the following sections we shall work on the subset Tr M of T M consisting
of spheres of constant radius r, we shall consider only the tangent vectors y for which
2
the energy density t is equal to r2 , and the coefficients from the definition (2.2) of the
e become constant. So we may consider them constant from the beginning.
metric G
Then the M -tensor fields involved in the expression of the Levi-Civita connection
become simpler:
Q(X V , Y V )
=
d2
g(X, Y
c2 +r 2 d2
)y V ,
P (X V , Y H )
=
d1
g(Y, y)X H
2c1
+
d1
[c g(X, Y
2c1 (c1 +r 2 d1 ) 1
c2
− 2c
(R(X, y)Y )H −
1
(2.5)
S(X H , Y H )
) − d1 g(X, y)g(Y, y)]y H
c2 d1
g(X, R(Y, y)y)y H ,
2c1 (c1 +r 2 d1 )
d1
[g(X, y)Y V + g(Y, y)X V ]
= − 2c
2
d2
g(y, X)g(y, Y )y V − 21 (R(X, Y )y)V .
+ c2 (cd21+2td
2)
3
Tangent sphere bundles of constant radius r
Let Tr M = {y ∈ T M : gτ (y) (y, y) = r2 }, with r ∈ (0, ∞), and the projection
τ : Tr M → M , τ = τ ◦ i, where i is the inclusion map.
The horizontal lift of any vector field on M is tangent to Tr M , but the vertical
lift is not always tangent to Tr M . The tangential lift of a vector X to (p, y) ∈ Tr M ,
used in some recent papers like [4], [11], [12], [18], [25], is defined by
XyT = XyV −
1
gτ (y) (X, y)yyV .
r2
The tangent bundle to Tr M is spanned by δi and ∂jT = ∂j − r12 g0j y k ∂k , i, j, k = 1, n.
Remark that the vector fields {∂jT }j=1,n are not independent; there is the relation
y j ∂jT = 0, which can be directly checked. In any point of Tr M the vectors ∂iT ; i =
1, . . . n span an (n − 1)-dimensional subspace of T Tr M . The vector field
N = y i ∂i
57
Tangent sphere bundles of natural diagonal lift type
is normal to Tr M in T M .
e defined on T M . If
Denote by G the metric on Tr M induced from the metric G
we consider only the tangent vectors y ∈ Tr M , the energy density t defined by them
2
e become
is equal to r2 , and the coefficients from the definition (2.2) of the metric G
constants. Thus, the metric G from Tr M has the form
G(XyH , YyH ) = c1 gτ (y) (X, Y ) + d1 gτ (y) (X, y)gτ (y) (Y, y),
G(XyT , YyT ) = c2 [gτ (y) (X, Y ) − r12 gτ (y) (X, y)gτ (y) (Y, y)],
(3.1)
G(XyH , YyT ) = G(YyT , XyH ) = 0,
for every vector fields X, Y on M and every tangent vector y, where c1 , d1 , c2 are
constants. The conditions for G to be positive are c1 > 0, c2 > 0, c1 + r2 d1 > 0.
Proposition 3.1. The Levi-Civita connection ∇, associated to the Riemannian metric G on the tangent sphere bundle Tr M of constant radius r has the expression
h
∇∂ T ∂jT = Ahij ∂hT , ∇δi ∂jT = Γhij ∂hT + Bji
δh ,
i
∇ T δ = B h δ , ∇ δ = Γh δ + C h ∂ T ,
δi j
∂i j
ij h
ij h
ij h
where the M -tensor fields involved as coefficients have the expressions:
Ahij =
h
=
(3.2) Bij
− r12 g0j δih ,
Pijh −
d1
h
h
Cij
= − 2c
(δjh g0i − δih g0j ) − 21 R0ij
,
2
1
h
r 2 gi0 P0j
c2
h
− 2c
Rjik
yk −
1
=
d1 h
2c1 δi g0j
+
d1
2(c1 +r 2 d1 ) (gij
−
2c1 +r 2 d1
g0i g0j )y h
r 2 c1
c2 d1
h k l
2c1 (c1 +r 2 d1 ) Rikjl y y y ,
h
where g0j = gj0 = y i gij and P0j
= y i Pijh .
We mention that A and C have these quite simple expressions, since they are
the coefficients of the tangential part of the Levi-Civita connection. All the terms
containing y h ∂hT = 0 have been canceled.
The invariant form of the Levi-Civita connection from Tr M is
∇X T Y T = − r12 g(Y, y)X T ,
d1
d1
H
g(Y, y) X H + 2(c1 +d
∇X T Y H = 2c
2 g(X, Y )y
1
1r )
2
d1 (2c1 +r d1 )
c2
c2 d1
H
H
H
− 2r2 c1 (c1 +r2 d1 ) g(X, y)g(Y, y)y − 2c1 (R(X, y)Y ) − 2c1 (c1 +r2 d1 ) g(X, R(Y, y)y)y ,
d1
H
˙ X Y )T + d1 g(X, y) Y H +
∇X H Y T = ( ∇
2c1
2(c1 +r 2 d1 ) g(X, Y )y
2
c2
c2 d1
1 +r d1 )
H
H
H
− 2rd12 (2c
c1 (c1 +r 2 d1 ) g(X, y)g(Y, y)y − 2c1 (R(Y, y)X) − 2c1 (c1 +r 2 d1 ) g(Y, R(X, y)y)y ,
˙ X Y )H − d1 [g(X, y) Y T − g(Y, y) X T ] − 1 (R(X, Y )y)T .
∇X H Y H = ( ∇
2c2
2
˙ k Qh , ∇
˙ kP h , ∇
˙ k S h of the M -tensor fields Q, P, S are
The covariant derivatives ∇
ij
ij
ij
defined by some formulas similar to those from the classical cases. The covariant
derivative of P is given by
˙ k Pijh = δk Pijh + Γhkl Pijl − Γlki Pljh − Γlkj Pilh .
∇
58
S. L. Druţă and V. Oproiu
Remark that instead of the partial derivative ∂x∂ k we use the operator δk = δxδ k =
∂
˙ k S h . It can be
˙ k Qh and for ∇
− Γhk0 ∂y∂ k . Similar expressions are obtained for ∇
ij
ij
∂xk
˙ k S h are M -tensors
˙ kP h , ∇
˙ k Qh , ∇
proved by a straightforward computation that ∇
ij
ij
ij
of type (1, 3). The expression of the covariant derivative of an arbitrary M -tensor can
˙ k y i = 0, ∇
˙ k gij = 0. In the case where Q, P, S
be obtained easily. It follows that ∇
are given by (2.5), their covariant derivatives are:
h l
˙ k Qh = 0, ∇
˙ kSh = − 1 ∇
˙
∇
ij
ij
2 k Rlij y
h l
˙ k P h = − c2 ∇
˙
∇
ij
2c1 k Rjil y −
c2 d1
l r h
˙
2c1 (c1 +r 2 d1 ) ∇k Riljr y y y ,
˙ k Rh , ∇
˙ k Rhkij are the usual local coordinate expressions of the covariant
where ∇
kij
derivatives of the curvature tensor field and the Riemann-Christoffel tensor field of
˙ on the base manifold M . If the base manifold (M, g) is
the Levi Civita connection ∇
˙ k Qh = 0, ∇
˙ k P h = 0, ∇
˙ k S h = 0.
locally symmetric then, obviously, ∇
ij
ij
ij
In a similar way we obtain
h l
˙
˙ k Ah = 0, ∇
˙ kCh = − 1 ∇
∇
ij
ij
2 k Rlij y ,
h l
˙ k B h = − c2 ∇
˙
∇
ij
2c1 k Rjil y −
c2 d1
l r h
˙
2c1 (c1 +r 2 d1 ) ∇k Riljr y y y ,
The curvature tensor field K of the connection ∇ from Tr M is defined by the formula
K(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,
X, Y, Z ∈ T01 (Tr M ).
We obtain by a standard straightforward computation the horizontal and tangential
components of the curvature tensor field K
h
h
K(δi , δj )δk = HHHHkij
δh + HHHTkij
∂hT ,
h
h
K(δi , δj )∂kT = HHT Hkij
δh + HHT Tkij
∂hT ,
h
h
K(∂iT , ∂jT )δk = T T HHkij
δh , K(∂iT , ∂jT )∂kT = T T T Tkij
∂hT ,
h
h
h
K(∂iT , δj )δk = T HHHkij
δh + T HHTkij
∂hT , K(∂iT , δj )∂kT = T HT Hkij
δh ,
where the non-zero M -tensor fields which appear as coefficients are given by
h
h
l
h l
h l
HHHHkij
= Rkij
+ Blih Cjk
− Blj
Cik + Blk
R0ij ,
h
˙ j Rh y l − ∇
˙ i Rh y l ),
HHHTkij
= 12 (∇
lik
ljk
h
HHT Hkij
=
c2 ˙
h
2c1 (∇j Rikl
˙ i Rh )y l +
−∇
jkl
c2 d1
˙
2c1 (c1 +r 2 d1 ) (∇j Rilkm
˙ i Rjlkm )y l y m y h ,
−∇
h
h
l
h l
l
)∂hT ,
HHT Tkij
∂hT = (Rkij
+ Cilh Bkj
− Cjl
Bki + Ahlk R0ij
h
h
h
l
h l
T T HHkij
= ∂iT Bjk
− ∂jT Bik
+ Bilh Bjk
− Bjl
Bik −
h
T T T Tkij
=
1
h
r 2 (gjk δi
h
T HHHkij
=
− gik δjh ) +
c2 ˙
h
l
2c1 ∇j Rkil y
+
1
h
r 4 (δj g0i g0k
1
h
r 2 (gi0 Bjk
h
− gj0 Bik
),
− δih g0j g0k ),
c2 d1
l r h
˙
2c1 (c1 +r 2 d1 ) ∇j Rilkr y y y ,
h
l
h l
h
˙ j Rh y l )∂ T ,
+ Ahil Cjk
− Cjl
Bik − ∇
T HHTkij
∂hT = (∂iT Cjk
lik
h
h
h
h
h
l
˙ j Ah = 0.
, T HT Tkij
= −∇
T HT Hkij
= ∂iT Bkj
+ Bkj
Bilh − Alik Blj
ik
Tangent sphere bundles of natural diagonal lift type
59
h
h
We mention that from the final values of HHT Tkij
∂hT and T HHTkij
∂hT , obtained by
h
h
replacing the expressions (3.2) of Ahij , Bij
, Cij
, we shall eliminate the terms containing
h T
y ∂h , because they vanish.
We want to find the manifolds (Tr M, G) of constant sectional curvature k, i.e the
manifolds for which the curvature tensor field K satisfies the relation
K(X, Y )Z − k[G(Y , Z)X − G(X, Z)Y ] = 0, ∀X, Y , Z ∈ T01 (Tr M ).
(3.3)
Let us consider X, Y , Z ∈ T01 (Tr M ) as being the tangential lifts of X, Y, Z ∈
We have to study the vanishing conditions for the difference
T01 (M ).
K(X T , Y T )Z T − k[G(Y T , Z T )X T − G(X T , Z T )Y T ] = 0, ∀X, Y, Z ∈ T01 (M ),
which has the detailed expression
(3.4)
− g(X, Z)Y T ] + r14 g(y, Z)[g(y, X)Y T − g(y, Y )X T ]
−k[G(Y T , Z T )X T − G(X T , Z T )Y T ] = 0,
1
T
r 2 [g(Y, Z)X
for all the vector fields X, Y, Z and for every vector y tangent to M .
The local coordinate form of (3.4) is
¤
kr2 c2 − 1 £
1
gjk δih − gik δjh − 2 (δjh g0i g0k − δih g0j g0k ) = 0,
2
r
r
from which we obtain a first necessary condition for the manifold (Tr M, G) to have
constant sectional curvature k:
(3.5)
c2 =
1
.
kr2
If in the relation (3.3) we take instead of X, Y , Z ∈ T01 (Tr M ) the horizontal lifts of
X, Y, Z ∈ T01 (M ), we obtain that the following vanishing condition must be satisfied:
(3.6)
K(X H , Y H )Z H − k[G(Y H , Z H )X H − G(X H , Z H )Y H ] = 0,
for every tensor fields X, Y, Z ∈ T01 (M ).
Since
d2
K(X H , Y H )Z H = (R(X, Y )Z)H − 4c11c2 [g(y, X)Y H − g(y, Y )X H ]g(y, Z)
2
d1
+ 4c2 (cd1
[g(Y, Z)g(y, X) − g(X, Z)g(y, Y )]y H + 4c
[(R(y, Y )Z)H g(y, X)
2
1 +r d1 )
1
3d1
H
H
−(R(y, X)Z) g(y, Y ) + (R(X, Y )y) g(y, Z)] + 4(c1 +r2 d1 ) g(X, R(Z, y)Y )y H
d2
H
1
− 4c1 (c1 +r
2 d ) [g(Y, R(Z, y)y)g(X, y) − g(X, R(Z, y)y)g(Y, y)]y
1
c2
c2
[R(R(Y, Z)y, y)X − R(R(X, Z)y, y)Y ]H − 2c
[R(R(X, Y )y, y)Z]H
+ 4c
1
1
c2 d1
+ 4c1 (c1 +r2 d1 ) [g(X, R(R(Y, Z)y, y)y) − g(X, R(R(Y, y)y, y)Z)
˙ Z R(X, Y )y)T ,
−2g(X, R(R(Z, y)y, y)Y )]y H + 21 (∇
the coefficient of the horizontal part of the difference (3.6) has the following local
60
S. L. Druţă and V. Oproiu
coordinate expression:
h
HHHHkij
− k[(c1 gjk + d1 g0j g0k )δih − (c1 gik + d1 g0i g0k )δjh ] =
h
= Rkij
+ c1 k(gik δjh − gjk δih ) +
d2
(3.7)
d1 (4c1 c2 k−d1 ) h
(δj g0i g0k
4c1 c2
h
1
+ 4c2 (c1 +r
+
2 d ) (gjk g0i − gik g0j )y
1
h
1
+ 4(c13d
+r 2 d1 ) Rijk0 y −
d1
h
4c1 (Rk0j g0i
d21
4c1 (c1 +r 2 d1 ) (Rj0k0 g0i
− δih g0j g0k )
h
h
− Rk0i
g0j + R0ij
g0k )
− Ri0k0 g0j )y h
c2
l
h
l
h
l
h
+ 4c
(R0jk
Ril0
− R0ik
Rjl0
− 2R0ij
Rkl0
)
1
d1
l
l
l
h
+ 4c1 (cc12+r
2 d ) (R0i0 Rjkl0 − R0ik Rj0l0 − 2R0ij Rk0l0 )y .
1
h
For the indices h, i, j, k fixed, the condition for HHHHkij
− k[(c1 gjk + d1 g0j g0k )δih −
h
(c1 gik + d1 g0i g0k )δj ] to be zero leads to an equation of the type A + Al1 l2 y l1 y l2 +
Al1 l2 l3 l4 y l1 y l2 y l3 y l4 = 0, in the tangential coordinates y i ; i = 1, n, where the coefficients A’s are obtained from the above expression by full symmetrization. Differentiating the obtained expression four times, it follows that Al1 l2 l3 l4 = 0. Then differentiating the remaining expression A + Al1 l2 y l1 y l2 two times, it follows Al1 l2 = 0. We
conclude that the constant term A is zero, whence the base manifold must be a space
form, with the curvature of the form:
h
Rkij
= c1 k(gjk δih − gik δjh ).
(3.8)
Replacing this expression and the value (3.5) of c2 into (3.7), we obtain the following
vanishing condition:
i
k(3c1 − r2 d1 ) h c1 + r2 d1 h
h
h
= 0,
(δ
g
−
δ
g
)g
−
(g
g
−
g
g
)y
0i
0j
0k
jk
0i
ik
0j
j
i
4r2
c1
which is satisfied if and only if c1 has the form
(3.9)
c1 =
r 2 d1
.
3
h
The same values (3.5) of c2 and (3.9) of c1 lead to the identity T T HHkij
= 0, which
is another necessary condition for (Tr M, G) to be a space form. More precisely, after
imposing the expression (3.8) for the curvature of the base manifold, and the value
h
(3.5) of c2 into the expression of T T HHkij
, this becomes
h
T T HHkij
=
3c1 − r2 d1
[gjk δih − gik δjh + (δjh g0i − δih g0j )g0k − (gjk g0i − gik g0j )y h ],
4c1 r2
which vanishes if and only if c1 takes the value (3.9).
Due to the condition for the base manifold to be a space form, the components
h
h
h
HHHTkij
, HHT Hkij
, T HHHkij
of the curvature tensor field of (Tr , G) and the corh
responding components of the difference from (3.3) become zero. Since HHT Tkij
is
also vanishing when we substitute the expression of the curvature (3.8) and the values
(3.9) for c1 and (3.5) for c2 , we have immediately that
K(X H , Y H )Z T = k[G(Y H , Z T )X H − G(X H , Z T )Y H ], ∀X, Y, Z ∈ T01 (M ).
Tangent sphere bundles of natural diagonal lift type
61
The identity
K(X T , Y H )Z H = k[G(Y H , Z H )X T − G(X T , Z H )Y H ], ∀X, Y, Z ∈ T01 (M ),
reduces, after imposing the conditions (3.8), (3.9), (3.5), to the relation
2
T
T
+ g(X, Y )Z T
− d1 kr
3 [g(Y, Z)X + g(X, Z)Y
−g(y, X)g(y, Y )Z T − g(y, X)g(y, Z)Y T − g(y, Y )g(y, Z)X T ] = 0,
which is true for every vector fields X, Y, Z ∈ T01 (M ), and for every tangent vector
y ∈ T M , if and only if d1 = 0 or k = 0 (i.e the manifold (Tr , G) is flat).
Finally, we have that the manifold (Tr M, G) may never be a space form, since the
relation
K(X T , Y H )Z T = k[G(Y H , Z T )X T − G(X T , Z T )Y H ], ∀X, Y, Z ∈ T01 (M ),
which has the final form
1
T
r 2 [g(Y, Z)X
(3.10)
+ g(X, Z)Y T + g(X, Y )Z T ]
− r14 {g(y, X)g(y, Y )Z T + g(y, X)g(y, Z)Y T + g(y, Y )g(y, Z)X T
+[g(Y, Z)g(y, X) + g(X, Z)g(y, Y ) + g(X, Y )g(y, Z)
− r32 g(y, X)g(y, Y )g(y, Z)]y h } = 0,
may never be satisfied. Hence we may state
Theorem 3.2. The tangent sphere bundle Tr M , with the Riemannian metric G
e of diagonal lift type on the tangent bundle T M , has never
induced from the metric G
constant sectional curvature.
Corollary 3.3. The tangent sphere bundle Tr M , endowed with the metric induced
by the Sasaki metric g s from the tangent bundle T M is never a space form.
In fact, the Sasaki metric can be obtained as a particular case of our natural
diagonal lifted metric, with the coefficients c1 = c2 = 1, d1 = 0.
4
Einstein tangent sphere bundles
We shall find the conditions under which the manifold (Tr M, G), with G given by
the relations (3.1), is Einstein. To this aim, we shall compute the Ricci tensor of the
manifold (Tr M, G).
First, let us remark some facts concerning the obtaining of the Ricci tensor field
for the tangent bundle T M . We have the well known formula
Ric(Y, Z) = trace(X → K(X, Y )Z),
where X, Y, Z are vector fields on T M . Then we get easily the components of the
Ricci tensor field on T M
h
h
g
g
RicHH
jk = Ric(δj , δk ) = HHHHkhj + V HHVkhj ,
g Vjk = Ric(∂
g j , ∂k ) = V V V V h − V HV H h ,
RicV
khj
kjh
62
S. L. Druţă and V. Oproiu
h
h
h
where the components V V V Vkij
, V HV Hkij
, V HHVkij
, are obtained from the curh
h
vature tensor field on T M in a similar way as the components T T T Tkij
, T HT Hkij
,
h
T HHTkij are obtained from the curvature tensor field on Tr M . In the expression
g
g
g
g
of RicHV
jk = Ric(δj , ∂k ) = RicV Hjk = Ric(∂j , δk ) there are involved the covariant derivatives of the curvature tensor field R. If (M, g) is locally symmetric then
RicHVjk = RicV Hjk = 0. In particular we have RicHVjk = RicV Hjk = 0 in the
case where (M, g) has constant sectional curvature.
Equivalently, we may use an orthonormal frame (E1 , . . . , E2n ) on T M and we may
use the formula
2n
X
g
Ric(Y, Z) =
G(Ei , K(Ei , Y )Z).
i=1
We may choose the orthonormal frame (E1 , . . . , E2n ) such that the first n vectors
E1 , . . . , En are the vectors of a (orthonormal) frame in HT M and the last n vectors
En+1 , . . . E2n are the vectors of a (orthonormal) frame in V T M . Moreover, we may
assume that the last vector E2n is the unitary vector of the normal vector N = y i ∂i
to Tr M .
The components of the Ricci tensor field of Tr M can be obtained in a similar
way by using the above traces. However the vector fields ∂1T , . . . , ∂nT are not independent. On the open set from Tr M , where y n 6= 0 we can consider the basis
T
for T Tr M . The last vector ∂nT is expressed as
δ1 , ..., δn , ∂1T , ..., ∂n−1
∂nT = −
n−1
1 X i T
y ∂i .
y n i=1
T
Remark that the basis δ1 , ..., δn , ∂1T , ..., ∂n−1
can be completed with the normal vector
V
h
N = y = y ∂h .
h
h
For the components HHHHkij
and HT T Hkij
the traces can pe computed easily,
just like in the case of T M . The components for which we must compute more
h
h
carefully the traces are T T T Tkij
, and T HHTkij
. We have:
h
T HHTkij
∂hT
=
=
Pn−1
l=1
Pn−1
l=1
l
n
T HHTkij
∂lT + T HHTkij
∂nT
l
n 1
T HHTkij
∂lT − T HHTkij
yn
h
n 1 h T
= T HHTkij
∂hT − T HHTkij
y n y ∂h .
Pn−1
l=1
y l ∂lT
Thus the trace involved in the definition of RicHH on Tr M is
h
T HHTkhj
−
1 h
n
y T HHTkhj
.
yn
h
A short computation made by using the above expression of of T HHTkij
gives
h
y i T HHTkij
=
thus we get
d1 (4c1 + r2 d1 )
g0j g0k y h ,
4c1 c2 r2
1 i
d1 (4c1 + r2 d1 )
n
y
T
HHT
=
g0j g0k .
kij
yn
4c1 c2 r2
63
Tangent sphere bundles of natural diagonal lift type
It follows that
h
h
−
+ T HHTkhj
RicHHjk = HHHHkhj
d1 (4c1 + r2 d1 )
g0j g0k .
4c1 c2 r2
In a similar way we get that the trace involved in the definition of RicT T on Tr M is
h
T T T Tkhj
−
Then
n
y h T T T Tkhj
=
1 h
n
y T T T Tkhj
.
yn
1
1
gjk y n − 4 g0j g0k y n .
2
r
r
Hence
h
h
RicT Tjk = Ric(∂jT , ∂kT ) = HT T Hkhj
+ T T T Tkhj
−
1
1
gjk + 4 g0j g0k .
2
r
r
After the computations we obtain the detailed expressions of RicHHjk and RicT Tjk :
RicHHjk =
Ricjk −
d1 (2c1 +r 2 d1 )
2c2 (c1 +r 2 d1 ) gjk
c2
− 2c
Rh0kl Rl jh0 +
1
RicT Tjk =
+
d1 (2c1 +r 2 d1 )[c1 n+r 2 d1 (n−1)]
g0j g0k
2r 2 c1 c2 (c1 +r 2 d1 )
c2 d1
h
2c1 (c1 +r 2 d1 ) R 0k0 Rh0j0 ,
r 4 d21 −2c1 (c1 +r 2 d1 )(n−2) 1
( r2 g0j g0k
2c1 r 2 (c1 +d1 r 2 )
− gjk ) +
c22
R
Rhij0
4c21 hik0
c2 d
− 2c2 (c12+r1 2 d1 ) Rh0j0 Rh0k0 .
1
The components Rh0kl , Rl jh0 , Rh0k0 , Rh0j0 , etc. are obtained as usual from the com˙ by transvecting with y’s. E.g. Rh =
ponents of the curvature tensor field R of ∇
0kl
h
i
l
l
i
h
h
R ikl y , R jh0 = R jhi y , R 0k0 = R ikj y i y j , Rh0j0 = Rhijk y i y k . Recall that
˙
Rhijk = ghl Rl ijk are the components of the Riemann-Christoffel tensor field of ∇.
Taking the above relations into account, we obtain that the differences between the
components of the Ricci tensor and the corresponding components of the metric G
multiplied by a constant ρ have the following forms:
(1)
RicHHjk − ρGjk = Ricjk −
+ d1 (2c1 +r
2
d1 (2c1 +r 2 d1 )+2ρc1 c2 (c1 +r 2 d1 )
gjk
2c2 (c1 +r 2 d1 )
d1 )[c1 n+r 2 d1 (n−1)]−2ρr 2 c1 c2 d1 (c1 +r 2 d1 )
g0j g0k
2c1 c2 r 2 (c1 +r 2 d1 )
c2
l
h
l
h
(Rkh0
R0jl
+ Rjh0
R0kl
)+
+ 4c
1
(2)
RicT Tjk − ρGjk =
+
c22
R
Rhi
4c21 hik0 j0
c2 d1
h
2c1 (c1 +r 2 d1 ) Rh0j0 R0k0 ,
−
c22 d1
R
Rh
2c21 (c1 +r 2 d1 ) h0j0 0k0
r 4 d21 +2c1 (c1 +r 2 d1 )(n−2)+ρ2c1 c2 r 2 ) 1
( r2 g0j g0k
2r 2 c1 (c1 +r 2 d1 )
− gjk ).
When the base manifold (M, g) has constant sectional curvature c, i.e. when
(4.1)
h
Rkij
= c(gjk δih − gik δjh ),
64
S. L. Druţă and V. Oproiu
the above differences take the forms
(1)
RicHHjk − ρGjk =
+
cc2 [2(n−1)(c1 +r 2 d1 )−cc2 r 2 ]−d1 (2c1 +r 2 d1 )−2ρc1 c2 (c1 +r 2 d1 )
gjk
2c2 (c1 +d1 r 2 )
d1 [2c21 n+r 4 (d21 −c2 c22 )(n−1)]+r 2 c1 [d21 (3n−2)−c2 c22 (n−2)]−2ρr 2 c1 c2 d1 (c1 +r 2 d1 )
g0j g0k ,
2r 2 c1 c2 (c1 +d1 r 2 )
(2)
RicT Tjk − ρGjk =
r 4 (d21 −c2 c22 )+2c1 (c1 +r 2 d1 )(2−n+ρc2 r 2 ) ¡ 1
2r 2 c1 (c1 +r 2 d1 )
r 2 g0j g0k
− gjk
(2)
¢
The difference RicT Tjk − ρGjk vanishes if and only if the constant ρ has the value
ρ=
2c21 (n−2)+r 2 [2c1 d1 (n−2)+r 2 (c2 c22 −d21 )]
.
2r 2 c1 c2 (c1 +r 2 d1 )
(1)
After replacing this expression, the difference RicHHjk − ρGjk becomes
(1)
(4.2)
RicHHjk − ρGjk =
2(cc2 r 2 −c1 )
{c1 (n
r2
− 2) − r2 [cc2 − d1 (n − 1)]}gjk
+{4c21 d1 + r2 c1 [d21 (n + 2) − c2 c22 (n − 2)] + r4 d1 (d21 − c2 c22 )n}g0j g0k .
By doing a detailed analysis of all the situations in which the difference expressed by
(4.2) vanishes, we may prove the following theorem.
Theorem 4.1. The tangent sphere bundle Tr M of an n-dimensional Riemannian
(M, g) of constant sectional curvature c is Einstein with respect to the metric G ine defined on T M , i.e. it exists a real
duced from the natural diagonal lifted metric G
constant ρ such that
Ric(X, Y ) = ρG(X, Y ), ∀X, Y ∈ T01 (Tr M ),
if and only if
c1 =
r 2 d1 n
d1 n
c(n − 1)2 (n − 2)
, c2 =
, ρ=
.
n−2
c(n − 2)
r 2 d1 n 2
Proof. The difference expressed by (4.2) vanishes if and only if the both coefficients
are equal to zero. The vanishing condition for the coefficient of gjk leads to two cases
which must be studied:
I)
c1 = cc2 r2 , II) c1 =
r2 [cc2 − d1 (n − 1)]
.
n−2
In the case I, the coefficient of g0j g0k from (4.2) becomes
−(cc2 + d1 )2 [cc2 (n − 2) − d1 n]r4 = 0,
so we have two possible values for c2 :
(4.3)
c2 = −
d1 n
d1
, or c2 =
.
c
c(n − 2)
Replacing the first one into the expression of c1 , we obtain that c1 + r2 d1 = 0, thus
the constant ρ, some components of K as well as the Levi Civita connection are not
defined. So this situation should be excluded.
65
Tangent sphere bundles of natural diagonal lift type
When c2 takes the second value from (4.3), c1 and ρ have the forms presented in
the case I from the theorem.
If c1 has the expression from case II the coefficient of g0j g0k given in (4.2) decomposes as
(4.4)
−
r4 (cc2 − d1 ){c2 c22 (n − 2)2 + d1 [d1 n2 + 2cc2 (n2 − 4n + 2)]}
= 0.
(n − 2)2
The vanishing condition for the first factor and the value of c1 from case II lead us
again to the situation where c1 + r2 d1 = 0, so it should be excluded.
The second factor which appears in (4.4) is a second degree function of c2 . The
discriminant of the attached equation is
∆ = 16c2 d21 (1 − n)(n2 − 3n + 1).
Since the dimension n of the base manifold is a natural number bigger then one, ∆ is
positive only for n = 2, value which makes vanish the denominator of the expression
(4.4), so the case must be treated separately, starting with the expressions of the
(1)
(2)
differences RicHHjk − ρGjk and RicT Tjk − ρGjk , which become of the forms
(1)
RicHHjk − ρGjk =
cc2 [2(c1 +r 2 d1 )−cc2 r 2 ]−d1 (2c1 +r 2 d1 )−2ρc1 c2 (c1 +r 2 d1 )
gjk
2c2 (c1 +d1 r 2 )
+
(2)
RicT Tjk − ρGjk =
d1 [4c21 +r 4 (d21 −c2 c22 )]+4r 2 c1 d21 −2ρr 2 c1 c2 d1 (c1 +r 2 d1 )
g0j g0k ,
2r 2 c1 c2 (c1 +d1 r 2 )
r4 (d21 − c2 c22 ) + 2ρr2 c1 c2 (c1 + r2 d1 )
2r2 c1 (c1 + r2 d1 )
µ
¶
1
g
g
−
g
0j 0k
jk .
r2
From these relations we get
ρ=
r2 (c2 c22 − d21 )
,
2c1 c2 (c1 + r2 d1 )
and then
(1)
RicHHjk − ρGjk = 2(cc2 r2 − c1 )(d1 − cc2 )gjk
+2d1 [2c1 (c1 + r2 d1 ) + r4 (d21 − c2 c22 )]g0j g0k .
This last difference vanishes if and only if c1 = r2 cc2 and d1 = −cc2 , or d1 = cc2 and
c1 = −r2 cc2 , but the both cases must be excluded, since they lead to c1 + r2 d1 = 0.
Thus the theorem is proved.
¤
Corollary 4.2. The tangent sphere bundle Tr M of an n-dimensional Riemannian
space form (M, g) may never be Einstein with respect to the metric induced by the
Sasaki metric g s from T M .
In fact, in this case, the coefficients should be c1 = c2 = 1, d1 = 0, and they do
not satisfy the condition from theorem 4.1.
66
S. L. Druţă and V. Oproiu
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Authors’ addresses:
Simona-Luiza Druţă
Faculty of Mathematics, ”AL. I. Cuza” University of Iaşi,
Bd. Carol I, No. 11, RO-700506 Iaşi, Romania.
E-mail: [email protected]
Vasile Oproiu
Faculty of Mathematics, ”AL. I. Cuza” University of Iaşi,
Bd. Carol I, No. 11, RO-700506 Iaşi, Romania.
Institute of Mathematics ”O. Mayer”,
Romanian Academy, Iaşi Branch.
E-mail: [email protected]