We study positive definite quaternionic contact (4n+3)-manifolds (qc-manifold for short). Just like the CR-structure contains the class of Sasaki manifolds, the qc-structure admits a class of 3-Sasaki manifolds with integrable distribution isomorphic to su(2). A big difference concerning the integrable complementary qc-distribution V of the qc-structure from 3-Sasaki structure is the existence of Lie algebra not isomorphic to su(2). We take up non-compact qc-manifolds to find out a salient feature of topology and geometry in case V generates the qc-transformations R3.
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TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Black Holes and Theoretical Physics
We study positive definite quaternionic contact (4n+3)-manifolds
(qc-manifold for short).
Just like the CR-structure contains the class of Sasaki manifolds,
the qc-structure admits a class of
3-Sasaki manifolds with integrable distribution isomorphic to su(2).
A big difference concerning the integrable complementary qc-distribution
V of the qc-structure from 3-Sasaki structure
is the existence of Lie algebra not isomorphic to su(2).
We take up non-compactqc-manifolds to find out a
salient feature of topology and geometry in case
V generates the qc-transformations R3.
It is known that the quaternionic contact structure (qc-structure for short)
on a 4n+3-dimensional manifold (M,D)
contains a class of 3-Sasaki manifolds
as a quaternionic CR-structure.
In this case the complementary integrable distribution V is locally
isomorphic to su(2) (≅sp(1)).
In order to obtain the Reeb field as in CR-structure,
the qc-automorphism group Autqc(M)
is relatively too big to act properly on M. Like as the
pseudo-Hermitian transformation subgroup extracted from
the CR-transformation group of a CR-manifold,
we may find a subgroup Pshqc(M) from Autqc(M).
V is said to the qck-distribution
if it generates a 3-dimensional subgroup of Pshqc(M).
The idea of reduction to this group was performed in our paper
[5] by using the vanishing of equivariant smooth
cohomology groups. Geometrically this means,
given an ImH-valued
1-form ω representing the qc-structure on M,
there is a smooth function v∈C∞(M,R+) such that
the one-form η=v⋅ω gives rise to
the subgroup Pshqc(M,η)≤Autqc(M).
With the aid of the work [29], the following theorem is obtained
along the same method of [5] which
clarifies how
Pshqc(M) interacts the qc-structure.
(See Theorem 3.5.)
Theorem A**.**
Let (M,(D,{Jα}α=13)) be a
positive definite qc-manifold.
Then either one of the following holds:
(i)
There exists a positive definite
qc-structure (η,{Jα}α=13) with D=kerη
such that
[TABLE]
2. (ii)
M* has a spherical qc-structure isomorphic to
either the standard sphere S4n+3 or the quaternionic Heisenberg Lie group M.*
In case (ii),
(Pshqc(M),Autqc(M),M) is exactly the following:
[TABLE]
A qc-manifold (X,D) with kerω=D
is positive definite if the Levi form
dωα∘Jα:D×D⟶R defined by
dωα∘Jα(x,y)=dωα(Jαx,y) is a
positive definite symmetric bilinear form (α=1,2,3).
A qc-manifold may be assumed to be positive definite throughout this paper.
A qc-manifold X with the qck-distribution V
generalizes the notion of 3-Sasaki (quaternionic CR-) manifold
(with the Killing Reeb fields), that is V generates R≤Pshqc(X),
called the qck-group.
Then
[TABLE]
where NPshqc(X)(R)
is the normalizer of R in Pshqc(X).
(See Corollary 4.3, also [4].)
In case M is a compact qc-manifold with a qck-group
T≤Pshqc(M), T is locally isomorphic to Sp(1) or
T is isomorphic to the toral group T3 (cf. Proposition 4.5).
For qCR-manifolds (3-Sasaki manifolds),
see [9], [15].
In general see [17] for compact qc-manifolds and
[7], [18], [19] for the work on the qc-structure
and the references therein.
In this paper we study
non-compact positive definite qc-manifolds X with a non-compact qck-group R,
mainly R=R3. (Compare Proposition 5.1.)
We discuss Riemannian submersions obtained from X.
Compare Theorem 5.4 (also Proposition 5.6).
Theorem B**.**
Let X be a simply connected
non-compact qc-manifold with an ImH-valued 1-form η and
the qck-distribution V
generating R3≤Pshqc(X).
Then there is a principal Riemannian
submersion R3→X⟶pY where (Y,Ω) is a simply connected hyperKähler
manifold such that p∗Ω=dη.
For each R≤R3, the quotient
X/R is a complex contact manifold which
admits a holomorphic principal bundle over
a hyperKähler manifold Y=X/R3:
[TABLE]
For each R2≤R3,
the quotient X/R2 is
a strictly pseudoconvex CR-manifold admitting
a pseudo-Hermitian (Sasaki) bundle:
[TABLE]
*Similarly suppose M is a compact qc-manifold X/Γ
whose qck-group T3 lifts to an R3-action to X. Then
(i) For each S1≤T3,
the quotient oribifold M/S1
supports a complex contact structure such that
M/S1 is the holomorphic orbibundle
over the hyperKähler orbifold Z=M/T3:*
[TABLE]
(ii)* For a torus T2≤T3,
the quotient orbifold M/T2
admits a strictly pseudoconvex CR-structure. Furthermore
this gives the pseudo-Hermitian (Sasaki ) orbibundle:*
[TABLE]
(iii)* The fundamental group Γ=π1(M)
to (1.3)
induces a nontrivial group extension:
1→Z2→Γ/Z⟶Q→1 where Y/Q=Z.
Γ also assigns to (1.4)
a nontrivial group extension:
1→Z→Γ/Z2⟶Q→1.
Then (1.3)
is a nontrivial TC1-orbibundle.*
In case X is a simply connected qc-manifold
with the qck-group R3,
the qc-Hermitian group (respectively
hyperKähler group) can be described exactly
as \displaystyle\mathop{\rm Psh}\nolimits_{qc}(X,\omega,\{J_{\alpha}\}_{\alpha=1}^{3})=\bigl{\{}f\in\mathop{\rm Diff}\nolimits(X)\,\big{|}\ f^{*}\omega=a\cdot\omega\cdot\bar{a}\,,\ a\in\mathop{\rm Sp}\nolimits(1)\bigl{\}},
\displaystyle\mathop{\rm Isom}\nolimits_{hK}(Y,\Omega,\{{\sf J}_{\alpha}\}_{\alpha=1}^{3})=\bigl{\{}h\in\mathop{\rm Diff}\nolimits(Y)\,\big{|}\ h^{*}\Omega=b\cdot\Omega\cdot\bar{b}\,,\ b\in\mathop{\rm Sp}\nolimits(1)\big{\}}.
In general note that a,b are smooth functions on X, Y respectively.
(Compare Corollary 8.3 for the precise description.)
The following concerns
the structure of Pshqc(X) (cf. Corollary 8.4).
Proposition C**.**
Let X be a simply connected qc-manifold
with qck-group R3.
There is a natural exact sequence:
[TABLE]
For example if X=M is a quaternionic Heisenberg
Lie group as a qc-homogeneous manifold,
then it follows
Pshqc(X)=M⋊Sp(n)⋅Sp(1) where Y is
the quaternionic space Hn
such that IsomhK(Y)=Hn⋊Sp(n)⋅Sp(1).
Given a qCR-manifold X, the product
R+×X with the cone metric
is known to admit an (imcomplete) hyperKähler structure (cf. [3] for instance).
This construction cannot be applied to qc-manifolds X with
the qck-group R3 (see Note 7.3).
However we shall construct a hyperKähler metric on R×X
suitable for any qc-manifold X with qck-group R3.
(Compare Proposition 7.1.) Applying this construction to
the quaternionic Heisenberg Lie group M, we obtain
Theorem D**.**
There is a complete hyperKähler metric g0 on the quaternionic space
Hn+1(n≥1)
such that the quaternionic isometry group is
IsomhK(Hn+1,g0)=H⋊(Sp(n)×Sp(1)).
In particular, g0 is not equivalent to the standard quaternionic metric
of Hn+1 up to a quaternionic isometry.
Our next purpose is to classify spherical homogeneous qc-manifolds.
Definition 1.1**.**
A 4n+3-dimensional positive definite qc-manifold M is spherical
(or uniformizable)
if it is locally modeled on S4n+3
with coordinate changes lying in PSp(n+1,1).
Equivalently there exists a qc-developing map of the universal covering M~
to S4n+3.
Compare [19].
The classification of homogeneous qc-manifolds is
a difficult subject in its own right.
The spherical homogeneous qc-manifolds
are comparatively nontrivial examples appropriate to Proposition C.
(Refer to [10] for the spherical CR case.) See Theorem 9.6
(cf. Propositions 9.2, 9.5).
Theorem E**.**
Any simply connected spherical homogeneous qc-manifold M is qc-isomorphic to
S4n+3, S4n+3−S4m−1(1≤m≤n)
or M, X(k), X(0)(1≤k≤n).
In particular each one of M, X(k) or X(0) is a principal qc-bundle
over a domain of Hn with qck-fiber R3.
It is important to verify the geometric properties of G-structure on qc-manifolds
(cf. [7]).
It is worthwhile pointing out
the vanishing theorem of Biquard’s quaternionic conformal geometry
obtained by S. Ivanov and D. Vassilev [19].
They proved the vanishing of a qc-conformal (curvature) tensor implies
a qc-manifold is spherical, which eventually arrives at Theorem A
(followed by the works of [29], [13]).
Associated with the canonical Riemannian metric g of (3.6),
we simply calculate the curvatures of
positive definite qc-manifolds with qck-group R3
by applying O’Neill’s formula to the fiberings of Theorem B.
Then X is no longer an Einstein manifold unlike a 3-Sasaki manifold.
Compare Propositions 10.1, 10.2.
Finally we prove the geometric uniqueness of qc-manifolds X
with qck-group R3 concerning the model quaternionic
Heisenberg Lie group M under several conditions.
This assertion is brought together
by Propositions 10.4, 10.5,
10.6, 10.7.
(See also [21, Theorem 4.4].)
The paper is organized as follows.
In Section 2, we prepare several basic facts of qc-structures,
especially the equivalence relation of qc-conformal change.
Section 3 is a review of the qc-conformal invariant
(cf. [4]) whose vanishing on a qc-manifold M reduces the qc-automorphism group
Autqc(M,ω) to
the qc-Hermitian group Pshqc(M,η).
In Section 4 we introduce a qck-distribution on qc-manifolds.
In general there is a complementary distribution V to
the qc-subbundle D of a qc-manifold M. We study
a qc-manifold whose distribution generates
a qck-group R in Pshqc(M).
In Section 5, we discuss fiberings on a positive definite
qc-manifold X with qck-group R3.
Section 6 reviews the the standard qc-structure D0 on
the quaternionic Heisenberig Lie group M.
In Section 7 we construct a complete hyperKähler metric
on the product R×X for a complete simply connected
noncompact qc-manifold X.
In Section 8,
we determine the structure of Pshqc(X) when X has the qck-group
R3.
It may be useful to determine Pshqc(X)
for several qc-manifolds like homogeneous spaces.
In Section 9, we classify spherical homogeneous qc-manifolds.
In Section 10, we calculate Ricci curvatures on
qc-manifolds X with qck-group R3
using the associated Riemannian metric.
2. Quaternionic contact group
This section is mainly concerned with the equivalence of quaternionic contact structures.
Let H be the field of quaternions {1,i,j,k}.
A quaternionic contact structure
is a codimension 3-subbundle D on
a 4n+3-dimensional smooth manifold X such that
D with
[D,D] generates TX. Moreover the following conditions are required:
There exists a non-degenerate ImH-valued 1-form
ω=ω1i+ω2j+ω3k which represents D, that is
kerω=∩α=13kerωα=D
such that
ω∧ω∧ω∧dω∧⋯∧dωn=0 on X.
The 1-form ω is said to be a quaternionic contact form on X.
Then the bundle of
endomorphisms {J1,J2,J3}
defined by
[TABLE]
constitutes a hypercomplex structure on D,
that is Jα2=−1, JαJβ=Jγ.
The Levi form
dωα∘Jα:D×D⟶R
is a positive definite symmetric bilinear form on D (cf. Introduction).
Then (X,D,ω,{Jk}k=13) is called a
positive definite quaternionic contact manifold
(qc-manifold for short). See [7], [17],
[2] for the definition and the reference therein.
Choose an ImH-valued 1-form ω such that D=kerω.
There is no canonical choice of ω representing D.
If ω′ represents D, then it is easy to see that
there is a map v⋅b:X→H∗=R+×Sp(1)
such that ω′=v⋅bωbˉ.
Definition 2.1**.**
Two ImH-valued 1-forms ω, ω′ are
qc-conformal if ω′=v⋅bωbˉ
for some map v⋅b∈C∞(X,R+×Sp(1)).
A quaternionic contact transformation is a diffeomorphism
α:X→X preserving D and the hypercomplex structure {Jα}α=13.
More precisely, choose an ImH-valued 1-form ω such that D=kerω.
The quaternionic contact groupAutqc(X)=Autqc(X,D,{Jk}k=13) is defined by
[TABLE]
for some map
λα=uα⋅aα∈C∞(X,R+×Sp(1)) and
the matrix (akj)∈C∞(X,SO(3))
is given by the conjugate of aα on ImH,
that is z→aα⋅z⋅aα.
This definition may depend on the choice of
ω.
Suppose ω′=v⋅b⋅ω⋅bˉ(∃v⋅b:X→R+×Sp(1)).
The matrix (bij):X→SO(3) is defined by the conjugation
z↦b⋅z⋅bˉ.
Then the quaternionic structure {J1′,J2′,J3′}
is obtained as
[TABLE]
Lemma 2.2**.**
Let \displaystyle B=\left[\begin{array}[]{ccc}b_{11}&b_{12}&b_{13}\\
b_{21}&b_{22}&b_{23}\\
b_{31}&b_{32}&b_{33}\\
\end{array}\right]\in C^{\infty}(X,{\rm SO}(3)) defined by the conjugate of b∈C∞(X,Sp(1).
Then
[TABLE]
Proof.
Since dωβ′=v∑jbjβdωj on D as above,
using (2.3), it follows
v∑jbjβdωj(Jγ′u,v)=v∑jbjαdωj(u,v). Thus
[TABLE]
Noting dω1∘J1=dω2∘J2=dω3∘J3,
(2.5) is described:
[TABLE]
By the non-degeneracy of dω1∘J1 for any v∈D, we obtain
[TABLE]
Equivalently
[TABLE]
Multiply (2.7) by
J1b1β,J2b2β,J3b3β respectively:
[TABLE]
Sum up these equations.
[TABLE]
Noting α=β,
b1β2+b2β2+b3β2=1,b1βb1α+b2βb2α+b3βb3α=0,
it follows
[TABLE]
As (α,β,γ)∼(1,2,3),
we obtain the following equation from (2.8):
[TABLE]
Represent tB=[b1′,b2′,b3′] as
column vectors. The above equation turns to
Autqc(X,D,ω,{Jk}k=13)=Autqc(X,D,ω′,{Jk′}k=13)*
whenever ω′=v⋅b⋅ω⋅bˉ, that is, it is uniquely determined by
the qc-conformal class of ω.
Henceforth the definition of Autqc(X,D) makes sense.*
Proof.
Let α∈Autqc(X,D,ω,{Jk}k=13).
By the definition of (2.2) for ω,
(1)
α∗ω=uα⋅aα⋅ω⋅aα.
2. (2)
α∗[J1J2J3]=aα[J1J2J3]aα⋅α∗.
From Lemma 2.2 note
[J1′J2′J3′]=b[J1J2J3]bˉ.
Suppose ω′=v⋅b⋅ω⋅bˉ.
Then α∗ω′=α∗v⋅α∗b⋅α∗ω⋅α∗b=(α∗v⋅uα⋅v−1)⋅(α∗b⋅aα⋅bˉ)⋅ω′⋅(α∗b⋅aα⋅bˉ).
[TABLE]
Noting b⋅aα⋅α∗b=α∗b⋅aα⋅bˉ,
this implies α∈Autqc(X,D,ω′,{Jk′}k=13) by the definition.
Conversely let α∈Autqc(X,D,ω′,{Jk′}k=13).
(1’)
α∗ω′=uα′⋅aα′⋅ω′⋅aα′.
2. (2’)
α∗[J1′J2′J3′]=aα′[J1′J2′J3′]aα′⋅α∗.
Similarly as above,
[TABLE]
[TABLE]
Hence
α∈Autqc(X,D,ω,{Jk}k=13). It follows
[TABLE]
∎
3. Conformal invariant
Let Autqc(X)=Autqc(X,D,{Jα}α=13).
If C∞(X,R+) is the module
consisting of smooth functions of X to the positive numbers R+,
then it is
endowed with an action of α∈Autqc(X).
For f∈C∞(X,R+),
[TABLE]
Thus C∞(X,R+) is a smooth G-module (Compare [4].)
For a form ω=ω1i+ω2j+ω3k, denote the norm as
∣ω∣=ω12+ω22+ω32=⟨ω,ω⟩.
Let G be a closed subgroup of Autqc(X).
There is a cocycle μqc∈Hd1(G,C∞(X,R+))
which is a qc-conformal invariant.
Proof.
Let α∈Autqc(X) such that α∗ω=uα⋅aαω⋅aα.
For α,β∈Autqc(X), calculate
[TABLE]
(Note that β∗aα(x)=aα(βx)=aα(βx)=β∗aα(x)).
Since each aγ(γ∈G) is a map from X
to Sp(1), we have
∣(αβ)∗ω∣=uαβ∣ω∣=β∗uα⋅uβ∣ω∣
by the above equation.
Thus the smooth maps uα,uβ,uαβ:X→R+ satisfy
In particular, λω(α)(x)=uα(α−1x).
It suffices to show λω is a crossed homomorphism.
Calculate
[TABLE]
Hence
λω(αβ)=λω(α)⋅α∗λω(β),
that is λω is a crossed homomorphism so that
[λω]∈Hd1(G,C∞(X,R+)).
We show [λω] is a quaternionic conformal invariant.
Suppose ω′=u⋅bωbˉ for some positive function u on X and
a map b:X→Sp(1).
For α∈G, let
α∗ω′=uα′⋅aα′ω′aα′.
Then α∗ω′=uα′⋅aα′u⋅bωbˉaα′=(uα′⋅u)⋅(aα′⋅b)ω⋅(aα′⋅b),
while
[TABLE]
Taking the norm ∣α∗ω′∣, it follows
uα′⋅u=α∗u⋅uα, that is
uα′=α∗u⋅u−1⋅uα
on X.
For α−1x∈X, as uα′(α−1x)=α∗u(α−1x)⋅u−1(α−1x)⋅uα(α−1x), it follows α∗uα′(x)=u(x)⋅(α∗u)−1(x)α∗uα(x), that is
α∗uα′(x)⋅(α∗u)(x)⋅u(x)−1=α∗uα(x)(∀x∈X). This shows
α∗uα′⋅δ0(u)(α)=α∗uα.
Hence
[λω]=[λω′] and so
[λω] is a quaternionic conformal invariant.
We may put
[TABLE]
∎
Remark 3.2**.**
(i)* It is shown in [4] that
Hdi(G,C∞(X,R+))=0(i≥1)
provided that G acts properly on X. In particular, μqc=0.
(ii) Since we evaluate the norm of ω,
λω is in fact a conformal invariant
whenever ω′=u⋅ω.*
Definition 3.3**.**
The qc-Hermitian group is denoted by
[TABLE]
where
aα∈C∞(X,Sp(1)) induces
(aij)∈C∞(X,SO(3)) as
its conjugate.
Using the equality
dω1∘J1=dω2∘J2=dω3∘J3 on D (cf. (4.3), also [2]),
each α∈Pshqc(X,ω,{Jk}k=13) satisfies
[TABLE]
[TABLE]
Since dω1∘J1:D×D→R
defined by dω1(J1a,b)
is positive definite in our case,
a qc-manifold (X,D(=kerω),{Jα}α=13)
assigns to D a Riemannian metric (cf. (4.3))
[TABLE]
The isometry group Isom(X,gω) is not related to
Autqc(X) in general.
Proposition 3.4**.**
Pshqc(X,ω,{Jα}α=13)≤Isom(X,gω).
Then Pshqc(X,ω) acts properly on X.
When X is compact, Pshqc(X,ω,{Jα}α=13) is a
compact Lie group.
Proof.
If α∈Pshqc(X,ω,{Jα}α=13), then
α∗ω=aα⋅ω⋅aα by the definition.
As ∑i=13ωi⋅ωi=−ω⋅ω∈R,
it follows α∗ω⋅α∗ω=aα⋅ω⋅aα⋅aα⋅ω⋅aα=ω⋅ω.
By (3.5),
α∗(dω1∘J1)=dω1∘J1.
This implies α∗g=g such that
Pshqc(X,ω,{Jα}α=13)≤Isom(X,gω).
∎
Theorem 3.5**.**
Let G be a closed subgroup of
Autqc(X,D,{Jk}k=13).
Then μqc=0 in Hd1(G,C∞(X,R+))
if and only if G acts properly on X.
In that case, there is an ImH-valued 1-form
ηconformal to ω, such that
[TABLE]
Proof.
Suppose μqc=[λω]=0 in Hd1(G,C∞(X,R+))
for an ImH-valued 1-form ω.
Then λω=δ0v for v∈C∞(X,R+).
It follows α∗uα(x)=α∗v(x)v(x)−1(∀α∈G).
So uα(α−1x)v(x)=v(α−1x). In particular,
uα(x)v(αx)=v(x)(∀x∈X),
that is uα⋅α∗v=v.
If α∈G, then α∗ω=uα⋅aα⋅ω⋅aα as above.
Put
[TABLE]
Then
α∗η=α∗v⋅uα⋅aα⋅ω⋅aα=v⋅aα⋅ω⋅aα=aα⋅η⋅aα.
Moreover,
if {Jj′}j=13 is a hypercomplex structure for η=η1i+η2j+η3k,
then dηi(Jj′a,b)=dηk(a,b)(∀a,b∈D) by the definition.
It implies vdωi(Jj′a,b)=vdωk(a,b)
on D, thus dωi(Jj′a,b)=dωk(a,b)=dωi(Jja,b). Replace b by Jib.
The non-degeneracy of dωi∘Ji on D(i=1,2,3)
implies Jj′=Jj(j=1,2,3).
Hence
α∈Pshqc(X,η,{Jα}α=13),
that is G≤Pshqc(X,η,{Jα}α=13).
By Proposition 3.4, Pshqc(X,η,{Jα}α=13)
acts properly on X, so does G .
Conversely, if G acts properly on X, then it follows from [5, Theorem 14] that
Hd1(G,C∞(X,R+))=0.
In particular, μqc=0 by Theorem 3.1.
∎
4. 3-dimensional complementary distribution V
When (X,ω,{Jα}α=13) is a qc-manifold,
there is a qc-structure (η,{Jα}α=13)qc-conformal to ω by Theorem A. Let η=η1i+η2j+η3k be the 1-form
with D=kerη.
As dηα:D×D→R is non-degenerate,
there exist three vector fields
ξα (cf. [7]) such that
[TABLE]
Put V={ξ1,ξ2,ξ3} associated with η.
Then the above equations determine V uniquely.
Definition 4.1**.**
Let V be the 3-dimensional distribution
on a qc-manifold (X,η,{Jα}α=13)
with D=kerη. (cf. \eqreffix).
If V generates a subgroup R of Pshqc(X,η,{Jα}α=13),
then V is said to a
qck-distribution on X associated with η. R is called a qck-group.
Suppose V is a qck-distribution on X. Noting
[ξα,u]∈D(α=1,2,3), the above equation shows
[TABLE]
There is a decomposition:
[TABLE]
Denote Jαξβ=0(α,β=1,2,3) as usual.
Let x=∑αaαβξα+u,y=∑αbαβξα+v as above.
For any such x,y∈TX, it follows
dηα(Jαx,y)=dηα(Jαu,v)
by (4.1).
In particular, from (2.1)
[TABLE]
Proposition 4.2**.**
Let V={ξ1,ξ2,ξ3} be the
3-dimensional qck-distribution on X.
If h∈Pshqc(X,D,η,{Jα}α=13), then
(i)
dη1(J1h∗x,h∗y)=dη1(J1x,y)* (∀x,y∈TX).*
2. (ii)
h∗V=V.
Proof.
(1) First similarly to the equation (3.5), we obtain
[TABLE]
Then we prove (ii) h∗V=V.
By (1),
dη1(J1h∗u,h∗ξα)=dη1(J1u,ξα)=0 for ∀u∈D.
Suppose h∗ξα=∑bαβξβ+v
for some v∈D.
Since dη1(J1h∗u,h∗ξα)=dη1(J1h∗u,∑bαβξβ+v)=dη1(J1h∗u,v),
the non-degeneracy
of dη1∘J1 on D implies v=0.
Hence h∗ξα∈V.
We prove (i) for any x,y∈TX.
If x=∑aαβξα+u as above, then
h∗x=∑aαβh∗ξα+h∗u where
h∗ξα∈V by (ii). Thus
dη1(J1h∗x,h∗y)=dη1(J1h∗u,h∗y)=dη1(J1u,y)=dη1(J1x,y).
∎
In particular, there is an analogy of
the Killing (Reeb) fields of Sasaki CR-manifolds to qc-manifolds from Proposition 4.2.
We may put Pshqc(X)=Pshqc(X,D,η,{Jα}α=13)
for short. By Proposition 4.2 it follows
Corollary 4.3**.**
Suppose a qck-distribution V generates a subgroup R≤Pshqc(X).
If NPshqc(X)(R)
is the normalizer of R in Pshqc(X), then
[TABLE]
Remark 4.4**.**
Even though ω generates the qck-distribution,
any ηqc-conformal to ω does not necessarily generate a qck-distribution.
Suppose a smooth 4n+3-dimensional compact manifold M admits
a qc-structure (D,{Jα}α=13).
By Theorem A, there exists
an ImH-valued 1-form η such that
kerη=D. Then
Autqc(M,D,{Jα}α=13)=Pshqc(M,η,{Jα}α=13)
unless M is qc-conformal to S4n+3.
Suppose V is a qck-distribution on M.
V generates a qc-subgroup T≤Pshqc(M) (cf. (2) Definition 4.1).
As Pshqc(M) is a compact Lie group by Proposition 3.4,
so is the closure Tˉ of T.
Since Tˉ is connected,
it follows from (4.2) that
Tˉ=T.
Then T is isomorphic to either Sp(1)
or T3. The compact case falls into the following dichotomy.
Proposition 4.5**.**
Let (M,η,{Jα}α=13) be a compactqc-manifold with the qck-distribution V={ξ1,ξ2,ξ3}.
If T≤Pshqc(M) is
the subgroup generated
by V, then either one of the following
occurs exactly:
(1)
T* is isomorphic to Sp(1)≤Pshqc(M).
M is a quaternionic CR-manifold (3-Sasaki manifold ) for which
V
coincides with the common kernel of ρα=dηα+2ηβ∧ηγ(α=1,2,3), (α,β,γ)∼(1,2,3);*
[TABLE]
2. (2)
T* is isomorphic to T3≤Pshqc(M).
M is a T3-orbibundle over a hyperKähler orbifold Z.
In this case,*
[TABLE]
Proof.
(1) If T=Sp(1), then V={ξα,ξβ,ξγ}((α,β,γ)∼(1,2,3))
satisfies [ξα,ξβ]=2ξγ.
If kerρα={ξ∈TM∣ρα(ξ,x)=0∀x∈TM},
then V=∩α=13kerρα. M is
a quaternionic CR-manifold (cf. [2]).
(2) If T=T3, then
V={ξ1,ξ2,ξ3} is the commutative algebra and so
dωγ(ξα,x)=0(∀x∈TM).
If p:M→Z=M/T3 is the projection, then this equation gives a hyperKähler form Ωα on the orbifold Z such that p∗Ωα=dωα,
(see (5.9) for the detail, also [21]).
∎
Remark 4.6**.**
If M is a qCR(3-Sasaki) manifold, then
for each S1≤S3
the orbifold M/S1 is the twistor space over the quaternionic Kähler orbifold
S2→M/S1⟶M/S3. Moreover,
it is shown in [9], [15]
that M/S1 is a
Kähler-Einstein orbifold of positive scalar curvature.
For (2), M is a qc-Einstein manifold with zero qc-scalar curvature
by the result of [18].
5. Non-compact qc-manifold with qck-distribution R3
Since a qc-manifold X is non-compact simply connected in our case,
we may assume the 3-dimensional qck-subgroup R≤Pshqc(X)
is also a non-compact simply connected Lie group.
From Definition 3.3,
r∗η=ar⋅η⋅ar for r∈R
where ar∈C∞(X,Sp(1)) in general. We note
the following.
Proposition 5.1**.**
If each ar is a constant map, that is ar is
an element of Sp(1), then R=R3. In addition
t∗η=η, t∗Jα=Jαt∗ for any t∈R3.
Proof.
The correspondence ν(r)=ar is a homomorphism of R to
Sp(1) ([ar]∈SO(3) if necessary). If R is semisimple, then it is isomorphic to
the universal covering SL(2,R).
As Sp(1) is simply connected, ν:R→Sp(1)
turns to an isomorphism, which is impossible by our hypothesis,
or ν(R)={1}. If R has the nontrivial radical,
then R is solvable. It follows
either ν(R)={1} or ν(R)=S1≤Sp(1). In case
ν(R)=S1,
the identity component of kerν has 2-dimension.
Let V={ξα,ξβ,ξγ} be the qck-distribution which generates R.
We may assume (kerν)0 induces {ξβ,ξγ} for example.
Noting r∗η=η for r∈(kerν)0,
it follows Lξβη=0.
By (4.1), dη(ξβ,x)=0(∀x∈TX).
Taking x=ξα,
η([ξβ,ξα])=0.
Then [ξβ,ξα]∈D=kerη.
Since [ξk,J1v]∈D by (4.1)
(k=α,β,∀v∈D),
Jacobi identity implies
dη1([ξβ,ξα],J1v)=0.
The non-degeneracy of dη1∘J1
shows [ξβ,ξα]=0.
Similarly taking x=ξγ, we have [ξβ,ξγ]=0.
As Lξγη=0 also, [ξγ,ξα]=0 as above.
Hence V generates R=R3 which shows Lξαη=0.
As a matter of fact, ν(R)={1}.
When R is simply connected semisimple, ν(R)={1}
so the above argument applies to show R=R3 which were
impossible.
In addition, let dηβ(Jγ(t∗u),t∗v)=dηα(t∗u,t∗v)(∀u,v∈D) by (2.1).
Since t∗η=η,
dηβ(t∗−1Jγt∗u,v)=dηα(u,v)=dηβ(Jγu,v). The non-degeneracy of dηβ
implies t∗−1Jγt∗u=Jγu,
that is Jαt∗=t∗Jα on D(α=1,2,3).
∎
Remark 5.2**.**
When ar∈C∞(X,Sp(1)) is a nontrivial smooth map,
we do not know whether
a solvable group (=R3) or
SL(2,R)
may occur as R.
We discuss the fiberings of a non-compact simply connected
qc-manifold X with a qck-group R3.
Let TX=V⊕D where V=⟨ξ1,ξ2,ξ3⟩
is the qck-distribution.
Take J1 from {Jα}α=13 on D.
Put E=⟨ξ2,ξ3⟩⊕D.
Define an almost complex structure Jˉ1 on the distribution
E to be
[TABLE]
If E⊗C=E1,0⊕E0,1 is the eigenspace decomposition for Jˉ1,
then E1,0=⟨ξ2−iξ3⟩⊕D1,0.
In order to prove Theorem 5.4 below, we prepare the
following lemma, (the
proofs are essentially the same as those of
[22, Lemma 3.2, Proposition 3.3], [2, Lemma 2.9, Section 2.1].)
Lemma 5.3**.**
The following equality holds.
(i)
For any x,y∈D1,0, there is an element
u∈D⊗C such that
[x,y]=a(ξ2−iξ3)+u
for some a∈R. Furthermore,
2. (ii)
u∈D1,0⊗C, that is
J1u=iu.
Theorem 5.4**.**
Let X be a simply connected non-compact
qc-manifold with the qck-distribution V.
Suppose V generates R3≤Pshqc(X). Then
(1)* For each R≤R3, the quotient
X/R is a complex contact manifold.
There is a holomorphic bundle over
a hyperKähler manifold Y=X/R3:*
[TABLE]
(2)* For each R2≤R3,
the quotient manifold X/R2 is
a strictly pseudoconvex CR-manifold which
has a pseudo-Hermitian (Sasaki)
bundle:*
[TABLE]
Let M be a closed qc-manifold with the qck-group T3.
Suppose T3 lifts to an R3-action to the universal covering of M. Then the following holds.
(1)′* For each S1≤T3,
the quotient oribifold M/S1
supports a complex contact structure such that
M/S1 is the holomorphic orbibundle
over the hyperKähler orbifold Z=M/T3:*
[TABLE]
(2)′* For a torus T2≤T3,
the quotient orbifold M/T2
admits a strictly pseudoconvex CR-structure. Furthermore
this gives the pseudo-Hermitian
(Sasaki)
orbibundle:*
[TABLE]
The fundamental group Γ=π1(M)
induces a nontrivial group extension:
1→Z2→Γ/Z⟶Q→1.
In particular (5.4) is a nontrivial TC1-bundle.
Γ also assigns to (5.5) a nontrivial group extension:
1→Z→Γ/Z2⟶Q→1.
Proof.
Since R3 acts properly on X, the orbit spaces
[TABLE]
are smooth manifolds respectively for Rk≤R3(k=1,2,3).
For the qc-structure (η,{Jα}α=13) on X,
we prove (1), (1)′.
First, (i) of Lemma 5.3 implies
[D1,0,D1,0]⊂E1,0.
For u∈D1,0,
as ξ2 generates t2∈R3,
J1[ξ2,u]=i[ξ2,u]. Similarly J1[ξ3,u]=i[ξ3,u].
Since [ξ2−iξ3,u]=[ξ2,u]−i[ξ3,u]∈D⊗C, it follows
[ξ2−iξ3,u]∈D1,0.
Noting (5.1), it follows
[TABLE]
that is Jˉ1 is integrable on E.
The projection p1:X→X1 maps isomorphically E onto TX1
at each point of X1, so p1 induces an almost complex structure
J^1 on X1 such that p1∗J1ˉ∣E=J^1p1∗.
For A,B∈E1,0, noting [A,B]∈E1,0,
it follows p1∗[A,B]=[p1∗A,p1∗B]∈TX11,0, that is J^1
is a complex structure on X1.
Moreover, as R3 leaves each ξα invariant,
both p1∗ξ2 and p1∗ξ3 induce the vector fields
ξ^2,ξ^3 respectively on X1.
Then it follows
TX11,0=p1∗E1,0=⟨ξ^2−iξ^3⟩⊕p1∗D1,0
so that
[TABLE]
By the definition,
p1∗D1,0 is an (invariant) complex contact bundle on the complex manifold
(X1,J^1).
For (1)′, let M=X/Γ (we use X as the universal covering of M
with the same notations as those of X).
There is the commutative diagram of principal bundles:
[TABLE]
such that Z=Γ∩R
and the bottom sequence is an orbibundle.
Taking the quotient of X1 by Γ1,
M/S1=X1/Γ1 is a complex contact orbifold induced
from that of X1.
Putting Z3=R3∩Γ, consider
the equivariant principal bundle
(Z3,R3)→(Γ,X)⟶q~(Q,X3)
whose quotient gives an orbibundle: T3→M⟶Z=X3/Q.
Recall t∗η=η,t∗Jα=Jαt∗∣D(∀t∈R3,α=1,2,3) (cf. Proposition 5.1).
Noting dη(V,u)=0,dη(V,V)=0(u∈D) from (4.1),
there is a Kähler form Ωα such that Ω=Ω1i+Ω2j+Ω3k on X3
satisfying (cf. [21]):
[TABLE]
As q~∗:D→TX3 is an isomorphism at each point of X3,
there is an almost complex structure Jα on X3 such
that q~∗Jα=Jαq~∗ on D.
It follows from (5.9) that
Ωα(Jαx,Jαy)=Ωα(x,y)(x,y∈TX3). Moreover,
Ω1(J1x,y)=Ω2(J2x,y)=Ω3(J3x,y).
Each Jα
turns to a complex structure on X3 such that
g=Ωα∘Jα(α=1,2,3)
is a hyperKähler metric on the complex manifold Y=X3.
As ⟨ξ^2−iξ^3⟩ generates C,
the above complex contact structure on (X1,J^1) admits a holomorphic
principal bundle
C→(X1,J^1)⟶q(X3,J1)
over the Kähler manifold (X3,J1).
The following commutative diagram of principal bundles
associates the commutative diagram of group extensions:
[TABLE]
[TABLE]
Taking the quotients gives a holomorphic orbibundle:
[TABLE]
This proves (1), (1)′.
Next let
(Z2,R2)→(Γ,X)⟶μ(Γ2,X2)
be the equivariant principal bundle
where R2=⟨ξ2,ξ3⟩.
μ∗ maps D isomorphically to μ∗(D).
Put μ∗(D)=F which is the codimension 1-subbundle of
TX2. If J1′
is the induced almost complex structure on F such that μ∗J1=J1′μ∗, then μ∗:(D1,0,J1)→(F1,0,J1′)
is an isomorphism.
Since [D1,0,D1,0]⊂⟨ξ2−iξ3⟩⊕D1,0 by Lemma 5.3,
it implies μ∗[D1,0,D1,0]=[F1,0,F1,0]⊂F1,0, that is
J1′ is integrable on F.
As η1({ξ2,ξ2})=0, η1 induces
a 1-form η^1 on X2
such that μ∗η^1=η1. Put ξ^1=μ∗(ξ1) so that
η^1(ξ^1)=1. Since
μ∗(η^1∧(dη^1)2n)=η1∧(dη1)2n∣{ξ1,D}=0,
it follows η^1∧(dη^1)2n=0.
(η^1,J1′) is a strictly pseudoconvex
pseudo-Hermitian structure on X2 with Reeb field ξ^1.
The equivariant principal bundle:
(Z,R)→(Γ2,X2)⟶(Q,X3)
gives rise to a pseudo-Hermitian Sasaki orbibundle:
[TABLE]
This proves (2), (2)′.
Finally taking into account the commutative diagram,
[TABLE]
the group extension in the middle
gives a 2-cocycle [f]∈H2(Q,Z2).
As the projection p2:Z2→Z induces a 2-cocycle
[p2(f)]∈H2(Q,Z) which represents the
group extension 1→Z→Γ2⟶q1Q→1.
Since this group extension is obtained from the pseudo-Hermitian Sasaki orbibundle
of (5.12), it does not split, that is [p2(f)]=0.
Thus [f]=0, the above holomorphic bundle (5.4) is
a nontrivial smooth bundle.
∎
Example 5.5**.**
The quaternionic Heisenberg Lie group M
has the qck-group R3=Ri+Rj+Rk=ImH(cf. Section 6).
Taking R(=Ri),
the quotient Lie group L=M/R is a
complex Lie group which admits a complex contact structure
(cf. [23]). A holomorphic fibering gives an exact sequence:
1→C→L⟶Hn→1.
L is called Iwasawa complex Lie group. Let
Sim(L)=L⋊(Sp(n)⋅S1×R+)
be the holomorphic subgroup preserving the complex contact structure of L.
A (4n+2)-manifold M locally modeled on L with coordinate changes
lying in Sim(L) is said to a complex contact similarity manifold.
The following is shown similarly as in the proof of [23].
Proposition 5.6**.**
Let M be a compact complex contact similarity manifold.
If the developing map dev:M~⟶L is injective,
then M is holomorphically isometric to
the infranilmanifold L/Γ(Γ≤L⋊(Sp(n)⋅S1))
or an infra-Hopf manifold
L−{0}/Γ(Γ≤Sp(n)⋅S1×R+).
6. The standard qc-structure D0 for (M,ω0)
Recall the 4n+3-dimensional (positive definite)
quaternionic Heisenberg Lie group M
from [2].
Put t=(t1,t2,t3),s=(s1,s2,s3)∈R3=\mboxImH, and
z=t(z1,…,zn),w=t(w1,…,wn)∈Hn and so on.
Then M is the product R3×Hn with group law:
[TABLE]
where ⟨z,w⟩=tzˉw
is the Hermitian inner product.
As M is nilpotent such that the center is the commutator subgroup
[M,M]=R3 consisting of elements (t,0).
Let ω0 be an ImH-valued 1-form on M defined by
[TABLE]
Put ω0=ω1i+ω2j+ω3k.
Denote by D0
the codimension 3-subbundle kerω0 on M.
As in (2.1), three endomorphisms {Jα}α=13
are defined by
Jγ=(dωβ∣D0)−1∘(dωα∣D0):D0→D0
on D0((α,β,γ)∼(1,2,3)).
For the projection π:M→Hn,
π∗:D0→THn is
the isomorphism at each point of Hn.
Let {i,j,k} be the standard quaternionic structure on Hn.
We define a quaternionic structure
{J~α}α=13 on D0 by
π∗J~1u=(π∗u)iˉ, π∗J~2u=(π∗u)jˉ, π∗J~3u=(π∗u)kˉ respectively.
A calculation shows
that the quaternionic structure
{J~α}α=13 on D0
coincides with {Jα}α=13,
that is Jα=J~α(α=1,2,3). (Compare [3, Section 6.2].)
By the definition, the pair (D0,{Jα}α=13) is
the standard qc-structure on M with
the qck-distribution
V0=⟨dt1d,dt2d,dt3d⟩ generating the center R3.
By Theorem A,
Autqc(M)=M⋊(Sp(n)⋅Sp(1)×R+)
is the full group of qc-automorphisms of M.
Recall [21] that the action of an element
α=((s,z),λ⋅A⋅a)∈Autqc(M) on
M is obtained as (∀(t,w)∈M) :
[TABLE]
For the standard qc-form ω0 (cf. (6.1)), it follows
[TABLE]
Then Pshqc(M,ω0,{Jα}α=13)=M⋊(Sp(n)⋅Sp(1)) by Definition 3.3.
7. Complete hyperKähler structure on R×X
If X is a qCR-manifold, then it is known that the product
R+×X with the cone metric
admits an (incomplete) hyperKähler structure.
This construction is not applicable to qc-manifolds with
the qck-distribution R3. (See Note 7.3.)
However we shall construct a hyperKähler metric on the product
R×X
suitable for a qc-manifold X with qck-group R3.
Let (X,D,ω,{Jα}α=13) be a simply connected non-compact
qc-manifold with the qck-group R3 where
ω=ω1i+ω2j+ω3k is an ImH-valued 1-form.
By (5.9) of Theorem 5.4, there is a principal
bundle R3→X⟶πY
over a hyperKähler manifold Y.
R3 induces
the distribution V=⟨dt1d,dt2d,dt3d⟩ as before.
As ωα(dtβd)=δαβ
with (6.1),
each dual form satisfies
[TABLE]
Choose the coordinate t0 for R in
the product R×X. Then
the canonical distribution R4
is endowed with the standard quaternionic structure such as
[TABLE]
Replace R4 by
H=⟨dt0d,dt1d,dt2d,dt3d⟩.
As there is a decomposition
[TABLE]
extend a quaternionic structure {Jˉα} on
H⊕D to be
[TABLE]
The product group R×Pshqc(X) acts on
R×X as usual:
[TABLE]
There is also a principal bundle
H→R×X⟶π~Y.
Denote the subgroup
[TABLE]
The normal subgroup H≤R×Pshqc(X)
is obviously contained in G (see (i) of Note 7.3).
Proposition 7.1**.**
Let X be a simply connected non-compact qc-manifold with
the qck-group R3. Suppose Y is complete.
Then there is a complete hyperKähler metric g0 on the
quaternionic manifold
(R×X,{Jˉα}α=13)
such that the quaternionic isometry group
IsomhK(R×X) is R×G.
Proof.
According to the decomposition, an arbitrary element
is described as a+u(a∈H,u∈D).
Define a Riemannian metric on R×X to be
[TABLE]
Note
dωα(dt0d,H⊕D)=0 by the decomposition.
Since V=⟨dt1d,dt2d,dt3d⟩
is R3, dωα(dtid,dtjd)=0(i,j=1,2,3)
and dωα(V,D)=0 (cf. (4.1)).
Using the projection
p:T(R×X)=H⊕D⟶H on the first factor, it follows
[TABLE]
Since Jˉα preserves the decomposition which also
leaves invariant ∑i=03dti⋅dti,
dωα∘Jα respectively, we obtain
[TABLE]
In order to prove g0 is a hyperKähler metric on R×X,
put
[TABLE]
It is easy to check that
[TABLE]
A calculation shows
[TABLE]
[TABLE]
Each Θα is closed (α=1,2,3) and
Θα(Jˉαx,y)=g0(x,y) by (7.6). Thus
g0 is a hyperKähler metric on R×X.
In particular, the principal bundle
[TABLE]
turns to a Riemannian submersion where g^0 is the induced hyperKähler
metric from dωα∘Jα (cf. (5.9)).
If (Y,g^0) is complete, then (R×X,g) will be also complete
by the construction.
Finally in view of (7.4),
the subgroup R×G
is the quaternionic isometry group IsomhK(R×X,g0).
∎
Theorem 7.2**.**
There is a complete hyperKähler metric g0 on the quaternionic space
Hn+1(n≥1) such that the quaternionic isometry group is
IsomhK(Hn+1,g0)=H⋊(Sp(n)⋅Sp(1)).
In particular, g0 is not equivalent to the standard quaternionic metric
up to a quaternionic isometry.
Proof.
Take M with qck-group R3=C(M).
We apply Proposition 7.1 to R×M.
In the decomposition T(R×M)=H⊕D0,
the quaternionic structure {Jα}α=13 on D0
maps quaternionically onto the standard quaternionic structure on Hn (cf. Section 6). Noting (5.1),
the quaternionic structure (R×M,{Jˉα}α=13)
is equivalent with the standard quaternionic structure (Hn+1,{i,j,k})
where an identification is
R×M=R×R3×Hn=H×Hn=Hn+1.
By the complete Kähler metric g^0
on the base space Hn, using
the projection π~ of (7.8),
(Hn+1,g0) will be a complete hyperKähler manifold.
As Pshqc(M,ω0)=M⋊(Sp(n)⋅Sp(1))
by (6.3),
the subgroup G preserving both ∑i=13dti⋅dti
and dω1∘J1
is isomorphic to R3⋊(Sp(n)⋅Sp(1))(see Note 7.3).
Hence
IsomqK(Hn+1,g0)=H⋊(Sp(n)⋅Sp(1)).
Since
IsomqK(Hn+1,gH)=Hn+1⋊(Sp(n)⋅Sp(1))
for the standard metric gH,
the hyperKähler metric g0 is not equivalent with
gH on Hn+1.
∎
Note 7.3**.**
(i)* Put dt=dt1i+dt2j+dt3k for brevity.
Then Re⟨dt,dt⟩=dt1⋅dt1+dt2⋅dt2+dt3⋅dt3.
The conjugate of an element a∈Sp(1) leaves
Re⟨dt,dt⟩ invariant, while
Sp(n) acts trivially on dt (see (6.2)). In fact,
Re⟨a⋅dt⋅aˉ,a⋅dt⋅aˉ⟩=Re(adtaˉ⋅adtaˉ)=Re(dt⋅dt)=Re⟨dt,dt⟩.
By (6.2), the elements of the form
(0,z)∈((0,0,0),Hn)≤M
act on ((t1,t2,t3),w)∈M as
(0,z)((t1,t2,t3),w)=((t1,t2,t3)−Im⟨z,w⟩,z+w), which
do not preserve dt1⋅dt1+dt2⋅dt2+dt3⋅dt3.
(ii)
We shall
explain that the aforementioned cone-construction does not work.
Given a non-compact simply connected qc-manifold
(X,g,ω,{Jα}α=13)
with the qck-group R3, g(x,y)=∑i=13ωi(x)⋅ωi(y)+dω1(J1x,y) is the Riemannian metric
invariant under Pshqc(X,ω,{Jα}α=13) on X
*(cf. (3.6)).
Recall g′=dt2+t2g is the cone metric on R+×X(t∈R+)(cf. [3]).
The exact two-form for g′ is
Ωα′=d(t2⋅ωα)=2tdt∧ωα+t2dωα.
Noting V={d/dt1,d/dt2,d/dt3} is
the qck-distribution generating R3,
the quaternionic structure on R+×X is defined by
Jˉαd/dtα=td/dt,Jˉαd/dt=−d/dtα(α=1,2,3).
In particular
Jˉαd/dtβ=d/dtγ.
If g′ happened to be a hyperKähler metric, then
it would satisfy
Ωα′(Jˉαx,y)=g′(x,y)(α=1,2,3).
However taking x=y=d/dtβ,
it follows g′(x,y)=(dt2+t2g)(d/dtβ,d/dtβ)=t2g(d/dtβ,d/dtβ)=t2, while
Ωα′(Jˉαx,y)=2tdt∧ωα(d/dtγ,d/dtβ)+t2dωα(d/dtγ,d/dtβ)=0.
The cone-metric is not hyperKähler for
any qc-manifold (X,g,ω,{Jα}α=13,R3).
(iii) Given a non-compact simply connected qc-manifold
(X,g,ω,{Jα}α=13)
with V=R3(cf. Theorem \refhyperM), there is
a Riemannian metric
g1=dt0⋅dt0+ω1⋅ω1+ω2⋅ω2+ω3⋅ω3+dω1∘J1 on R×X.
Defining a quaternionic structure on R×X
as in (7.2), (5.1), we obtain 2-forms
Ωα=−4(ωα∧dt0+ωβ∧ωγ)+dωα((α,β,γ)∼(1,2,3)) by a calculation, so Ωα is not closed. However,
Isomhk(R×X,g1)=R×Pshqc(X) is the isometry group
which may act transitively on R×X.
8. Pshqc(X) with the abelian qck-distribution
Let Pshqc(X)=Pshqc(X,(η,{Jα}α=13)) be
the qc-Hermitian group defined in Definition 3.3.
When the qck-distribution V generates R3,
we observe that Pshqc(X) is determined
exactly by using the Boothby-Wang fibering. (Compare [4, Proposition 3.4]
for the case of pseudo-Hermitian Sasaki manifolds.)
As NPshqc(X)(R3)=Pshqc(X)
by Corollary 4.3,
there is a principal bundle :
[TABLE]
for which p∗Ω=dη (cf. (5.9)).
As in Definition 3.3, we introduce
hyperKähler isometry group as
Definition 8.1**.**
[TABLE]
for
b∈C∞(Y,Sp(1)), and
(bij)∈C∞(Y,SO(3)) is its conjugate by b.
Take h∈Pshqc(X) such that
[TABLE]
for some map a∈C∞(X,Sp(1)) (cf. Definition 3.3).
For t∈R3, hth−1∈R3 as above,
there induces a map h^:Y→Y with the commutative diagram:
[TABLE]
Applying t∈R3 to (8.2),
t∗a⋅η⋅t∗a=t∗h∗η=h∗(hth−1)∗η=h∗η=a⋅η⋅aˉ,
it follows
t∗a=±a∈C∞(X,Sp(1))
since η(t[ξ1,ξ2,ξ3])=t[i,j,k]. Thus t∗a=a and so a induces a map
b∈C∞(Y,Sp(1))
such that p∗b=a.
Differentiate (8.2), then
h∗dη=a⋅dη⋅aˉ on D.
Since h∗p∗Ω=a⋅p∗Ω⋅aˉ on D,
the commutativity shows
p∗h^∗Ω=p∗(b⋅Ω⋅bˉ)
on D, that is
[TABLE]
As h∗Jα=∑βaαβJβh∗ on D,
it follows
h^∗Jα=∑βbαβJβh^∗
on Y. Here bαβ is induced from b.
By the definition, we have
h^∈Isomhk(Y,Ω,{Jα}α=13).
By (8.4), let h^∗Ωα=∑βbαβΩβ where bαβ∈C∞(Y,SO(3)).
Lemma 8.2**.**
Assume dimY≥8.
Then bαβ is constant, that is an element of Sp(1).
In particular h∗η=a⋅η⋅aˉ
for an element a∈Sp(1).
Proof.
Differentiate (8.4) such that
0=dh^∗Ωα=∑βdbαβ∧Ωβ.
So we may put θ1∧Ω1+θ2∧Ω2+θ3∧Ω3=0
where θk=dbαk for each α(k=1,2,3).
It suffices to show θk=0 on Y. Let g=Ωα∘Jα.
Choose x∈TY such that g(x,x)=1.
There is a decomposition :
TY={x,J1x,J2x,J3x}⊕{x,J1x,J2x,J3x}⊥.
Case 1. Let y∈{x,J1x,J2x,J3x}⊥.
[TABLE]
The above equality shows θ1(y)=0
for all y∈{x,J1x,J2x,J3x}⊥.
Case 2. For x∈{x,J1x,J2x,J3x},
choose y∈{x,J1x,J2x,J3x}⊥
such that g(y,y)=1. Note
J1y,J2y,J3y∈{x,J1x,J2x,J3x}⊥.
Then
[TABLE]
Thus θ1(x)=0.
This calculation implies similarly
θ1(J1x)=θ1(J2x)=θ1(J3x)=0.
Hence it follows θ1=dbα1=0 on Y.
Thus bα1∈C∞(Y,SO(3)) is a constant.
The above argument shows also θ2=θ3=0, that is
dbα2=dbα3=0 on Y.
The matrix (bαβ)∈C∞(Y,SO(3)) is constant and so
is b∈C∞(Y,Sp(1)).
Since a=p∗b, a∈C∞(X,Sp(1)) is also constant.
∎
Corollary 8.3**.**
When the qck-distribution V is R3,
the qc-Hermitian group (respectively
hyperKähler isometry group) is described as follows:
[TABLE]
Proposition 8.4**.**
Suppose H1(Y;R3)=0.
There associates a natural exact sequence
with the principal bundle
R3→X⟶pY of (8.1):
[TABLE]
Proof.
The proof is almost similar to the argument of Section 4 of [4].
As usual put
η=η1i+η2j+η3k,Ω=Ω1i+Ω2j+Ω3k.
Then we have the equality p∗Ω=dη.
Suppose V=R3 consists of one-parameter groups {φt(1),φt(2),φt(3);t∈R}.
If we choose a section s:Y→X(p∘s=idY) such that
X is equivalent with the trivial bundle R3×Y by
a bundle map f:R3×Y→X such as
f((t1,t2,t3),y)=φt1(1)∘φt2(2)∘φt3(3)(s(y)).
This gives a commutative diagram:
[TABLE]
Define a 1-form θ=s∗η which satisfies
[TABLE]
Let η0=∑i=13dti+pr∗θ
be the 1-form on the product R3×Y satisfying dη0=pr∗Ω
compatible with the regular hyperKähler structure. (See [4, Proposition 3.1].)
D0=kerη0 admits a quaternionic structure {Jα}α=13
which is the pullback of the quaternionic structure {Jα}α=13
on Y. (See (3.5), (3.6) of [4].)
Then it is easy to check that
[TABLE]
Then the qc-structure
(R3×Y,η0,{Jα}α=13,{d/dti}i=13) is equivalent through the bundle map f with
(X,η,{Jα}α=13,{ξi}i=13).
(Compare [4, Proposition 3.1].)
By the commutative diagram (8.3),
define ϕ(h)=h^ for h∈Pshqc(X).
Then ϕ:Pshqc(X)→IsomhK(Y) is a homomorphism.
In order to prove
the exactness of (8.5), take h^∈Isomhk(Y).
By Corollary 8.3, h^∗Ω=b⋅Ω⋅bˉ
for some b∈Sp(1).
Define h1:R3×Y→R3×Y to be
h1(t,y)=(t,h^(y)) where t=(t1,t2,t3)∈R3.
Put
[TABLE]
By a calculation,
dh1∗η0=h1∗pr∗Ω=pr∗h^∗Ω=b⋅pr∗Ω⋅bˉ, it follows
[TABLE]
Letting the obvious section s′:Y→R3×Y, put θ′=(s′)∗η′
on Y. As s′∘pr∣0×Y=id,
it follows pr∗θ′∣0×Y=η′∣0×Y.
Noting
[TABLE]
it implies
η′=bˉ⋅dt⋅b+pr∗θ′
on R3×Y.
(Here dt=dt1i+dt2j+dt3k.)
Note by (8.10)
dθ′=(s′)∗dη′=(s′)∗pr∗Ω=Ω.
Since dθ′=Ω=dθ on Y from (8.7),
[θ−θ′]∈H1(Y,R3).
By the hypothesis H1(Y;R3)=0,
there is a map λ:Y→R3 such that
[TABLE]
Let G:R3×Y→R3×Y
be a gauge transformation defined by
G(t,y)=(b⋅(t+λ(y))⋅bˉ,y).
Noting pr∘G=pr,
[TABLE]
Put h′=h1∘G:R3×Y→R3×Y.
In fact, h′(t,y)=(b⋅(t+λ(y))⋅bˉ,h^(y)).
Noting h1∗η0=b⋅η′⋅bˉ by (8.9),
a calculation shows
h∗η0=b⋅η0⋅bˉ.
Since f∗η=η0,
we obtain (fh′f−1)∗η=b⋅η⋅bˉ.
Put h=fh′f−1 and so
ph∘f=pf∘h′.
By (8.6), it follows
p∘h=h^∘p.
As
h^∗Jα=∑β=13bαβJβh^∗ and p∗Jα=Jαp∗,
we have
p∗h∗Jα=∑bαβJβh^∗p∗=p∗(∑bαβJβh∗):D→TY,
thus
h∗Jα=∑β=13bαβJβh∗.
This implies h∈Pshqc(X) such that ϕ(h)=h^ by (8.3).
This shows ϕ is surjective.
Let ϕ(h)=h^ as above.
Put h0=f−1∘h∘f:R3×Y→R3×Y.
As in the argument of surjectivity, there is a map λ:Y→R3
such that h0(t,y)=(h0th0−1+λ(h^(y)),h^(y)).
If h∗η=a⋅η⋅aˉ(a∈Sp(1)), then note
h^∗Ω=a⋅Ω⋅aˉ (cf. (8.4)).
Suppose ϕ(h)=1. Then Ω=a⋅Ω⋅aˉ.
Noting each Ωi is a Kähler form for
Ω=Ω1i+Ω2j+Ω3k, this equation implies a=±1
so that h∗η=η.
Since f∗η=η0 by (8.8),
this shows h0∗η0=η0.
Moreover h0(0,y)=(λ(y),y) as above.
For any (0,wy)∈(0,TY)⊂TR3×TY, it follows
(h0)∗(wy)=(λ∗(wy)+wy). A calculation shows
η0(wy)=h0∗η0(wy)=η0(λ∗(wy)+wy)=η0(λ∗(wy))+η0(wy).
Thus η0(λ∗(wy))=0, so λ∗(wy)∈D0=kerη0.
Since λ∗(wy)∈TR3=V, it follows λ∗(wy)=0,
that is λ∗(TY)=0. Hence λ is a constant s∈R3.
Then h0(t,y)=(h0th0−1+s,y).
As η0=dt+pr∗θ,
[TABLE]
Noting h0∗η0=η0, h0th0−1−t=c for some constant c∈R3.
Taking t=0, c=0 and so
h0th0−1=t.
Hence h0(t,y)=(t+s,y)=s(t,y) so that
h0=s∈R3. Since f is R3-equivariant,
the equation h=f∘h0∘f−1 implies h=s.
This proves the exactness of ϕ.
∎
9. Spherical homogeneous qc-manifolds
We determine spherical (uniformizable) homogeneous qc-manifolds
following the method of spherical homogeneous CR-manifolds [10].
Let (ρ,dev):(Autqc(M),M)→(Autqc(S4n+3),S4n+3)
be a developing pair. Suppose that M is homogeneous by a
subgroup G of Autqc(M).
Put G=ρ(G)≤Autqc(S4n+3)=PSp(n+1,1).
We may assume G is a non-compact closed subgroup
taking the closure if necessary.
Then G⋅p(=dev(M)) is a homogeneous domain of S4n+3 for some p∈dev(M).
For a closed submanifold L⊂S4n+3,
denote by Autqc(S4n+3−L)
the subgroup of PSp(n+1,1) whose elements leave L invariant.
If G has the radical,
then G belongs to the maximal amenable Lie subgroup
in PSp(n+1,1), that is Autqc(M) up to conjugate.
Otherwise, G is semisimple.
It follows ([12], [24, Lemma 3.1])
Lemma 9.1**.**
If G is non-compact semisimple but not PSp(n+1,1),
then G is one of the following groups up to conjugate.
In each case G acts properly on S4n+3−L where L is a sub-sphere.
(1)
G=Autqc(S4n+3−Sm−1)=P(O(m,1)⋅Sp(1)×Sp(n−m+1))* (1≤m≤n+1).
Sp(n−m+1) is the maximal compact subgroup
which fixes Sm−1(=∂HRm).*
2. (2)
G=Autqc(S4n+3−S2m−1)=P(U(m,1)⋅U(1)×Sp(n−m+1))* (1≤m≤n+1).
Sp(n−m+1)) is the maximal compact subgroup which fixes
S2m−1(=∂HCm).*
3. (3)
G=Autqc(S4n+3−S4m−1)=Sp(m,1)⋅Sp(n−m+1)* (1≤m≤n).
Sp(n−m+1) is the maximal compact subgroup
which fixes S4m−1(=∂HHm).*
4. (4)
G=Autqc(S4n+3−S2)=SL2(C)⋅Sp(n).
Sp(n) is the maximal compact subgroup
which fixes S2=∂HImH.
In this case, it fixes S3=∂HH1⊃S2 so this case reduces to case (3).
Proposition 9.2**.**
Let G be a non-compact semisimple subgroup of PSp(n+1,1).
(1) Suppose G is not the whole group PSp(n+1,1).
Only G=Sp(m,1)⋅Sp(n−m+1) acts transitively
on S4n+3−S4m−1(1≤m≤n).
The positive definite homogeneous qc-manifold
S4n+3−S4m−1 supports a principal bundle
S3→S4n+3−S4m−1⟶HHm×HPn−m in which S3 is not a qck-group.
(2) Suppose G=PSp(n+1,1). Then S4n+3=PSp(n+1,1)/Autqc(M)=Sp(n+1)⋅Sp(1)/Sp(n)⋅Sp(1)
in which Sp(1) is the qck-group of Sp(n+1)⋅Sp(1)=Pshqc(S4n+3)=NPshqc(S4n+3)(Sp(1)).
Suppose G has the non-compact radical in PSp(n+1,1).
Then G belongs to the maximal amenable Lie subgroup
M⋊(Sp(n)⋅Sp(1)×R+)=Autqc(M) up to conjugate.
In particular the closed subgroup G is also amenable so that
G is an extension of a solvable group by a compact group.
Let G=R⋊K be the semidirect product
where R is solvable and K is compact.
It is noted from [6] that there is a simply connected characteristic
solvable subgroup R0 such that R=R0⋅T where
T is a maximal compact subgroup of R. Letting H=T⋅K,
we have a semidirect product
G=R0⋊H.
Let Pshqc(M)=M⋊(Sp(n)⋅Sp(1))≤Autqc(M) and
M=S4n+3−{∞} be as above.
Suppose M=G/K is a simply connected homogeneous qc-manifold
where G≤Autqc(M) and K is a closed subgroup of G.
Let (ρ,dev):(Autqc(M),M)→(Autqc(M),M))
be the developing pair. Put ρ(G)=G.
Proposition 9.3**.**
If G≤Pshqc(M) up to conjugate, then
dev(M)=M for which M is a homogeneous
qc-manifold by the qc-group M⋊(Sp(n)⋅Sp(1)).
Proof.
The developing pair reduces to
(ρ,dev):(G,M)→(Pshqc(M),M).
Taking an M⋊(Sp(n)⋅Sp(1))-invariant Riemannian metric
on M, the pullback metric by dev
induces a homogeneous Riemannian metric on M.
Noting M is geodesically complete (cf. [34]),
dev:M→M is an isometry. Hence M is qc-homogeneous by the group
M⋊(Sp(n)⋅Sp(1)).
∎
In general case for G=R0⋊H≤Autqc(M),
consider the exact sequence:
[TABLE]
where pr(G)=R+ from Proposition 9.3.
Then there exists a one-parameter subgroup A≤R0
such that pr(A)=R+. It follows
R0=N⋊A
where N is a nilpotent subgroup
such that N=(kerpr)∩R0=M∩R0.
Then G=(N⋊A)⋅H.
Since A≤PU(n+1,1) is of non-elliptic type which
stabilizes {0,∞}⊂S4n+3 up to conjugate,
there is a geodesic segment between
0 and ∞ translated by A. As H≤Sp(n)⋅Sp(1)
fixes {0,∞},
the conjugate h⋅A⋅h−1 for h∈H
fixes {0,∞} also, there is a geodesic segment between
0 and ∞ translated by h⋅A⋅h−1. Then
these two geodesic segments between the same endpoints spans a
(geodesically) flat plane in HHn+1.
Since HHn+1 is a complete simply connected
Riemannian manifold of constant negative quaternionic curvature,
there is no such flat plane so that h⋅A⋅h−1=A.
In particular, the elements of A and H commute.
We have
[TABLE]
Let q:M→Hn be the projection which is equivariant
with respect to the homomorphism q in the exact sequence:
[TABLE]
where q(G)=q(N)⋊(A×H) by (9.2).
Denote the point 1 in M=R3×Hn by
[TABLE]
Thus q(1)=(01)∈Hn.
Since D0 on M maps to q∗(D0)=THn,
the homogeneous quaternionic manifold q(G)⋅0
becomes (q(N)⋅0(A×H)⋅1)⊂Hn.
Here either (A×H)⋅1=A×H/H1=R+×S4k−1=Hk−{0}
and q(N)⋅0=Hn−k(1≤k≤n), or
(A×H)⋅1=A⋅1=R+ and
q(N)⋅0=(Hn−1ImH).
Lemma 9.4**.**
The orbit G⋅1 at 1∈M becomes:
[TABLE]
Proof.
According to each orbit, G is isomorphic to either G(k) or G(0)
respectively:
such that Re(u+λ)=λ>0.
Putting H+={z∈H∣Re(z)>0}, it follows
[TABLE]
∎
Let G(k)1
be the stabilizer at 1. Suppose
\displaystyle(\boldsymbol{t},\left[\begin{array}[]{c}\boldsymbol{0}\\
z\\
\end{array}\right])\cdot\left(\left(\begin{array}[]{cc}A&\\
&B\end{array}\right)\cdot\lambda\cdot a\right)\cdot\left[\boldsymbol{0},\binom{1}{\boldsymbol{0}}\right]=\left[\boldsymbol{0},\binom{1}{\boldsymbol{0}}\right].
Then
t=0,z=0.
The equation
λA⋅1⋅aˉ=1 shows
λ∣A⋅1⋅aˉ∣=1 where 1=t[1,0,…,0k]. So λ=1.
As A⋅1⋅aˉ=t[a11⋅aˉ,a21⋅aˉ,…,ak1⋅aˉ]=t[1,0,…,0], it follows a11=a,a21=⋯=ak1=0.
Then
\displaystyle A\cdot a=\left(\begin{array}[]{cc}a&0\\
0&A_{k-1}\\
\end{array}\right)\cdot a\ \ ({}^{\exists}\,A_{k-1}\in\mathop{\rm Sp}\nolimits(k-1)).
Letting a∈Sp(1),
[TABLE]
Similarly if g⋅1=1 for g∈G(0), then
note Re(u+λ)=λ=1 from (9.4). As u+1=1, u=0.
It follows
[TABLE]
Since both G(k)1 and G(0)1 are compact subgroups,
we obtain
Proposition 9.5**.**
The qc-manifolds G(k)/G(k)1,G(0)/G(0)1
are homogeneous Riemannian domains in M:
Let X(k)(0≤k≤n) be
the orbit G(k)⋅1(k=0)
or G(0)⋅1.
We obtain the following. (Compare Proposition 9.5, Proposition 9.2, (9.10).)
Theorem 9.6**.**
Any simply connected spherical homogeneous qc-manifold M is qc-isomorphic to
S4n+3, S4n+3−S4m−1(1≤m≤n),
or M, X(k)(0≤k≤n).
In particular, only S4n+3 admits the qck-group Sp(1)
and each M or X(k)
admits the qck-group R3.
Proof.
The proof divides into two cases whether the holonomy image
G=ρ(G)≤PSp(n+1,1) is semisimple or not. Noting
M is simply connected, there is a ρ-equivariant
developing map dev:M→S4n+3.
Put X=dev(M)=G⋅p.
When G is semisimple, Proposition 9.2 shows
the only homogeneous qc-manifold X is
S4n+3−S4m−1=Sp(m,1)⋅Sp(n−m+1)/(Sp(m)×(ΔSp(1)⋅Sp(n−m))(1≤m≤n) (cf. [24, Lemma 3.3 (ii)]).
In addition only X=S4n+3=Sp(n+1)/Sp(n) admits the qck-group R=Sp(1).
Since all of these X are Riemannian homogeneous,
the pullback metric by dev implies M is qc-isomorphic to X as before.
When G has the nontrivial radical, G is
either G(k) or G(0) by Proposition 9.5.
Let A be the one-parameter subgroup of G from (9.2).
Since A stabilizes {0,∞} in S4n+3,
the limit set L(G) contains {0,∞} (cf. [12]).
The complement
S4n+3−X is a closed subset containing more than one point
(otherwise X=S4n+3−{∞}=M).
Then it follows from [12] that
L(G)⊂S4n+3−X, that is
[TABLE]
Let 0=[0,(00)] be the origin of
M. According to whether
G is G(k) or G(0),
the orbit G⋅0 becomes
(i)
\displaystyle G(k)\cdot\boldsymbol{0}={\mathbb{R}}^{3}\times\left[\begin{array}[]{c}\boldsymbol{0}\\
{\mathbb{H}}^{n-k}\end{array}\right],
(ii) \displaystyle G(0)\cdot\boldsymbol{0}={\mathbb{R}}^{3}\times\left[\begin{array}[]{c}{\rm Im}\,{\mathbb{H}}\\
{\mathbb{H}}^{n-1}\\
\end{array}\right].
In each case the union G⋅0∪{∞}
is a closed subset in S4n+3. (In fact, it is the
sphere diffeomorphic to either
G(k)⋅0∪{∞}=S4(n−k)+3 or
G(0)⋅0∪{∞}=S4n+2.)
Since the limit set L(G) is G-invariant
with {0,∞}⊂L(G), it follows
L(G)⊂G⋅0∪{∞}⊂L(G), that is L(G)=G⋅0∪{∞}.
From (9.7),
By Proposition 9.5, G⋅1 is homogeneous Riemannian.
Since dev(M)=X is homogeneous, X=G⋅1.
As above M is qc-isometric to X=G⋅1.
∎
By Theorem 9.6 (cf. Proposition 9.5), it has a principal bundle:
[TABLE]
where Y(k) is a domain of Hn.
The standard qc-form ω0 on M
restricts an invariant qc-structure to X(k).
Since G(k) has ⟨λ⟩=R+,
λ∗ω0=λ2ω0 by (6.3),
though G(k) preserves the qc-structure.
Thus Autqc(X(k))=G(k).
Putting Psh(X(k))=Pshqc(X(k),ω0,{Jα}α=13)(0≤k≤n),
[TABLE]
However
Pshqc(X(k),ω0,{Jα}α=13) is not transitive on X(k).
Of course, there is a qc-form η
such that Pshqc(X(k),η,{Jα}α=13)=Autqc(X(k))=G(k) by Theorem 3.5.
Since η=v⋅ω0 for some non-constant v∈C∞(X(k),R+),
R3 does not induce a qck-distribution for η.
10. Curvature criterion of qc-manifolds with qck-group R3
Let (X,g,η,{Jα}α=13) be
a 4n+3-dimensional simply connected non-compact positive definite
qc-manifold with qck-distribution
V=R3 where g=gη (cf. (3.6)). (X,g) is
D-Einstein, that is
Proposition 10.1**.**
Ric(x,y)=−6g(x,y)* (∀x,y∈D).*
Proof.
Let D={v1,…,v4n}={v1,…,vn,Jαv1,…,Jαvn,α=1,2,3}. Choose x,y∈D satisfying
g(x,x)=g(y,y)=1.
O’Neill’s formula [11, (3.30)] shows (x,y,v∈D) :
Consider the qc-bundle (8.1) (Riemannian submersion):
R3→(X,g)⟶p(Y,g^)
where ⟨ξ1,ξ2,ξ3⟩⊕D=TX.
As Y is hyperKähler,
\displaystyle{\rm Ric}(\hat{\boldsymbol{x}},\hat{\boldsymbol{y}})=\sum_{i=1}^{4n}\hat{g}\bigl{(}\hat{R}(\hat{\boldsymbol{v}}_{i},\hat{\boldsymbol{x}})\hat{\boldsymbol{y}},\hat{\boldsymbol{v}}_{i}\bigr{)}=0.
Noting (10.1),
[TABLE]
On the other hand,
take R2={ξ2,ξ3} for which
there is the Riemannian submersion
R2→(X,g)⟶μ(X2,g2)
where
(X2,η^1,J1′,ξ^1) with μ∗ξ1=ξ^1
is the pseudo-Hermitian (Sasaki) manifold such that
[TABLE]
Put E=⟨ξ1⟩⊕D⊂TX. Then
μ∗:E⟶TX2=⟨ξ^1⟩⊕μ∗D
is an isometry such that ⟨ξ2,ξ3⟩=E⊥.
Let R2 be the Riemannian curvature on (X2,g2).
In general, a Sasaki manifold (X2,g2,η^1,J1′) satisfies
R2(ξ^1,x^)y^=g2(x^,y^)ξ^1−η^1(y^)x^(x^,y^∈TX2). (See [31, (2.5)] for instance.)
If x,y∈D⊂TX with μ∗(x)=x^,
μ∗(y)=y^, then
g(x,y)=dη1(J1x,y)=g2(x^,y^) by (10.3) (cf. (5.9)) and so the above equation becomes
R2(ξ^1,x^)y^=g(x,y)ξ^1.
O’Neill’s formula shows
g(R(ξ1,x)y,ξ1)=g2(R2(ξ^1,x^)y^,ξ^1)
since [ξ1,x]V=0 where V={ξ2,ξ3}.
Noting g2(ξ^1,ξ^1)=1,
substitute the above equation:
[TABLE]
Applying this argument to the other submersion
R2→(X,g)⟶μ′(X2′,g2′)
for which (X2′,η^2,J2′,ξ^2)
is the pseudo-Hermitian (Sasaki) manifold with R2={ξ3,ξ1}
(respectively so is
(X2′′,η^3,J3′,ξ^3) with R2={ξ1,ξ2} ).
Similarly
[TABLE]
Using (10.2), (10.4), (10.5),
for any x,y∈D,
we obtain
[TABLE]
∎
Proposition 10.2**.**
Let (X,g,η,{Jα}α=13) be as above.
Unlike the 3-Sasaki structure, (X,g) is not Einstein.
Indeed we obtain
[TABLE]
Proof.
Let R2={ξ2,ξ3}→(X,g)⟶μ(X2,g2)
be the Riemannian submersion as in (10.3).
Take an orthonormal basis
{u1,…,u4n+1}=W
such that
⟨ξ2,ξ3⟩⊕W(=⟨ξ1,ξ2,ξ3⟩⊕D)
forms a basis of TX. Then
Ric(ξ1,x)=∑i=14n+1g(R(ui,ξ1)x,ui)+∑α=23g(R(ξα,ξ1)x,ξα)(ξ1,x∈TX).
Since ξ1 is Killing,
[ξ1,ui]V=0 where V=⟨ξ2,ξ3⟩.
O’Neill’s formula implies
[TABLE]
As (X2,g2,η^1,J1′,ξ^1)
is a 4n+1-dimensional Sasaki manifold,
the fundamental property of Sasaki manifold
shows
Ric(ξ^1,x^)=4ng2(ξ^1,x^),
(see [32, (1.6)] for example.)
By (10.8),
Ric(ξ1,x)=4ng2(ξ^1,x^)+g(R(ξ2,ξ1)x,ξ2)+g(R(ξ3,ξ1)x,ξ3).
Replace x by ξ1.
Noting R3=⟨ξ1,ξ2,ξ3⟩ spans a flat geodesic subspace of X,
g(R(ξ2,ξ1)ξ1,ξ2)=K(ξ2,ξ1)=0,
g(R(ξ3,ξ1)ξ1,ξ3)=K(ξ3,ξ1)=0.
Hence
Ric(ξ1,ξ1)=4ng2(ξ^1,ξ^1)=4n.
Apply the same argument to
each Sasaki manifold
(X2′,g2′,η^2,J2′,ξ^2),
(X2′′,g2′′,η^3,J3′,ξ^3), it follows
Ric(ξ2,ξ2)=4ng(ξ2,ξ2)=4n,
Ric(ξ3,ξ3)=4ng(ξ3,ξ3)=4n respectively.
∎
Remark 10.3**.**
By Theorem 5.4, letting X1=X/R,X2=X/R2,
there is a holomorphic bundleR2→(X1,g1)⟶q(Y,g^),
a pseudo-Hermitian Sasaki bundleR→(X2,g2)⟶ν(Y,g^)
respectively. The similar argument shows that
(i)(X1,g1) is p1∗D-Einstein, that is
Ric(x^,y^)=−4g1(x^,y^)(x^,y^∈p1∗D)
where ⟨ξ2,ξ3⟩⊕p1∗D=TX1,
(ii)(X2,g2) is μ∗D-Einstein,
Ric(x^,y^)=−2g2(x^,y^)(x^,y^∈μ∗D)
where ⟨ξ^1⟩⊕μ∗D=TX2.
Proposition 10.4**.**
Let (X,g,η,{Jα}α=13) be
a 4n+3(≥11)-dimensional
simply connected non-compact positive definite qc-manifold with qck-group
R3.
For some α∈{1,2,3},
if the sectional curvature K(x,Jαx) is constant for every unit vector x∈D at all points of X,
then there is a qc-isometric immersion dev:(X,g)→(M,g0) with dev∗g0=g
such that dev∗ω0=η, dev∗Jα=Jαdev∗∣D and
dev∘t=t∘dev(∀t∈R3).
Moreover, if there exists a discrete subgroup Γ≤Pshqc(X,η)
such that M=X/Γ is compact, then dev induces
a qc-isometry of M onto the quaternionic Heisenberg infranilmanifold M/ρ(Γ)
for the holonomy group ρ(Γ)≤M⋊(Sp(n)⋅Sp(1)).
Proof.
Let R3→(X,g)⟶p(Y,g^)
be a Riemannian submersion where (Y,g^) is a hyperKähler manifold as before.
In particular, each induced endomorphism J^α is a complex structure on Y.
For every unit vector x^∈TY,
choose x∈D such that p∗x=x^.
By [11, (3.20) Theorem], the holomorphic sectional curvature
has the relation K(x,Jαx)=K^(x^,J^αx^)−3.
As K(x,Jαx) is constant by the hypothesis,
so is K^(x^,J^αx^) on Y.
Let K^(x^,J^αx^)=c
for every unit vector x∈D at all points of Y.
Then (Y,g^,J^α) is
locally holomorphically isometric to
a Kähler complex space form of constant curvature c. Since
(Y,g^,{J^α}α=13) is hyperKähler,
c=0. It follows from [26, II. Proposition 7.3 IX]
that (Y,g) is of zero curvature and hence
Y is locally isometric to Hn for dimY≥8 by the uniformization
(cf. [16, Theorem 3.9, also Theorem 5.2, Theorem 3.5], [1, Corollary 3]).
Let Dev:Y→Hn be a quaternionically
isometric immersion.
If (g0,Ω0,{J^α}α=13) is
the standard quaternionic structure, then
Dev satisfies
Dev∗Ω0=Ω and
Dev∗∘J^α=J^α∘Dev∗
up to conjugate by an element of Sp(1)(α=1,2,3). Since
R3→X⟶Y is the principal qc-bundle
as well as the standard principal qc-bundle
R3=C(M)→M⟶Hn,
applying Proposition 8.4 to Dev,
there is a lift of qc-immersion dev:X→M
satisfying dev∗ω0=η,dev∗∘Jα=Jα∘dev∗(α=1,2,3).
(In fact, noting H1(Y;R3)=0,
there is a map λ:Y→R3 for which
\displaystyle\mathop{\rm dev}\nolimits\bigl{(}(\boldsymbol{t},y)\bigr{)}=\bigl{(}(\boldsymbol{t}+\lambda(y),\mathop{\rm Dev}\nolimits(y)\bigr{)}
is a gauge transformation from X=R3×Y onto M.)
By the uniformization principle, we have the equivariant qc-developing pair
(ρ,dev):(Autqc(X),X)⟶(Autqc(M),M)
where ρ is the holonomy homomorphism.
Noting dev∗ω0=η, Corollary 8.3
implies the holonomy homomorphism
reduces to ρ:Pshqc(X,η)→Pshqc(M,ω0)(=M⋊(Sp(n)⋅Sp(1))).
As g0=∑α=13ωα⋅ωα+dω1∘J1,
dev:(X,g)→(M,g0) turns to a qc-local isometry, that is
dev∗g0=g.
If M=X/Γ is compact, then dev:X→M
is an isometry and hence M≅M/ρ(Γ)
where ρ(Γ) is discrete uniform in M⋊(Sp(n)⋅Sp(1)).
By the generalized Bieberbach theorem,
M is finitely covered by a quaternionic Heisenberg nilmanifold
M/Δ(Δ≤M).
∎
Proposition 10.5**.**
Let (X,g,η,{Jα}α=13) be
a positive definite 4n+3(≥11)-dimensional homogeneous qc-manifold G/H
with qck-group R3.
If G≤Pshqc(X,η) is a unimodular group,
then X is qc-isometric to M.
Proof.
As R3 is normal in G, the quotient manifold
Y=X/R3 is homogeneous by the unimodular group G/R3.
Then the homogeneous hyperKähler manifold
(Y,Ω,{J^α}α=13)
has a positive definite Hermitian metric g^
for each J^α. If a direct factor of (Y,g^,J^α)
is not flat, then it is a homogeneous Kähler manifold of non-compact type
whose Hermitian form has negative Ricci tensor (cf. [14], [26]).
Since Y is homogeneous hyperKähler, applying the proof of Proposition 10.4,
there is a quaternionic isometry Dev:Y→Hn
which lifts to a qc-isometry dev of X onto M.
As dev is equivariant, there is an isomorphism
ρ:Pshqc(X,η)→M⋊(Sp(n)⋅Sp(1)).
Since G is transitive on X,
ρ induces a qc-isometry ρ^:X=G/H⟶M.
∎
Proposition 10.6**.**
Let M be a 4n+3(≥11)-dimensional
positive definite locally homogeneous closed asphericalqc-manifold X/Γ(Γ≤Pshqc(X))
with a qck-distribution V^.
Then M is qc-isometric to
M/ρ(Γ) where ρ(Γ)≤M⋊Sp(n). Moreover,
V^ generates T3.
Proof.
Since Pshqc(M) is compact,
V^ generates a compact qck-subgroup
K≤Pshqc(M).
By the result of
[27, Theorem 2.4.2, Corollary 3.1.12],
K is isomorphic to T3.
Moreover, the orbit map ι(t)=t⋅x(∀t∈T3)
at any point x∈M
induces an injective homomorphism ι∗:Z3=π1(T3)→C(Γ)≤Γ=π1(M)
where C(Γ) is the center of Γ.
X inherits a covering group action Z3→K~⟶K=T3
and hence K~=R3 which
is a qck-subgroup of Pshqc(X) since so is K≤Pshqc(M)0.
As Γ≤Pshqc(X) is cocompact, Pshqc(X) is unimodular.
Applying Proposition 10.5,
there is a qc-isometry dev:X→M which induces
a qc-isometry of X/Γ onto M/ρ(Γ).
Since K≤Pshqc(M)0,
K~=R3≤CPshqc(X)(Γ). Thus
ρ(Γ) centralizes ρ(R3)=C(M)=R3.
As the Sp(1)-action conjugates R3,
it follows ρ(Γ)≤M⋊Sp(n).
∎
Proposition 10.7**.**
Let R→X/R2⟶q1Y
be the pseudo-Hermitian
bundle as in (2) of Theorem 5.4 where
(X/R2,η^α,J^α′)
is a 4n+1(≥9)-dimensional CR-manifold
with Reeb field ξ^α
generating R≤R3(α=1,2,3).
If X/R2 is spherical, then X is locally qc-isometric to M.
Proof.
Since X/R2 is spherical CR,
the pseudo-Hermitian bundle
shows (Y,g) is a
Bochner flat Kähler manifold where g=Ωα∘Jα,
dη^α=q1∗Ωα
(cf. [33], [20]).
By the result of [30],
any Bochner flat Kähler Einstein manifold
is the space of constant holomorphic sectional curvature.
As (Y,g) is hyperKähler,
Y is locally holomorphically
isometric to the flat space C2n.
Applying the proof of Proposition 10.4,
X is locally qc-isometric to M.
∎
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