Bochner and Conformal Flatness of Normal Metric Contact Pairs

0.1 MB·English

by  Gianluca Bande

BOCHNER AND CONFORMAL FLATNESS OF NORMAL METRIC CONTACT
PAIRS
5 GIANLUCABANDE,DAVIDE.BLAIR,ANDAMINEHADJAR
1
0
2

r
a ABSTRACT. Weprovethatthenormalmetriccontactpairswithorthogonalcharacteristicfoliations,
M whichareeitherBochner-flatorlocallyconformallyflat,arelocallyisometrictotheHopfmanifolds.
Asa corollaryweobtaintheclassificationoflocallyconformallyflatandBochner-flatnon-Ka¨hler

0 Vaismanmanifolds.
3

] 1. INTRODUCTION
G
D In 1949 S. Bochner [13] introduced in Ka¨hler geometry a new tensor field as a formal ana-
. logue of the Weyl conformal curvature tensor. The Bochner tensor was extended from the Ka¨hler
h
t manifold setting to general almost Hermitian manifolds by Tricerri and Vanhecke [16]. Whilethe
a
m BochnertensorandBochner-flatnesshavebeenstudiedinKa¨hlergeometrybyanumberofauthors
over the years, there have not been many applications to Hermitian manifolds that are not always
[

Ka¨hler. In [12] V. Mart´ın-Molinaand the second author showed that there are no conformally flat
3
v normal complex contact metric manifolds but that a Bochner-flat normal complex contact metric
2 manifold must be Ka¨hler and locally isometric to CP2n+1(4). It should be remarked that the no-
0
tion ofnormality used is that due to Korkmaz[15]and includes such non-Ka¨hlernormal complex
6
6 contact metricmanifoldsas thecomplexHeisenberggroup.
0
A normal metric contact pair (with decomposable φ) carries two complex structures and hence
.
1 two Bochner tensors. In the present paper we show that if either Bochner tensor vanishes the
0
manifoldis locally isometricto theHopfmanifold S2m+1(1)×S1. As a corollary we will see that
5
1 a conformally flat normal metric contact pair must also be locally isometric to a Hopf manifold.
:
v Moreover,asabypassresult,werecovertheclassificationoflocallyconformallyflatandBochner-
i flat non-Ka¨hlerVaismanmanifolds.
X
r
a
2. PRELIMINARIES
Contact pairs were introduced by G. D. Ludden, K. Yano and the second author in [11] under
the name bicontact and by the first and third authors in [1, 4] with the name contact pair. A pair
of1-forms(α ,α ) onamanifoldM issaidto beacontactpairof type(m,n) if
1 2
α ∧(dα )m ∧α ∧(dα )n is avolumeform,
1 1 2 2
(dα )m+1 = 0and (dα )n+1 = 0.
1 2
Whileitispossibletoconsideracontactpairoftype(0,0),itseemsmostnaturaltorequireatleast
oneoftheforms toresembleacontactform, sothat at leastoneofmornwillbepositive.
Date:March31,2015;MSC2010classification:primary53C55;secondary53C25,53D10,53A30,53C15,53D15.
Keywordsandphrases. Normalmetriccontactpairs,Bochner-flat,locallyconformallyflat,Vaismanmanifolds.
ThefirstauthorissupportedbyP.R.I.N.2010/11–Varieta`realiecomplesse:geometria,topologiaeanalisiarmon-
ica–Italy.
1
2 GIANLUCABANDE,DAVIDE.BLAIR,ANDAMINEHADJAR
Wecan naturallyassociatetoacontact pairtwosubbundlesofthetangentbundleTM:
{X : α (X) = 0,dα (X,Y) = 0 ∀ Y}, i = 1,2
i i
Thesesubbundlesareintegrable[1,4]anddeterminethecharacteristicfoliationsofM,denotedF
1
andF respectively. Thecharacteristicfoliationsaretransverseandcomplementaryandtheleaves
2
of F and F are contact manifolds of dimension 2n + 1 and 2m + 1 respectively, with contact
1 2
forms induced by α and α . We also define the (2m+2n)-dimensional horizontalsubbundle H
2 1
tobetheintersectionofthekernels ofα and α .
1 2
Theequations
α (Z ) = α (Z ) = 1, α (Z ) = α (Z ) = 0,
1 1 2 2 1 2 2 1
i dα = i dα = i dα = i dα = 0
Z1 1 Z1 2 Z2 1 Z2 2
where i is the contraction with the vector field X, determine uniquely the two vector fields Z
X 1
and Z , called Reeb vector fields. Since they commute, they give rise to a locally free R2-action,
2
called theReeb action.
A contact pair structure [5] on a manifold M is a triple (α ,α ,φ), where (α ,α ) is a contact
1 2 1 2
pairand φatensorfield oftype(1,1)such that:
φ2 = −Id+α ⊗Z +α ⊗Z , φZ = φZ = 0.
1 1 2 2 1 2
Therank ofφis dimM −2 and α ◦φ = 0 fori = 1,2.
i
The endomorphism φ is said to be decomposable if φ(TF ) ⊂ TF , for i = 1,2. When φ is
i i
decomposable,(α ,Z ,φ)(respectively(α ,Z ,φ))induces,oneveryleafofF (respectivelyF ),
1 1 2 2 2 1
a contact form and the restriction φ of φ to the leaf forms an almostcontact structure (α ,Z ,φ ).
i i i i
It is important to note that there exists contact pair structures with decomposable φ which are not
locallyproducts[6].
In [6] the first and third authors introduced two natural almost complex structures on the mani-
foldM by
J = φ−α ⊗Z +α ⊗Z , T = φ+α ⊗Z −α ⊗Z .
2 1 1 2 2 1 1 2
The contact pair structure is said to be normal if both of these almost complex structures are
integrable.
Onmanifoldsendowedwithcontactpairstructuresitisnaturaltoconsiderthefollowingmetrics
[5]. Let (α ,α ,φ) be a contact pair structure on M. A Riemannian metric g on M is said to be
1 2
associated if
g(X,φY) = (dα +dα )(X,Y),
1 2
g(X,Z ) = α (X), i = 1,2.
i i
A metriccontact pair on a manifold M is a four-tuple (α ,α ,φ,g) where (α ,α ,φ) is a contact
1 2 1 2
pair structure and g an associated metric with respect to it. Such a manifold will also be called a
metriccontactpair. Wenotethatonanormalmetriccontactpair,theReebvectorfieldsareKilling.
For a metric contact pair, (α ,α ,φ,g), the endomorphism field φ is decomposable if and only
1 2
ifthecharacteristicfoliationsF ,F areorthogonal. Inthiscase(α ,φ,g)inducesacontactmetric
1 2 i
structure (α ,φ ,g) on the leaves of F , for j 6= i . Thus we assume the decomposability of φ
i i j
throughout. By thenormalityeach (α ,φ ,g) is a Sasakian structure on each leaf ofthe character-
i i
isticfoliations. Moreovertheleavesareminimalsubmanifolds[7].
BOCHNERANDCONFORMALFLATNESSOFNORMALMETRICCONTACTPAIRS 3
In the course of our work we will need the following lemmas. Some formulas from [2, 5] can
besummarizedas follows:
Lemma 1. Ona normalmetriccontactpairwith decomposableφ,forevery X we have
∇ Z = −φ X, ∇ Z = −φ X.
X 1 1 X 2 2
Usingthepreviouslemma,wecan restateCorollary 3.2of[3]as follows:
Lemma2. Onanormalmetriccontactpairwithdecomposableφ,thecovariantderivativeofφis
given by
2
g((∇Xφ)Y,V) = X(cid:0)dαi(φY,X)αi(V)−dαi(φV,X)αi(Y)(cid:1)
i=1
andhence forthealmostcomplexstructureJ wehave
2
g((∇XJ)Y,V) =X(cid:0)dαi(φY,X)αi(V)−dαi(φV,X)αi(Y)(cid:1)
i=1
−dα (X,Y)α (V)−dα (X,V)α (Y)
2 1 1 2
+dα (X,Y)α (V)+dα (X,V)α (Y).
1 2 2 1
OurconventionsforthecurvaturetensorofaRiemannian manifoldare
R(X,Y)V = ∇ ∇ V −∇ ∇ V −∇ V,
X Y Y X [X,Y]
R(X,Y,V,W) = g(R(X,Y)V,W).
Let Z = Z +Z ; from Theorem 2 of [8] or the proof of Lemma 2.1 of [2] one readily has the
1 2
followinglemma.
Lemma 3. Ona normalmetriccontactpairwith decomposableφ,
g(R Z,V) =dα (φV,X)α (Y)+dα (φV,X)α (Y)
XY 1 1 2 2
−dα (φV,Y)α (X)−dα (φV,Y)α (X).
1 1 2 2
Lemma4. OnanormalmetriccontactpairwithdecomposableφletX bealocalunitvectorfield
inTF ∩H. Then
2
R(X,Z ,Z ,X) = 1, R(X,Z ,Z ,X) = 0, R(X,Z ,Z ,X) = 0.
1 1 1 2 2 2
Proof. UsingLemmas1 and 2,wehave
R(X,Z ,Z ,X) =g(−∇ ∇ Z −∇ Z ,X)
1 1 Z1 X 1 [X,Z1] 1
=g(∇ φ X +φ [X,Z ],X)
Z1 1 1 1
=g((∇ φ )X +φ (∇ Z ),X)
Z1 1 1 X 1
=g(−φ2X,X)
1
=1.
Similarly
R(X,Z ,Z ,X) = g(−∇ ∇ Z −∇ Z ,X) = 0.
1 2 Z1 X 2 [X,Z1] 2
4 GIANLUCABANDE,DAVIDE.BLAIR,ANDAMINEHADJAR
Finally,sinceZ is Killing,
2
R(X,Z ,Z ,X) =g(−∇ Z ,X)
2 2 [X,Z2] 2
=g(∇ Z ,X)
∇Z2X 2
=−g(∇ Z ,∇ X)
X 2 Z2
=0.
(cid:3)
We denote by ρ(R) or ρ(X,Y) or simplyρ the Ricci tensor of R and by τ the scalar curvature.
For an almost Hermitian manifold with almost complex structure J, we recall the ∗-Ricci tensor
defined by ρ (X,Y) = R(X,e ,Je ,JY) where {e } is an arbitrary orthonormal basis. In
∗ Pi i i i
general ρ is not symmetric but it does satisfy ρ (X,Y) = ρ (JY,JX). The trace of ρ , denoted
∗ ∗ ∗ ∗
τ , iscalled the∗-scalarcurvature.
∗
A normal metriccontact pair carries two Hermitian structures (g,J) and (g,T), with respect to
thesamemetricg,givingriseto apairof∗-Ricci tensors. In thesequel,ρ willdenotethe∗-Ricci
∗
tensorof(g,J).
Lemma 5. Ona normalmetriccontactpairwith decomposableφ,
ρ (X,Y) =ρ(X,Y)−(2m−1)g(φ X,φ Y)−(2n−1)g(φ X,φ Y)
∗ 1 1 2 2
−2mα (X)α (Y)−2nα (X)α (Y).
1 1 2 2
Moreover ρ issymmetricand J-invariantand wehave
∗
τ −τ = 4(m2 +n2).
∗
Proof. On an almost Hermitian manifold we have the following relation between the Ricci tensor
and ∗-Ricci tensor,[18, p. 195]
(ρ −ρ )Jt = ∇ ∇ Jt −∇ ∇ Jt .
jt ∗jt i t j i j t i
From Lemmas1 and2 wehave
∇ Jt = −2m(α ) −2n(α )
t i 1 i 2 i
and differentiating
∇ ∇ Jt = −2m(φ ) −2n(φ ) .
j t i 1 ji 2 ji
In likemanner, differentiationoftheformulaofLemma2 alsoyields
∇ ∇ Jt = −φ −2m(α ) (α ) +2n(α ) (α ) .
t j i ji 2 i 1 j 1 i 2 j
Usingthesewehave
(ρ −ρ )Jt = (2m−1)(φ ) +(2n−1)(φ ) −2m(α ) (α ) +2n(α ) (α ) .
jt ∗jt i 1 ji 2 ji 2 i 1 j 1 i 2 j
Orinan invariantlanguage
ρ (Y,JX)−ρ(Y,JX) =−(2m−1)dα (Y,X)−(2n−1)dα (Y,X)
∗ 1 2
+2mα (Y)α (X)−2nα (X)α (Y).
1 2 1 2
BOCHNERANDCONFORMALFLATNESSOFNORMALMETRICCONTACTPAIRS 5
Replacing X byJX wehave
ρ(Y,X)−ρ (Y,X) =(2m−1)g(φ Y,φ X)+(2n−1)g(φ Y,φ X)
∗ 1 1 2 2
+2mα (X)α (Y)+2nα (X)α (Y)
1 1 2 2
as desired. By the previous relation and the symmetry of ρ we get the symmetry of ρ . Next, by
∗
the fact that ρ (X,Y) = ρ (JY,JX), we obtain the J-invariance of ρ . Contracting in the same
∗ ∗ ∗
relation,wehaveforthescalar curvatureτ −τ = 4(m2 +n2). (cid:3)
∗
Nowconsideralocal orthonormalbasis {E ,...,E ,E ,...,E ,Z ,Z } wherethefirst
1 m m+1 m+n 1 2
m vector fields are tangent to the leaves of F and the next n tangent to the leaves of F . From
2 1
Lemma3 wehavethefollowing.
m
ρ(Z1,Z1 +Z2) = Xdα1(φEi,Ei) = 2m.
i=1
Similarlyρ(Z ,Z +Z ) = 2n. Forsimplicityweabbreviateρ(Z ,Z )byρ andthesameforρ .
2 1 2 i j ij ∗
Wethen havethefollowinglemma.
Lemma 6. On a normal metric contact pair with decomposableφ, ρ is J-invariant on horizontal
vectors, i.e. ρ(JX,JY) = ρ(X,Y), and
ρ = 2m, ρ = 2n, ρ = 0, ρ = ρ = ρ = 0.
11 22 12 ∗11 ∗22 ∗12
Proof. From the symmetry of ρ and the formula of Lemma 5, the J-invariance of ρ restricted to
∗
H is immediate. By the J-invariance of ρ we have ρ (Z,Z) = ρ (Z − Z ,Z − Z ) giving
∗ ∗ ∗ 2 1 2 1
ρ = 0. The formula of Lemma 5 then gives ρ = 0. We noted above that ρ(Z ,Z) = 2m and
∗12 12 1
ρ(Z ,Z) = 2nand thereforeρ = 2mandρ = 2n. Finally Lemma5 givesρ = ρ = 0. (cid:3)
2 11 22 ∗11 ∗22
3. THE BOCHNER CURVATURE TENSOR
In [16] F. Tricerri and L. Vanhecke gave a complete decomposition of the space of curvature
tensorsoveraHermitianvectorspaceintoirreduciblefactorsundertheactionoftheunitarygroup
similar to the well known decompostion of curvature tensors in the Riemannian setting with the
action of the orthogonal group. In the real case the Weyl conformal curvature tensor emerges
in a natural manner and correspondingly the Bochner tensor for an almost Hermitian manifold
emerges as one factor ofthe decomposition. The Bochner tensor in almost Hermitiangeometry is
considerably more complicated than that in Ka¨hler geometry and as a result has not been studied
extensively.
Define(0,4)-tensorsπ , π and L R by:
1 2 3
π (X,Y,Z,W) = g(X,Z)g(Y,W)−g(Y,Z)g(X,W),
1
π (X,Y,Z,W) = 2g(JX,Y)g(JZ,W)+g(JX,Z)g(JY,W)−g(JY,Z)g(JX,W),
2
L R(X,Y,Z,W) = R(JX,JY,JZ,JW).
3
6 GIANLUCABANDE,DAVIDE.BLAIR,ANDAMINEHADJAR
Givena(0,2)-tensorS,we denoteby ϕ(S)and ψ(S):
ϕ(S)(X,Y,Z,W) =g(X,Z)S(Y,W)+g(Y,W)S(X,Z)
−g(X,W)S(Y,Z)−g(Y,Z)S(X,W),
ψ(S)(X,Y,Z,W) =2g(X,JY)S(Z,JW)+2g(Z,JW)S(X,JY)
+g(X,JZ)S(Y,JW)+g(Y,JW)S(X,JZ)
−g(X,JW)S(Y,JZ)−g(Y,JZ)S(X,JW).
Takingintoaccount oursignconventionfor thecurvaturetensorand complexdimensionofour
complexmanifold,thenormalmetriccontactpairM,theBochnertensorofTricerriandVanhecke
is given by the following. Given a metric contact pair of complex dimension m+n+1 > 2, the
BochnertensorcorrespondingtothecomplexstructureJ isdefined as
1 1
B =R+ ψ(ρ )(R−L R)+ ϕ(ρ)(R−L R)
∗ 3 3
4(m+n+2) 4(m+n)
1
+ (ϕ+ψ)(ρ+3ρ )(R+L R)
∗ 3
16(m+n+3)
1
+ (3ϕ−ψ)(ρ−ρ )(R+L R)
16(m+n−1) ∗ 3
τ +3τ τ −τ
∗ ∗
− (π +π )− (3π −π ).
16(m+n+2)(m+n+3) 1 2 16(m+n−1)(m+n) 1 2
We will work almost exclusively with this complex structure. When discussing both J and T we
willdenotethecorrespondingBochnertensors byB and B respectively.
J T
Whenthecomplexdimensionis2,theformulafortheBochnertensorissomewhatdifferentand
theformulagivenin [16] isinerror; thecorrect formulaisthefollowing.
1 1
B =R+ ψ(ρ )(R−L R)+ ϕ(ρ)(R−L R)
∗ 3 3
12 4
1 τ +3τ τ −τ
∗ ∗
+ (ϕ+ψ)(ρ+3ρ )(R+L R)− (π +π )+ (3π −π ).
∗ 3 1 2 1 2
64 192 32
4. MAIN RESULTS
We now turn to our main result on Bochner-flatness and then discuss thequestion of conformal
flatness as acorollary.
Theorem 7. Let (M,α ,α ,φ,g) be a normalmetric contact pair with decomposableφ. If either
1 2
of the two Bochner tensors, B or B , vanishes, then M is locallyisometricto the Hopf manifold
J T
S2m+1(1)×S1.
The proof will be given in three stages. We will first prove the theorem for the Bochner tensor
B to be denoted simply by B for complex dimension m + n + 1 > 2 (Stage 1). Then we will
J
provethetheoremforthecaseofcomplexdimension2(Stage2). Finallyweindicatetheprooffor
theBochnertensorB (Stage3).
T
Proof. Stage 1,Step 1:
We begin by evaluatingB on the Reeb vector fields, in particular we have 0 = B(Z ,Z ,Z ,Z ).
1 2 2 1
We proceed term by term in the definition of B. That R(Z ,Z ,Z ,Z ) = 0 is immediate since
1 2 2 1
∇ Z = 0. Thenext termin thedefinitionvanishesby virtueofρ beingJ-invariant(Lemma5).
Zi j ∗
BOCHNERANDCONFORMALFLATNESSOFNORMALMETRICCONTACTPAIRS 7
In the third term the Ricci tensors in ϕ(ρ)(R−L R) all cancel. We separate the fourth term into
3
twopartscorrespondingtotheactionof(ϕ+ψ)onρandon3ρ . Inthefifthtermweseparateinto
∗
the parts corresponding to 3ϕ(ρ − ρ ) and −ψ(ρ − ρ ) and these contribution cancel each other.
∗ ∗
Recallingthatτ−τ = 4(m2+n2)thesixthandseventhtermsareeasilyevaluated. Thuswehave
∗
1 τ −3(m2 +n2)
0 = B(Z ,Z ,Z ,Z ) = (−16m−16n)+ .
1 2 2 1
16(m+n+3) (m+n+2)(m+n+3)
Solvingforτ weobtain
(1) τ = 2m(2m+1)+2n(2n+1)+2mn.
Stage1, Step 2:
Since metric contact pairs of type (0,0) are not of interest, we suppose that m > 0. Then we may
choose the unit vector X in TF ∩H. Calculating as above using the results of Lemmas 4 and 6
2
wehavethefollowing.
1
0 =B(X,Z ,Z ,X) = 1+ (−(2m−2n))
1 1 4(m+n)
1 3
+ (−(2m+2n)−2ρ(X,X))+ (−2ρ (X,X))
∗
16(m+n+3) 16(m+n+3)
3
+ (−(2m+2n)−2ρ(X,X)+2ρ (X,X))
∗
16(m+n−1)
τ −3(m2 +n2) 3(m2 +n2)
+ +
4(m+n+2)(m+n+3) 4(m+n−1)(m+n)
m−n
(2) =1− +the additional terms.
2(m+n)
Similarly
0 =B(X,Z ,Z ,X)
2 2
1
= (−(2n−2m))+the same additional terms.
4(m+n)
Subtractingthesetwoequationswehave
m−n 2n
0 = 1− =
m+n m+n
givingn = 0and, from(1),τ = 2m(2m+1).
Stage1, Step 3:
Returningto equation(2)withn = 0 and usingLemma5 wehave
1 1
0 =B(X,Z ,Z ,X) = + (−2m−2ρ(X,X))
1 1 2 16(m+3)
3 3
+ (−2ρ(X,X)+4m−2)+ (−6m+2)
16(m+3) 16(m−1)
τ −3m2 3m2
+ + .
(m+2)(m+3) 4m(m−1)
8 GIANLUCABANDE,DAVIDE.BLAIR,ANDAMINEHADJAR
Withτ = 2m(2m+1) wemaysolveforρ(X,X) and wehave
ρ(X,X) = 2m and ρ (X,X) = 1.
∗
Stage1, Step 4:
ThevectorfieldZ restrictedtoaleafofF isthenormaltotheleafasasubmanifold. By Lemma
2 2
1, Z is parallel along the submanifold giving that the leaves are totally geodesic submanifolds.
2
MoreoveritsownintegralcurvesonM aregeodesics. ThereforeM isalocalRiemannianproduct.
FinallywithX as above,wecompute0 = B(X,φX,φX,X) obtaining
0 = B(X,φX,φX,X) = R(X,φX,φX,X)−1.
Therefore the leaves of F are Sasakian manifolds of constant φ-sectional curvature +1. With
2
n = 0 and m+n+1 > 2, the leaves are of dimension ≥ 5; but a Sasakian manifold of constant
φ-sectional curvature +1 and dimension ≥ 5 must be of constant curvature +1 (see e.g. [10, p.
139]).
Asaresult theuniversalcoverofM is S2m+1(1)×R, completingStage 1oftheproof.
Stage2:
The proof of the theorem in dimension 4 is very similar but using the formula for the Bochner
tensor in this dimension. Since m+n+1 = 2, one of m or n vanishes and we take n = 0 at the
outset. Thecomputationof0 = B(Z ,Z ,Z ,Z )isstraightforwardandgives,asinthederivation
1 2 2 1
ofequation(1),
τ
ρ +ρ = 1+ .
11 22
6
From Lemma 6, ρ = 2 and ρ = 0, therefore τ = 6. We also have for X unit and horizon-
11 22
tal ρ(X,X) = ρ(JX,JX) and we then obtain ρ(X,X) = 2. Since X was arbitrary as a unit
horizontalvectorwealso haveρ(X+JX, X+JX) = 2 and henceρ(X,JX) = 0.
√2 √2
As in Stage 1, Step 4, M is locally the Riemannian product of a Sasakian manifold N3 of R.
Since Z is the Reeb vector field of a Sasakian manifold, Z is pointwise an eigenvector of the
1 1
Ricci operator(see e.g. [10, p. 113]), whichgivesus ρ(Z ,X) = ρ(Z ,JX) = 0. Therefore, with
1 1
ρ(X,JX) = 0 and ρ = ρ(X,X) = ρ(JX,JX) = 2 we have that N3 is Einstein and in turn of
11
constantcurvature+1.
Stage3:
RecallingthatJ = φ−α ⊗Z +α ⊗Z andT = φ+α ⊗Z −α ⊗Z ,notethatonhorizontal
2 1 1 2 2 1 1 2
vectors these are the same and that they act with opposite signs on vertical vectors. In particular
this means that interchanging the roles of the type numbers m and n, we have that the Bochner
tensorsB andB areinterchanged. ThustoshowthatB = 0impliesthatM islocallyisometric
J T T
to the Hopf manifold, it is enough to return to Stage 1, Step 2 and work through the rest of the
proof starting with n > 0 instead of m > 0 and X ∈ TF ∩ H. This proceeds in the same way
1
givingm = 0,etc. In the4-dimensionalcase, work withm = 0 tobeginwith. (cid:3)
Remark 8. Observe that a priori the two Bochner tensors are not the same. For example on a,
not necessarily Bochner-flat, normal metric contact pair B (X,JX,Z ,Z ) is the negative of
T 1 2
B (X,JX,Z ,Z ) forahorizontalX.
J 1 2
IntheStage2oftheproofofTheorem7,weonlyusethenormalityconditionandtheassumption
B(Z ,Z ,Z ,Z ) = 0. Hence:
1 2 2 1
BOCHNERANDCONFORMALFLATNESSOFNORMALMETRICCONTACTPAIRS 9
Theorem 9. A normal metric contact pair of dimension 4 on which either B (Z ,Z ,Z ,Z ) or
J 1 2 2 1
B (Z ,Z ,Z ,Z ) vanishes,islocallyisometrictotheHopfmanifoldS3(1)×S1.
T 1 2 2 1
Turning to the conformally flat question, we have the following theorem as a corollary of our
Theorem 7.
Theorem 10. Let (M,α ,α ,φ,g)be anormalmetriccontactpair withdecomposableφ. If M is
1 2
conformallyflat, thenitis locallyisometricto theHopfmanifoldS2n+1(1)×S1.
Proof. As a complex manifold, if a normal metric contact pair is conformally flat, it is locally
conformal to Cn+1 and Cn+1 is Bochner-flat. Since the Bochner tensor is a conformal invariant
[16, Theorem11.1]themanifoldisBochner-flat and hencelocallyisometricto theHopfmanifold
by Theorem7. (cid:3)
Weend thepaperwithanapplicationtoVaismanmanifolds. Recall thatonaVaismanmanifold
the Lee form is parallel, then it has constant length c. The manifold is non-Ka¨hler if and only
if c 6= 0 and after a constant rescaling of the metric we can achieve that c = 1. In [9] it was
shownthat,uptoconstantrescalingofthemetric,thereisabijectionbetweennon-Ka¨hlerVaisman
manifoldsand normalmetriccontactpairs oftype(n,0) (thelattercorrespondingto c = 1).
Therefore we can apply our previous results to give a classification of locally conformally flat
and Bochner-flat non-Ka¨hlerVaismanmanifolds:
Corollary 11. Let M be a (2n + 2)-dimensional non-Ka¨hler Vaisman manifold. If M is either
Bochner-flat or locally conformally flat then, after a rescaling of the metric by the inverse of the
lengthoftheLee form,itislocallyisometricto theHopfmanifoldS2n+1(1)×S1.
For the locally conformally flat case, the result was already proven by Vaisman [17, Theorem
3.8]. TheBochner-flat casewasprovenbyKashiwada[14]whichshowedanequivalencebetween
locallyconformallyflatness and Bochner-flatness forVaismanmanifolds.
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