New structures on the tangent bundles and tangent sphere bundles

0.25 MB·English

by  Marian Ioan Munteanu

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New structures on the tangent bundles
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and tangent sphere bundles
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a
J Marian Ioan Munteanu ∗

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Abstract
D
Inthispaperwestudy aRiemanianmetriconthe tangentbundle T(M)ofaRieman-
.
h nian manifold M which generalizes Sasaki metric and Cheeger Gromoll metric and a
at compatiblealmostcomplexstructurewhichtogetherwiththe metricconferstoT(M)
m a structure of locallyconformalalmostK¨ahlerianmanifold.This is the naturalgener-
alizationofthewellknownalmostK¨ahlerianstructureonT(M).Wefoundconditions
[
under which T(M) is almost K¨ahlerian, locally conformal K¨ahlerian or K¨ahlerian or
2
when T(M) has constant sectional curvature or constant scalar curvature. Then we
v
will restrict to the unit tangent bundle and we find an isometry with the tangent
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7 spherebundle(notnecessaryunitary)endowedwiththerestrictionoftheSasakimet-
3 ric from T(M). Moreover, we found that this map preserves also the natural almost
1 contactstructuresobtainedfromthealmostHermitianambientstructuresontheunit
1
tangent bundle and the tangent sphere bundle, respectively.
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/ Mathematics Subject Classifications (2000): 53B35, 53C07, 53C25, 53C55.
h
t
a Key words and Phrases: Riemannian manifold, Sasaki metric, Cheeger Gromoll
m
metric, tangent bundle, tangent sphere bundle, locally conformal (almost) K¨ahlerian
: manifold, almost contact metric manifold.
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i
X
1 Introduction
r
a
A Riemannian metric g on a smooth manifold M gives rise to several Riemannian metrics
on thetangent bundleT(M)of M.Maybe thebestknownexample is theSasakimetricg
S
introduced in [21]. Although the Sasaki metric is naturally defined, it is very rigid in the
following sense. For example, O.Kowalski [13] has shown that the tangent bundle T(M)
with the Sasaki metric is never locally symmetric unless the metric g on the base manifold
is flat. Then, E.Musso & F.Tricerri [16] have shown a more general result, namely, the
Sasaki metric has constant scalar curvature if and only if (M,g) is locally Euclidian. In
the same paper, they have given an explicit expression of a positive definite Riemannian
metric introduced by J.Cheeger and D.Gromoll in [11] and called this metric the Cheeger-
Gromoll metric. In [22] M.Sekizawa computed the Levi Civita connection, the curvature
∗Beneficiary of a CNR-NATOAdvanced Research Fellowship pos. 216.2167 Prot. n.0015506.
1
tensor, the sectional curvatures and the scalar curvature of this metric. These results are
completed in 2002 by S.Gudmundson and E.Kappos in [12]. They have also shown that
the scalar curvature of the Cheeger Gromoll metric is never constant if the metric on the
base manifold has constant sectional curvature. Furthermore, M.T.K.Abbassi & M.Sarih
have proved that T(M) with the Cheeger Gromoll metric is never a space of constant
sectional curvature (cf. [3]). A more general metric is given by M.Anastasiei in [7] which
generalizes both of the two metrics mentioned above in the following sense: it preserves
the orthogonality of the two distributions, on the horizontal distribution it is the same
as on the base manifold, and finally the Sasaki and the Cheeger Gromoll metric can be
obtained as particular cases of this metric. A compatible almost complex structure is also
introduced and hence T(M) becomes an locally conformal almost Ka¨herian manifold.
V.Oproiuandhiscollaborators constructedafamilyof Riemannianmetrics onthetangent
bundles of Riemannian manifolds which possess interesting geometric properties (cf. [17,
18, 19, 20]). In particular, the scalar curvature of T(M) can beconstant also for a non-flat
base manifold with constant sectional curvature. Then M.T.K.Abbassi & M.Sarih proved
in [4] that the considered metrics by Oproiu form a particular subclass of the so-called
g-natural metrics on the tangent bundle (see also [1, 2, 4, 5, 6, 14]).
By thinking T(M) as a vector bundle associated with O(M) (the space of orthonormal
frames on M), namely T(M) O(M) Rn/O(n) (where the orthogonal group O(n)
≡ ×
acts on the right on O(M)), Musso & Tricerri construct natural metrics on T(M) (see 4
§
in [16]). The idea is to consider Q a symmetric, semi-positive definite tensor field of type
(2,0) and rank2n on O(M) Rn. Assumingthat Q is basic for ψ :O(M) Rn T(M),
× × −→
(u,ζ) p,ζiu , where u = (p,u ,...,u ) and ζ = (ζ1,...,ζn) (i.e. Q is O(n)-invariant
i 1 n
7→
and Q(X,Y)= 0 for all X tangent to a fiber of ψ) there is a unique Roiemannian metric
(cid:0) (cid:1)
g on T(M) such that ψ g = Q. Inthis paperwe willshow that the metric introduced in
Q ∗ Q
[7] can be construct by using the method of Musso and Tricerri and we study it. Then we
will give the conditions underwhich T(M) is locally conformal Ka¨hlerian and respectively
Ka¨hlerian(Theorems2.6and2.8).TheseresultsextendtheknownresultsayingthatT(M)
endowed with the Sasaki metric and the canonical almost complex structure is Ka¨hlerian
if and only if the base manifold is locally Euclidean.
Next we want to have constant sectional curvature and constant scalar curvature, re-
spectively on T(M). With this end in view, we compute the Levi Civita connection, the
curvaturetensor,thesectional curvatureandthe scalar curvatureof this metric. We found
relations between the sectional curvature (resp. scalar curvature) on T(M) and the cor-
responding curvature on the base M. We give an example of metric on T(M) of Cheeger
Gromoll typewhich isflat.(Recall thefactthatCheeger Gromoll metriccan nothave con-
stantsectional curvature.)SeeProposition2.17.We alsoobtain alocally conformalKa¨hler
structure (cf. Example 2.8/ ) and a Ka¨hler structure (cf. Remark 2.10/ ) on T(M). We
2 3
give some examples of metrics on T(M) (when M is a space form) having constant scalar
curvature. See Examples 2.22 and 2.23.
In section 3 we restrict the structure on the unit tangent bundle, obtaining an almost
contact metric. We will show that the unit tangent bundle is isometric with a tangent
sphere bundle T (M) (we find the radius r) endowed with the restriction of Sasaki metric
r
2
fromT(M)(seealso[9],Remark4,p.88).Moreover, thismappreserves thealmostcontact
structures. M.Sekizawa & O.Kowalski have studied the geometry of the tangent sphere
bundleswith arbitrary radii endowed with the induced Sasaki metric (see [15]). They have
also noticed that the unit tangent bundle equipped with the induced Cheeger Gromoll
metric is isometric to the tangent sphere bundle T (M), of radius 1 endowed with the
√12 √2
metric induced by the Sasaki metric. Some other generalizations concerning this fact are
given in [2]. In the end of the section we obtained some properties for T (M) as contact
1
manifold.Amongtheresultswestatethefollowing:The contact metric structure onT (M)
1
is K-contact if and only if the base manifold has positive constant sectional curvature. In
this case T (M) becomes a Sasakian manifold.
1
2 The tangent bundle T(M)
Let (M,g) be a Riemannian manifold and let be its Levi Civita connection. Let
∇
τ : T(M) M be the tangent bundle. If u T(M) it is well known the following
−→ ∈
decomposition of the tangent space TuT(M) (in u at T(M))
TuT(M) = VuT(M) HuT(M)
⊕
where VuT(M) = kerτ ,u is the vertical space and HuT(M) is the horizontal space in u
∗
obtained by using . (A curve γ : I T(M) , t (γ(t),V(t)) is horizontal if the vector
∇ −→ 7→
field V(t) is parallel along γ = γ τ. A vector on T(M) is horizontal if it is tangent to
◦
an horizontal curve and verticael if it is tangent to a fiber. Locally, if (U,xi), i = 1,...,m,
m = dimM, is a local chart in pe M, consider (τ−1(U),xi,yi) a local chart on T(M). If
∈
Γk(x) are the Christoffel symbols, then δ = ∂ Γk(x)yj ∂ in u, i = 1,...,m span the
ij i ∂xi − ij ∂yk
spaceHuT(M),while ∂ ,i= 1,...,mspantheverticalspaceVuT(M).)Wehaveobtained
∂yi
the horizontal (vertical) distribution HTM (VTM) and a direct sum decomposition
TTM = HTM VTM
⊕
of the tangent bundle of T(M). If X χ(M), denote by XH (and XV, respectively) the
∈
horizontal lift (and the vertical lift, respectively) of X to T(M).
If u ∈ T(M) then we consider the energy density in u on T(M), namely t = 21 gτ(u)(u,u).
The Sasaki metric is defined uniquely by the following relations
g (XH,YH)= g (XV,YV) = g(X,Y) τ
S S
(1) ◦
g (XH,YV) = 0,
S
(cid:26)
for each X,Y χ(M).
∈
On T(M) we an also define an almost complex structure J by
S
(2) J XH = XV, J XV = XH, X χ(M).
S S
− ∀ ∈
It is known that (T(M),J ,g ) is an almost Ka¨hlerian manifold. Moreover, the integra-
S S
bility of the almost complex structure J implies that (M,g) is locally flat (see e.g. [8]).
S
3
The Cheeger-Gromoll metric on T(M) is given by
gCG(p,u)(XH,YH)= gp(X,Y), gCG(p,u)(XH,YV)= 0
(3) 1
 gCG(p,u)(XV,YV) = 1+2t (gp(X,Y)+gp(X,u)gp(Y,u))
for any vectorsX and Y tangent to M.
Since the almost complex structure J is no longer compatible with the metric g , one
S CG
defines on T(M) another almost complex structure J , compatible with the Chegeer-
CG
Gromoll metric, by the formulas
1
J XH =rXV g (X,u)uV
CG (p,u) − 1+r p
(4) 
 1 1
 J XV = XH g (X,u)uH
CG (p,u) −r − r(1+r) p



where r = √1+2t and X T (M). Remark that J uH = uV and J uV = uH. We
p CG CG
∈ −
get an almost Hermitian manifold (T(M),J ,g ). Moreover, if we denote by Ω the
CG CG CG
Kaehler 2-form (namely Ω (U,V) = g (U,J V), U,V χ(T(M))) it is quite easy
CG CG CG
∀ ∈
to prove the following
Proposition 2.1 We have
(5) dΩ = ω Ω ,
CG CG
∧
whereω ∈Λ1(T(M))isdefinedbyω(p,u)(XH) = 0andω(p,u)(XV)= − r12 + 1+1r gp(X,u),
X Tp(M). (cid:16) (cid:17)
∈
Proof. A simple computation gives
Ω (XH,YH) = Ω(XV,YV)= 0
CG
Ω (XH,YV) = 1 g(X,Y)+ 1 g(X,u)g(Y,u) .
CG −r 1+r
(From now on we will omi(cid:16)t the point (p,u).) (cid:17)
The differential of Ω is given by
CG
dΩ (XH,YH,ZH)= dΩ (XH,YH,ZV) = dΩ (XV,YV,ZV)= 0
CG CG CG
dΩ (XH,YV,ZV) = 1 1 + 1 [g(X,Y)g(Z,u) g(X,Z)g(Y,u)]
CG r r2 1+r −
for any X,Y,Z χ(M). (cid:16) (cid:17)
∈
Hence the statement.
Remark 2.2 The almost Hermitian manifold (T(M),J ,g ) is never almost Kaehle-
CG CG
rian (i.e. dΩ = 0).
CG
6
4
Finally, we obtain a necessary condition for the integrability of J namely, the base
CG
manifold (M,g) should be locally Euclidian.
A general metric, let’s call it g , is in fact a family of Riemannian metrics (depending
A
on two parameters) and the Sasaki metric and the Cheeger-Gromoll metric are obtained
by taking particular values for the two parameters. It is defined (cf. [7]) by the following
formulas
gA(p,u)(XH,YH)= gp(X,Y)
(6)  gA(p,u)(XH,YV)= 0
 gA(p,u)(XV,YV) = a(t)gp(X,Y)+b(t)gp(X,u)gp(Y,u),
for all X,Y χ(M), where a,b : [0,+ ) [0,+ ) and a > 0. For a = 1 and b = 0 one
∈ ∞ −→ ∞
obtains the Sasaki metric and for a= b = 1 one gets the Cheeger-Gromoll metric.
1+2t
Proposition 2.3 The metric defined above can be construct by using the method described
by Musso and Tricerri in [16].
Proof. If we denote by θ = (θ1,...,θn) the canonical 1-form on O(M) (namely, if p :
O(M) M, θ is defined by dp (X) = θi(X)u , for u= (p,u ,...,u ) and X T (M))
u i 1 n p
−→ ∈
we have R (θi) = (a 1)iθh for each a O(n). The vertical distribution of ψ is defined by
u∗ − h ∈
θi = 0, Dζi := dζi+ζjωi
j
j
where ω = (ω ) denotes the so(n)-valued connection 1-form defined by the Levi Civita
i i,j
connection of g. Since R (ωi) = (a 1)iωhak we can also write R (Dζi) = (a 1)iDζh, for
a∗ j − h k j a∗ − h
all a O(n).
∈
Consider now the following bilinear form on O(M)
n n n 2
(7) Q = (θi)2+a(1 ζ 2) (Dζi)2+b(1 ζ 2) ζiDζi .
A 2|| || 2|| ||
i=1 i=1 (cid:18)i=1 (cid:19)
P P P
It is symmetric, semi-positive definite and basic. Moreover, since the following diagram
O(M) Rn ψ //T(M)
×
proj τ
1
(cid:15)(cid:15) (cid:15)(cid:15)
O(M) p // M
commutes, we have ψ g = Q . (See for details 4 in [16].)
∗ A A
§
Again, we have to find an almost complex structure on T(M), call it J , which is compat-
A
ible with the metric g . Inspired from the previous cases we look for the almost complex
A
structure J in the following way
A
J XH = αXV +βg (X,u)uV
A (p,u) p
(8)

 J XV = γXH +ρg (X,u)uH
 A (p,u) p


5
where X χ(M) and α, β, γ and ρ are smooth functions on T(M) which will be deter-
∈
mined from J2 = I and from the compatibility conditions with the metric g . Following
A − A
the computations made in [7] we get first α = 1 and γ = √a. Without lost of the
±√a ∓
generality we can take
1
α = and γ = √a.
√a −
Then one obtains
1 1 1 1
β = +ǫ and ρ = √a+ǫ√a+bt
−2t √a √a+2bt 2t
(cid:18) (cid:19)
(cid:16) (cid:17)
where ǫ = 1.
±
We have the almost complex structure J
A
1 1 1 1
J XH = XV +ǫ g(X,u)uV
A
(9) √a − 2t √a √a+2bt
 1(cid:18) (cid:19)
 J XV = √aXH + √a+ǫ√a+2bt g(X,u)uH
 A
− 2t
(cid:16) (cid:17)
and the almost Hermitian manifold (T(M),g ,J ).
A A
Remark 2.4 In this general case J is defined on T(M) 0 (the bundle of non zero
A
\
tangent vectors), but if we consider ǫ = 1 the previous relations define J on all T(M).
A
−
Remark 2.5 If we take ǫ = 1, a= 1 and b = 0 we get the manifold (T(M),g ,J ) and
S S
−
for ǫ = 1, a= b = 1 we obtain the manifold (T(M),g ,J ).
− 1+2t CG CG
If we denote by Ω the Ka¨hler 2-form (i.e. Ω (U,V) = g (U,J V), U,V χ(T(M)))
A A A A
∀ ∈
one obtains
Proposition 2.6 (see [7]) The almost Hermitian manifold (T(M),g ,J ) is locally con-
A A
formal almost Ka¨hlerian, that is
(10) dΩ =ω Ω
A A
∧
where ω is a closed and globally defined 1 form on T(M) given by
−
1 a 1
ω(XH) = 0 and ω(XV)= ′ + (√a+ǫ√a+2bt) g(X,u).
√a √a 2t
(cid:18) (cid:19)
As consequence one can state the following
Theorem 2.7 The almost Hermitian manifold (T(M),g ,J ) is almost Ka¨hlerian if and
A A
only if
2a(t)(ta(t)+a(t))
′ ′
b(t)=
a(t)
and for ǫ = 1, a(t) is an increasing function, while for ǫ = +1, ta(t) is a decreasing
−
function.
6
Proof. The condition ω = 0 is equivalent to
2ta(t)+a(t)= ǫ a(t) a(t)+2tb(t) .
′
− ·
p p
From here, we get b(t). Moreover it follows (a(t)√t is a monotone function, namely it is
increasing if ǫ = 1 and decreasing for ǫ = +1. Since b(t) is positive we conclude
−
if ǫ = 1: 2at+a> 0 2(at+a)> a at+a > 0
′ ′ ′
• − ←→ −→
a > 0 a increases (this implies a√t, at are also increasing functions);
′
−→ −→
if ǫ = +1: 2at+a< 0 at+a< at at+a< 0
′ ′ ′ ′
• ←→ − −→
at decreases (this implies a√t, a are also decreasing functions).
−→
The integrability of J .
A
In order to have an integrable structure J on T(M) we have to compute the Nijenhuis
A
tensor N of J and to ask that it vanishes identically.
JA A
For the integrability tensor N we have the following relations
JA
NJA(XH,YH) = −2aa′2 + aa+√taa′ A(t) g(X,u)Y −g(Y,u)X V +(RXYu)V
(11)  NJA(XV,YV1) =a(cid:16)′−+aRBX(tY)u+g(√Ya,uB)X(cid:17)(t(cid:0))g(gY(,Xu,)Ru)XYuuV−√a B(cid:1)(t)RYuu V−
−√a 2√(cid:0)a − (cid:1)
 (cid:16) (cid:17)(cid:0) (cid:1)

where A(t) = 1 1 +ǫ 1 and B(t) = 1 √a+ǫ√a+2bt . (The expression for
2t √a √a+2bt 2t
N (XH,YV) is v(cid:16)ery complicate(cid:17)d.)
JA (cid:0) (cid:1)
Thus if J is integrable then
A
a a+ta
R u = ′ + ′ A(t) g(Y,u)X g(X,u)Y
XY
−2a2 a√a −
(cid:18) (cid:19)
(cid:0) (cid:1)
for every X,Y χ(M) and for every point u T(M). It follows that M is a space form
∈ ∈
M(c) (c is the constant sectional curvature of M). Consequently,
a a+ta
′ ′
(12) + A(t) = c.
−2a2 a√a
So
X given a(t) and c we can easily find b(t);
X given b(t) and c we have to solve an ODE in order to find a(t);
X given a(t) and b(t) we have to check if c in (12) is constant.
Example 2.8
1. In Sasaki case (a(t) = 1,b(t) = 0,ǫ = 1) it follows c = 0 i.e. M is flat.
−
7
2. Looking for a locally conformal Ka¨hler structure on T(M) with the metric having
a(t) = b(t) we obtain
e2√1+2t
a(t) = b(t) =
2 ce2√1+2tt+(1+t+√1+2t)k
(cid:16) (cid:17)
with k a positive real constant and c must be nonnegative.
Replacing the expression of the curvature R in (11) we obtain again (12).
2
Question: Can (T(M),g ,J ) be a Kaehler manifold?
A A
If this happens then the base manifold is a space form M(c) and the functions a and b
satisfy
2a(ta +a)
′ ′
(13) b = and
a
(14) a = 2ca(2ta +a).
′ ′
If c =0 (M is flat) then a is a positive constant and b vanishes.
If c =0 the ODE (14) has general solutions
6
1 √1+κt
(15) a (t) = ±
1,2
4ct
with κ a real constant. Taking into account that a and b are positive functions, using (13)
one gets:
Case 1.
1+√1+κt κ(1+√1+κt)
(16) a= and b = .
4ct − 8ct(1+κt)
Here c> 0, t > 0, κ < 0, t < 1 and ǫ = +1.
−κ
Case 2.
κ κ2
(17) a= and b = .
−4c(1+√1+κt) 8c(1+κt)(1+√1+κt)
Here κc < 0, c< 0 (then κ > 0), t < 1 and ǫ = 1.
−κ −
Consider Bκ = v ∈T(M) : gτ(v)(v,v) < −κ2 and B˙κ = Bκ\M.
(cid:8) (cid:9)
Theorem 2.9 The manifolds B in Case 1 and B˙ in Case 2 are Kaehler manifolds.
κ κ
8
Remark 2.10 In order to have a positive definite metric g , the necessary and sufficient
A
conditions are a > 0 and a+2bt > 0 (b > 0 is too strong). Hence, the previous theorem
can be reformulated as:
1. (T(M) M,g ,J ) where a and b are given by (16), c > 0 and ǫ = +1 is a Kaehler
A A
\
manifold.
2. (B ,g ,J ) where a and b are given by (17), c > 0, k < 0 and ǫ = 1 is a Kaehler
κ A A
−
manifold.
3. (T(M),g ,J ) where a and b are given by (17), c < 0, k > 0 and ǫ = 1 is a Kaehler
A A
−
manifold.
Now we give
Proposition 2.11 Let (M,g) be a Riemannian manifold and let T(M) be its tangent
bundle equipped with the metric g . Then, the corresponding Levi Civita connection ˜A
A
∇
satisfies the following relations:
˜A YH = ( Y)H 1 (R u)V
∇XH ∇X − 2 XY

 ∇˜AXHYV = (∇XY)V + a2 (RuYX)H
(18) 


 ∇˜AXVYH = a2 (RuXY)H
 ˜A YV = L g(X,u)YV +g(Y,u)XV +Mg(X,Y)uV +Ng(X,u)g(Y,u)uV,
 ∇XV


 (cid:0) (cid:1)
where L= a′(t), M = 2b(t)−a′(t) and N = a(t)b′(t)−2a′(t)b(t).
2a(t) 2(a(t)+2tb(t)) 2a(t)(a(t)+2tb(t))
Proof. The statement follows from Koszul formula making usual computations.
HavingdeterminedLeviCivitaconnection,wecancomputenowtheRiemanniancurvature
tensor R˜A on T(M). We give
9
Proposition 2.12 The curvature tensor is given by
R˜XAHYHZH = (RXYZ)H + a4 [RuRXZuY −RuRYZuX +2RuRXYuZ]H+
+1 [( R) u]V
 2 ∇Z XY


 R˜XAHYHZV =+(cid:2)RMXg(YRZX+Yua4,Z(R)uYVR+uZXa2u[−(∇RXXRR)uuZZYYu)−(cid:3)V(∇+YLRg()ZuZ,Xu)](HRXYu)V+




 R˜XAHYVZH =+a2 [1(∇RXR)YuYZ]aHR+ u+Lg(Y,u)R u+Mg(R u,Y)u V
 2 XZ − 2 XRuYZ XZ XZ

(19)  R˜XAHYVZV = −a2 (cid:2)(RYZX)H − a42 (RuYRuZX)H+ (cid:3)
+a4′ g(Z,u)(RuYX)H −g(Y,u)(RuZX)H


 R˜XAVYVZH =a+(aR42X(cid:2)[RYuZX)RHu+YZa2′−[gR(XuY,uR)uRXuZY]ZH−g(Y,u)R(cid:3)uXZ]H+




 R˜A ZV = F (t)g(Z,u) g(X,u)YV g(Y,u)XV +
 XVYV +1F (t) g(X,Z)YV g(−Y,Z)XV +
 +F23(t)(cid:2)[g(X,(cid:2)Z)g(Y,−u)−g(Y,Z)g(cid:3)(X,(cid:3)u)]uV,


where F = L L2 N(1+2tL), F = L M(1+2tL) and F = N (M +M2+2tMN).
1 ′ 2 3 ′
− − − −
Remark 2.13
(a) In the case of Sasaki metric we have:
L = M = N = 0 , F = F =F = 0.
1 2 3
(b) In the case of Cheeger Gromoll metric we have (see also [12, 22]):
L = 1 , M = r+1 , N = 1 , L = 2 , M = 2(r+2) , 1+2tL = 1
−r r2 r2 ′ r2 ′ − r3 r
F = r 1 , F = r2+r+1 , F = r+2
1 −r3 2 − r3 3 r3
where r =1+2t.
In the following let Q˜A(U,V) denote the square of the area of the parallelogram with sides
U and V for U,V χ(T(M)),
∈
Q˜A(U,V) = g (U,U)g (V,V) g (U,V)2.
A A A
−
We have
Lemma 2.14 Let X,Y T M be two orthonormal vectors. Then
p
∈
Q˜A(XH,YH) = 1
(20) Q˜A(XH,YV) = a(t)+b(t)g(Y,u)2

 Q˜A(XV,YV)= a(t)2 +a(t)b(t) g(X,u)2 +g(Y,u) .
 (cid:0) (cid:1)
10