Goldberg conjecture and the related topics

Goldberg conjecture and the related topics

Takashi Oguro

Department of Mathematical Sciences, College of Science and Engineering, Tokyo Denki University, Saitama, 350-0394,

Japan.

e-mail : [email protected] Kouei Sekigawa

Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950- 2181,

Japan

e-mail : [email protected] Akira Yamada

Division of General Education, Nagaoka National College of Technology, Niigata, 940-8532,

Japan

e-mail : [email protected]

(2000AMS Mathematics Subject Classification: 53C15, 53C25.)

1. Introduction

Concerning the integrability of almost K¨ahler manifolds, the following conjec- ture proposed by S. I. Goldberg [6] is well-known.

Conjecture . The almost complex structure of a compact almost K¨ahler Einstein manifold is integrable.

The above conjecture is true in the case where the scalar curvature is non- negative. Part 2 will be devoted to the brief survey on the development on the Goldberg conjecture. More precisely, in §2.2, we recall the outline of the proof.

However, we have also to emphasize that the conjecture is still open in the remaining case where the scalar curvature is negative. In §2.3, we introduce several partial affirmative answers to the conjecture and also non-compact counter examples. Part

1

3 will be devoted to the related topics on contact geometry. More precisely, we shall introduce the work by Boyer and Galicki [4] relate to the partial answer in the case of non-negative scalar curvature for the Goldberg conjecture.

2. Goldberg Conjecture

2.1. Preliminaries and integral formulas

We shall prepare some fundamental formulas in almost K¨ahler manifolds. First, letM = (M, J, g) be an almost Hermitian manifold with almost Hermitian structure (J, g). We denote by Ω the K¨ahler form ofM defined by Ω(X, Y) =g(X, JY),X, Y ∈ X(M) (X(M) is the Lie algebra of smooth vector fields on M), and assume thatM is oriented by the voloume formdv= (−1)ⁿΩⁿ

n! (dimM = 2n). We denote byR,ρandτ the curvature tensor, the Ricci tensor and the scalar curvature ofM defined respectively by

R(X, Y)Z = [∇X,∇Y]Z− ∇[X,Y]Z, ρ(X, Y) = Trace{Z 7→R(X, Z)Y},

τ= Tracegρ= TraceQ

(Q is the Ricci transformation), for X, Y, Z ∈ X(M). Moreover, we define the Ricci∗-tensorρ^∗ by

ρ^∗(X, Y) = Trace{Z7→JR(Z, X)JY} and∗-scalar curvtatureτ^∗ by

τ^∗= Tracegρ^∗= TraceQ^∗

(Q^∗ is the Ricci ∗-transformation given by g(Q^∗X, Y) = ρ^∗(X, Y)). The Ricci

∗-tensorρ^∗ satisfies the following identity

ρ^∗(JX, JY) =ρ^∗(Y, X),

and coincides with the Ricci tensorρifM is K¨ahler (∇J = 0). An almost Hermitian manifold M is called a weakly ∗-Einstein manifold if ρ^∗ = τ^∗

2ng holds. Further, a weakly ∗-Einstein manifold is called a ∗-Einstein manifold if τ^∗ is constant. We

may also the curvature tensor as a (0,4)-tensor and an endomorphism of the vector bundle of 2-forms∧²(T^∗M) respectively as follows:

R(X, Y, Z, W) =g(R(X, Y)Z, W),

< R(φ1∧ψ1), φ2∧ψ2>=R(φ^#₁, ψ^#₁, φ^#₂, ψ₂^#),

for X, Y, Z, W ∈ X(M) and φi, ψi ∈Γ(M, T^∗M) (i= 1,2). Here the symbol # is the natural isomorphism T^∗M →T M and<·,·>is the inner product induced fromg. Let{ei}be a local orthonormal frame field andRijkl,ρij,ρ^∗_ijbe components ofR,ρandρ^∗ with respect to{ei} defined respectively by

Rijkl =R(ei, ej, ek, el), ρij =ρ(ei, ej), ρ^∗_ij =ρ^∗(ei, ej).

Indices with bar are the ones with respect to {Jei}, for example, R¯ijkl=R(Jei, ej, ek, el).

Using this notational convention, we get ρij =−X

a

Riaja,

ρ^∗_ij =−1 2

X

a

R_i¯ja¯a, τ =−X

a,b

R_abab,

and the Ricci∗-tensorρ^∗ satisfiesρ^∗_ij =ρ^∗¯j¯i. We assume that the indicesa,b,· · ·, i,j,· · · run over the range 1, 2,· · ·, 2nand the indicesα,β,· · · run over the range 1, 2, · · ·, n. We denoteJij and∇iJjk by

Jij=g(Jei, ej),

∇iJjk =g((∇eiJ)ej, ek).

{J_ij}and{∇_iJ_jk} satisfy

(2.1.1) Jij=−Jji=J¯i¯j,

(2.1.2) ∇_iJ_jk=−∇_iJ¯j¯k.

Next, we shall recall some fundamental formulas on almost K¨ahler manifolds.

LetM = (M, J, g) be a 2n-dimensional almost K¨ahler manifold. Then we have the following identities in addition to (2.1.1) and (2.1.2),

(2.1.3) ∇iJjk=−∇¯iJjk,

(2.1.4) X

a

∇aJia= 0,

(2.1.5) X

∇bJik· ∇aJjk· ∇¯aJil· ∇¯bJjl= 0,

(2.1.6) X

a

∇²_a¯iJ¯ja=−X

a

∇²_aiJja+X

a,b

∇aJ_b¯i· ∇aJbj,

(2.1.7) ρ^∗_ij+ρ^∗_ji=ρij+ρ¯i¯j+X

a,b

∇aJbi· ∇aJbj.

Here, we note that (2.1.7) can be derived from the following equality obtained by A. Gray

Rijkl−R_ijk¯¯l−R¯i¯jkl+R¯i¯j¯k¯l+R¯ijkl¯ +R¯ijk¯l+R_i¯jkl¯ +R¯i¯j¯k¯l

=2X

a

∇aJij· ∇aJkl. (2.1.8)

Now, we define smooth functionsfi onM by f1=X

Rabij(R_¯a¯bij−R_¯a¯b¯i¯j), f2=X

Ra¯aij(R_b¯bij−R_b¯b¯i¯j) = 2X

(ρ^∗_ij−ρ^∗_ji)², f3=X

Ra¯aij∇¯bJik· ∇bJjk=−2X

ρ^∗_ij∇bJik· ∇bJjk

=−X

(ρ^∗_ij+ρ^∗_ji)∇bJik· ∇bJjk

=−2X

ρij∇bJik· ∇bJjk−X

(∇aJci)· ∇aJcj· ∇bJik· ∇bJjk, f4=X

Rabij∇bJik· ∇aJjk, f5=X

(Rijab+R_i¯j¯a¯b−R_ij¯a¯b−R¯i¯jab)².

Letϕbe an endomorphism ofT M defined by ϕ(X) =−X

a

(∇eaJ)(∇eaJ)X

forX ∈TxM, where {ea} is an orthonormal frame ofTxM. The mapϕsatisfies ϕ(JX) =Jϕ(X), g(ϕ(X), Y) =g(X, ϕ(Y)).

Hence, there exists a unitary frame{eα, en+α=Jeα}atx∈M andλα=λn+α∈R such that ϕ(eα) = λαeα, ϕ(en+α) = λn+αen+α. We may define a non-negative continuous function f by

f(x) =X

a,b

(λa−λb)².

We here remark that eigenvalues{λa}ofϕare nonnegative because we get λα=g(ϕ(eα), eα) =−X

i

g((∇eαJ)(∇eαJ)ei, ei)

=X

k(∇eαJ)eik². Using the above unitary frame, we get

(2.1.9) X

a,b

∇aJbi· ∇aJbj=λiδij. In particular, we have

(2.1.10) k∇Jk²(x) =X

a

λa = 2X

α

λα

and hence

(2.1.11) f(x) = 4nX

a

λ²_a−2k∇Jk⁴(x).

From (2.1.9)∼(2.1.11), we obtain

(2.1.12) f3=−2X

i

λiρii− 1

2nk∇Jk⁴− 1 4nf.

Letξbe a vector field onM defined by ξ=X

R((∇ebJ)ek, Jek)eb

=X

a



X

b,i,j,k

R_ijab∇_bJ_ik·J_jk



e_a.

Then, by direct calculation, we get divξ=−X

(∇iρjb− ∇jρib)∇bJik·Jjk

+X

Rijab(∇²_abJik)Jjk+X

Rijab∇bJik· ∇aJjk

=−X

(∇iρjb− ∇jρib)∇bJik·Jjk−1 4

X(Rabij−R_ab¯i¯j)²+f4. (2.1.13)

On one hand, we get f₅=2X

(R_ijab−R_ij¯a¯b)²−2X

(R_ijab−R_ij¯a¯b)(R¯i¯jab−R¯i¯j¯a¯b)

=2X

(Rijab−R_ij¯a¯b)²−4f1. (2.1.14)

Thus, by (2.1.13) and (2.1.14), we have (2.1.15) divξ=−X

(∇iρjb− ∇jρib)∇bJik·Jjk−1 8f5−1

2f1+f4.

Next, we establish an integral formula on a compact almost K¨ahler manifold.

Let M = (M, J, g) be a compact almost K¨ahler manifold. We define the linear connection∇⁰ by

∇⁰_XY =∇XY −1

2J(∇XJ)Y.

The curvature tensorR⁰ of∇⁰ is then given by R⁰(X, Y)Z=1

2(R(X, Y)Z−JR(X, Y)JZ)

−1

4((∇XJ)(∇YJ)Z−(∇YJ)(∇XJ)Z) (2.1.16)

Ã

R⁰_ijkl=1

2(R_ijkl+R_ijk¯¯l) +X

a

(∇_iJ_ka· ∇_jJ_la− ∇_iJ_la· ∇_jJ_ka)

! . The first Pontrjagin formp1(∇) of the connection∇ is defined by

p1=− 1

8π²Trace(F∧F), whereF_b^a= 1

2 X

i,j

Rijabeⁱ∧e^j is the curvature form of∇((eⁱ)^#=ei) and we have

p1(∇) = 1 32π²

XRijabRklabeⁱ∧e^j∧e^k∧e^l.

We also denote the first Pontrjagin form p1(∇⁰) of ∇⁰. The first Pontrjagin class ofT M which is the de Rham cohomology class whose representative element is the

first Pontrjagin form of a certain linear connection on M is independent of linear connections onM. This implies that (p1(∇)−p1(∇⁰)) is an exact 4-form. Since the K¨ahler form Ω is closed, (p1(∇)−p1(∇⁰))∧Ωⁿ⁻²is an exact 2n-form. Thus, from Stokes’ theorem, we obtain

(2.1.17)

Z

M

(p1(∇)−p1(∇⁰))∧Ωⁿ⁻²= 0.

Let {eα, en+α = Jeα} be a local unitary frame field and {eⁱ} be the dual frame field. Then, the K¨ahler form Ω is given in the form

Ω =−X

α

e^α∧Je^α, where J acts onT^∗M by

Jθ(X) =−θ(JX).

A simple calculation shows p1(∇)∧Ωⁿ⁻²=(−1)ⁿ

32π² (n−2)!X

(R_i¯iabR_j¯jab−2RijabR¯i¯jab)dv.

Using (2.1.6), we have the following integral formula (2.1.18)

Z

M

µ f1−1

2f2+f3−2f4

¶ dv= 0.

On one hand, from (2.1.15), we have also (2.1.19)

Z

M

µ

f1−2f4+1

4f5+ 2X

(∇iρjb− ∇jρib)∇bJik·Jjk

¶ dv= 0.

2.2. The case of non-negative scalar curvature

First, we shall give the outline of the proof of the following result ([18]).

Theorem A. The conjecture is true in the case where the scalar curvature is non- negative.

By (2.1.18) and (2.1.19), we have 0 =

Z

M

n

−1

2f2+f3−1 4f5

−2X

(∇iρjb− ∇jρib)∇bJik·Jjk

o dv

and hence, by virtue of (2.1.12), Z

M

( 1 4nf+ 1

2nk∇Jk⁴+ 2X

i

λiρii

) dv

= Z

M

½

−f5

4 −f2

2 + 2X

(∇iρjb− ∇jρib)∇bJik·Jjk

¾ dv 5

Z

M

X(∇iρjb− ∇jρib)∇bJik·Jjkdv.

(2.2.1)

From (2.1.20), we have immediately Theorem A.

Now, we introduce an extension of Theorem A by H. Satoh. LetW be the Weyl conformal tensor of type (0,4). Then we have

(2.2.2) R=W + 1

4n−1B°∧g+ τ

4n(2n−1)g°∧g where B=ρ− τ

2ng is the traceless Ricci tensor and the symbol°∧is the Nomizu- Kulkarni product. The divergenceδW ofW is a (0,3)-tensor defined by

δW(X, Y, Z) =−Traceg∇W(·, X, Y, Z).

We remark thatδW = 0 is equivalent to

(2.2.3) (∇XC)(Y, Z)−(∇YC)(X, Z) = 0, where C=ρ− 1

2(2n−1)τ g. H. Satoh [17] proved the following by making use of the integral formulas (2.1.18), (2.1.19), (2.2.1)∼(2.2.3).

Theorem B. LetM = (M, J, g)be a compact almost K¨ahler manifold. If the Weyl conformal tensorW satisfiesδW = 0and the Ricci tensor is positive semi-definite, thenM is a K¨ahler manifold.

We shall write the outline of the proof. From (2.2.3), we have Z

M

X(∇iρjb− ∇jρib)∇bJik·Jjkdv

= 1

2(2n−1) X Z

M

((∇iτ)δjb−(∇jτ)δib)∇bJik·Jjkdv

=0.

(2.2.4)

Thus, from (2.2.1) with (2.2.4), we have immediately Theorem B.

Now, letM = (M, J, g) be a K¨ahler manifold whose Ricci tensor has two distinct constant eigenvaluesλandµ. We denote byDλandDµtheJ-invariant distributions on M corresponding to the eigenvalues λ and µ of Q (λ < µ), respectively. We assumeλµ >0. Let ¯gbe a Riemannian metric on M given by

¯

g(X, Y) =g(X, Y) for X, Y ∈Γ(Dλ),

¯

g(X, Y) =µ

λg(X, Y) for X, Y ∈Γ(Dµ),

¯

g(X, Y) =0 for X ∈Γ(Dλ), Y ∈Γ(Dµ).

Then, (M, J,g) is a K¨ahler Einstein manifold with scalar curvature 2nλ. Next, let¯ J¯be an almost complex structure onM given by

J¯=J on D_λ and J¯=−J on D_µ.

Then, we may easily check that (M,J,¯ ¯g) is an almost K¨ahler Einstein manifold.

Further, (M,J,¯ ¯g) is strictly if and only if (M, J, g) is irreducible.

Apostolov, Draghici and Moroianu [2] proved the following by making use of the integral formulas established in§2.1.

Theorem C. Let M = (M, J, g)be a compact K¨ahler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues λ and µ. Then the universal cover of M is the product of two simply connected K¨ahler Einstein manifolds of scalar curvatures λandµ, respectively.

The above arguments suggest us the possibility of the existence of counter examples to the conjecture. In fact, Apostolov, Draghici and Moroianu showed that there exist examples of non-compact, complete (homogeneous) strictly almost K¨ahler Einstein manifolds of negative scalar curvature. For example, they showed that such examples are constructed by deforming the given K¨ahler Einstein structure of the Hermitian symmetric spaces SO(n,2)/SO(2)×SO(n) (non-compact dual of complex quadratic spaces SO(n+ 2)/SO(2)×SO(n),n=3).

2.3. Four-dimensional cases

LetM = (M, J, g) be a 4-dimensional almost Hermitian manifold. It is known that the following identity holds on M.

(2.3.1) ρ^∗(X, Y) +ρ^∗(Y, X) =ρ(X, Y) +ρ(JX, JY) +τ^∗−τ

2 g(X, Y).

It is also known that the vector bundle∧²M is decomposed into the following (2.3.2) ∧²M =RΩ⊕LM⊕ ∧^1,1₀ M,

whereLM (resp. ∧^1,1₀ M) is the bundle ofJ-skew-invariant (J-invariant) 2-forms on M perpendicular to Ω. We can identify the subbundleRΩ⊕LM(resp. ∧^1,1₀ M) with the bundle∧²₊M (resp. ∧²₋M) of self-dual (resp. anti-self-dual) 2-forms onM. It is well-known thatMis Einstein if and only if the curvature operatorR:∧²M → ∧²M preserves∧²_±M and that the Weyl operator W is given by

(2.3.3) W =R− τ

12I.

Now, we assume that M = (M, J, g) is a 4-dimensional almost K¨ahler Einstein manifold. We define smooth functionsA,B,C,D,GandK onM by

A=X

(∇aJij)Rijkl∇aJkl, B=X

(∇aJij)(∇aJkl)(∇bJij)∇bJkl, C=X

RijklR_i¯jk¯l, D=X

(Rijkl−R_ijk¯¯l)² G=X

(ρ^∗_ij−ρ^∗_ji)² µ

= 1 2f2

¶ , K=(u−v)²+ 4w²,

whereu=< R(Φ),Φ>,v=< R(JΦ), JΦ>,w=< R(Φ), JΦ>, Φ = 1

√2(e¹∧e³− e²∧e⁴),JΦ = 1

√2(e¹∧e⁴+e²∧e³). Then, by direct calculation, we have

A= 1

2B= (τ^∗−τ)²

2 ,

C=−2K+(τ^∗−τ)²

8 ,

G= 4kρ^∗k²−(τ^∗)²= 16(ρ^∗₁₃²+ρ^∗₁₄²), K=(τ^∗−τ)²

16 −4 detR⁰_LM, kRLMk²= 1

16D, kR⁰_LMk²= 1

16(D−G), (2.3.4)

whereR_LM is the restriction ofRtoLMandR⁰_LM =p_LM◦R_LM, the composition ofRLM and the orthgonal projectionpLM:∧²M →LM. Further, the norm of the self-dual Weyl operatorW⁺ is given by

(2.3.5) kW⁺k²= 1

16 µ

G+D+ (τ^∗)²−τ² 3

¶ .

Now, letη be the vector field onM defined byηa=X

i,j

(∇aJij)ρ^∗_i¯j. Then, we have

(2.3.6) ∆τ^∗=G

2 + 4K+(3τ^∗−τ)(τ^∗−τ)

4 −4divη,

(2.3.7) D

2 −G

2 −4K−(τ^∗−τ)²

4 = 0.

Oguro, Sekigawa and Yamada [15] proved the following

Theorem D. LetM = (M, J, g)be a 4-dimensional strictly almost K¨ahler Einstein and weakly∗-Einstein manifold. Then,M is a Ricci-flat space of pointwise constant holomorphic sectional curvature τ^∗

8 (and hence, self-dual), whereτ^∗ is the ∗-scalar curvature ofM.

Corollary D. Let M = (M, J, g) be a 4-dimensional compact almost K¨ahler Ein- stein and weakly ∗-Einstein manifold. Then,M is a K¨ahler manifold.

The above theorem was also proved by J. Armstrong in his thesis independently.

The following example by Nurowski and Przanowski [13] is concerned with the result of Theorem D:

M ={(x1, x2, x3, x4)∈R⁴|x1>0},

g= (gij) =













x1 0 0 0

0 x₁+ x²₃ 4x1

−x2x3

4x1

x3

2x1

0 −x2x3

4x1

x1+ x²₂ 4x1

− x2

2x1

0 x3

2x1 − x2

2x1

1 x1











 ,

J = (Jji) =











0 0 −1 0

0 x3

2x1 −x2

2x1

1 x1

1 0 0 0

x2

2 −x1− x²₃ 4x1

x2x3

4x1

−x3

2x1









 .

Then, we get

Ω =−x1dx1∧dx3−x2

2 dx2∧dx3+dx2∧dx4

(and hencedΩ = 0). By direct calculation, we have ρ= 0 and ρ^∗= 1

x³₁g,

and hence (M, J, g) is a strictly almost K¨ahler Ricci-flat and weakly ∗-Einstein manifold with the∗-scalar curvatureτ^∗= 4

x³₁.

Quite recently, Apostolov, Calderbank and Gauduchon [1] obtained an exam- ple of 4-dimensional strictly almost K¨ahler, Ricci-flat and not weakly ∗-Einstein manifold.

Oguro and Sekigawa [14] also proved the following

Theorem E. Let M = (M, J, g)be a 4-dimensional almost K¨ahler Einstein mani- fold with constant∗-scalar curvature. ThenM is a K¨ahler manifold.

Recently, M. Itoh [8] proved the following by applying the Seiberg-Witten theory developed by Lebrun et al.

Theorem F. LetM = (M, J, g)be a 4-dimensional compact almost K¨ahler Einstein manifold with negative scalar curvature. If

Z

M

τ²dv= 32π²(2χ(M) +p1(M)), then M is a K¨ahler manifold.

As a corollary of Theorem F, he also proved the following

Corollary F. LetM = (M, J, g)be a 4-dimensinal almost K¨ahler Einstein manfold with negative scalar curvature. IfM satisfies

Z

M

τ²dv524 Z

M

kW⁺k²dv, or more strictly if |τ| 52√

6kW⁺k² at each point of M, then M must be K¨ahler Einstein.

As an application of Corollary F, we have the following

Theorem G. Let M = (M, J, g)be a 4-dimensional compact almost K¨ahler Ein- stein manifold with negative scalar curvature. If M satisfies

Z

M

{G+τ(τ^∗−τ)}dv=0, then M is K¨ahler Einstein.

We here introduce the proof of Theorem G ([9]). From (2.3.5), we get 24

Z

M

kW⁺k²dv− Z

M

τ²dv

=3 2

Z

M

{G+D+ (τ^∗−τ)(τ^∗+τ)}dv.

(2.3.8)

On one hand, from (2.3.6) and (2.3.7), we get 0 =

Z

M

½G

2 + 4K+(3τ^∗−τ)(τ^∗−τ) 4

¾ dv

= Z

M

½D

2 +τ^∗(τ^∗−τ) 2

¾ dv.

(2.3.9)

Thus, from (2.3.8) and (2.3.9), we obtain 24

Z

M

kW⁺k²dv− Z

M

τ²dv=2 3

Z

M

{G+τ(τ^∗−τ)}dv.

Therefore, from Corollary F, Theorem G follows immediately.

3. Contact Geometry

3.1. K-contact manifolds and the characteristic foliations

Let (M, η) be a contact manifold with a contact form η, and D = kerη. We denote by ω the restriction of dη to D. Then (D, ω) gives rise to a symplectic vector bundle overM. We may choose an almost complex structure J onD which is compatible with ω, nearly,

(3.1.1)

J²=−idD, dη(JX, JY) =dη(X, Y),

dη(X, JX)>0,

forX,Y ∈Γ(D). We note thatJ defines a Riemannian metricg_DonDby setting gD(X, Y) =dη(X, JY),

X,Y ∈Γ(D). We may easily check that

gD(JX, JY) =gD(X, Y)

for X, Y ∈ Γ(D). A choice ofJ givesM an almost CR-structure with a strictly pseudoconvex Levy formL(X, Y) =dη(X, JY). By extendingJ to all ofT M, we obtain almost contact structure (φ, ξ, η) (η(ξ) = 1,φξ = 0,φ² =−id +ξ⊗η) on M (the vector field ξis called the Reeb vector field). Further, by extending gD to a metric on all ofM by

(3.1.2) g(X, Y)≡dη(X, φY) +η(X)η(Y)

for X, Y ∈ X(M). We may also see that g satisfies the compatibility condition g(φX, φY) =g(X, Y)−η(X)η(Y) for X, Y ∈ X(M). Summing up, we see that M with the structure tensors (φ, ξ, η, g) gives rises to an almost contact metric manifold.

Now let us consider the chararcteristic foliationFξgenerated by the Reeb vector field. IfFξ is a Riemannian foliation, that is, the holonomy pseudogroup induces isometries on the local leaf spaces of the local submersions defining Fξ, then by pulling back the metrics on the local leaf spaces, we obtains a transverse metric gD on the vector bundle D that is invariant under the Reeb flow generated by ξ. This is equivalent to the metric g onM given by (2.1.2) being bundle-like. A contact metric manifold (M, η, g) is said to be bundle-like if the Riemannian metric g is bundle-like. A contact metric manifold (M, η, g) with bundle-like Riemannian metric g is called a K-contact manifold. On a complete contact metric manifold (M, η, g), the followings are equaivalent

(i) (M, η, g) is K-contact manifold, (ii) The Reeb flow is an isometry,

(iii) The Reeb flow leaves the almost complex structureJ andD in- variant (namely, almost CR-structure (J, D) invariant),

(iv) The Reeb flow leaves the (1,1)-tensorφinvariant.

Next, we discuss the integrability of the almost CR-structure on M associated with the contact metric structure (η, g). The almost CR-structure (D, J) is inte- grable (i.e., (D, J) defines a CR-structure onM) if and only if

(i) [X, JY] + [JX, Y]∈Γ(D),

(ii) [JX, JY]−[X, Y]−J[JX, Y]−J[X, JY] = 0 for X,Y ∈Γ(D).

In our case condition (i) follows automatically from the antisymmetry of the formdη.

Condition (ii) is the remaining of the Nijenhuis tensor ofJ. Let∇be the Livi-Civita connection with respect to the metricg. We denote by∇^D the induced connection from ∇ on D defined by ∇^D_XY = (∇_XY)^h if X ∈ Γ(D) and ∇^D_XY = [ξ, Y]^h, Y ∈Γ(D), where hdenote the projection onto D. Then we see that ∇^DJ = 0 if and only if the almost CR-structure (D, J) is integrable and Lξφ= 0. A contact metric structure (M, η, g) is said to be normal if the Nijenhuis tensorNφ defined by (3.1.3) Nφ(X, Y) = [φX, φY] +φ²[X, Y]−φ[φX, Y]−φ[X, φY]

satisfies

(3.1.4) N_φ=−dη⊗ξ.

A normal contact metric structure on M is also called a Sasakian structure. Let M˜ =R+×M, and ˜J be almost complex structure on ˜M defined by

J˜=

Ãφ ξ

−η 0

! .

Further, let ˜g be the Riemannian metric onR+×M defined by

˜ g=

Ãr²g 0

0 1

!

, r∈R+.

Then, we may observe that the cone ( ˜M ,J,˜ ˜g) is an almost K¨ahler manifold, and ( ˜M ,J,˜g) is integrable (namely, K¨ahler) if and only if (M, η, g) is normal. We˜ adopt this to our situation. Let (M, η, g) be a K-contact manifold and Fξ be the characteristic foliation. Further, letC(M,Fξ) be the commuting sheaf introduced by Molino [11] which is cosisted of germs of local transverse Killing vector fields with respect to the transverse metric gD, whose orbits are precisely the closures of the flows of Fξ. Carri`ere [5] proved that the leaf closures are diffeomorphic to tori and, that the flow is conjugate by the diffeomophism to a linear flow on the torus. We denote byI(M, g) the isometry group of (M, g) andA(M, η, g) the group of automorphisms of (M, η, g). In our case, I(M, g) is compact andA(M, η, g) is a closed Lie subgroup of I(M, g) and hence, compact, and Reeb flow belongs to A(M, η, g). Thus, the closureT of the Reeb flow is a compact commutative group,

i.e., a torus which lies inA(M, η, g). The Reeb flow is a strict contact transformation lying in the center of the group of strict contact transformations, and hence, it lies in the center of A(M, η, g). It follows thatT also lies in the center of A(M, η, g).

The dimension of the torusT is an invariant of the K-contact structure which are called the rank of (M, η, g) and denoted by rk(M, η). We may note that rk(M, η) is not an invariant of the contact structure (M, D) but only of the Pfaffian structure (M, η). The case rk(M, η) = 1 is the quasi-regular case. Rukimkira [16] showed that any K-contact formη can be approximated by a sequence of a quasi-regular K-contact forms in the same contact structure, namely, one can choose a sequence of vector fieldξj in t(the Lie algebra of T) with periodic orbits that converges to the Reeb vector fields ξ. Then 1-forms ηj are quasi-regular contact forms in the same contact structure. We here recall several curvature identities onM.

(3.1.5) R(X, ξ)ξ=X, ρ(ξ, ξ) = 2n, ρ(ξ, X) = 0, ρ(X, X) =ρD(X, Y)−2g(X, Y),

forX, Y ∈Γ(D), whereρDis the transverse Ricci tensor.

3.2. K-contact Einstein manifolds

In this section, we introduce the outline of the proof of the following result by Boyer and Galicki [4].

Theorem H. LetM = (M, η, g)be a compact K-contact Einstein manifold. Then M is a Sasakian Einstein manifold.

First, we assume that η is quasi-regular. Then, M is the total space of a principalS¹v-bundle over a compact almost K¨ahler orbifoldZ(Thomas [20], Boyer and Galicki [4]). Furthermore, the induced metrichonZis almost K¨ahler Einstein which has positive scalar curvature since g has positive scalar curvature. Since the proof of Theorem A only involves local curvature computations together with a Bochner type argument using Stokes’ Theorem, it carries over to the case of a compact orbifold. So, the almost complex structureJonZis integrable, and hence, (Z, J, h) is K¨ahler-Einstein. It then follows from the orbifold version of Hatakeyama [7] that (M, η, g) is normal, and hence, Sasakian-Einstein. This proves the Theorem H under the assumption of quasi-regularity.

Next, assume that (M, η, g) is K-contact and Einstein, but not quasi-regular.

Then, the Reeb vector fieldξlies int(M, Fξ)⊂a(M, g) which has dimensionk >1.

Thus, there exists a sequence of quasi-regular contact forms η_m and Reeb vector fields ξm ∈ t(M, Fξ) that approximate (η, ξ) in the compact-open C^∞ topology.

Explicitly, there is a monotonically decreasing sequence {εm} with lim

m→∞εm = 0 such that

(3.2.1) ηm=f(εm)η, ξm=ξ+σm, f(εm) = 1 1 +η(σ_m), where f(εm) are positive functions onM satisfying lim

m→∞f(εm) = 1. Clearlyσm∈ t(M, Fξ) and lim

m→∞σm = 0. Moreover, kerηm = kerη = D, so we have the same underlying contact strucure. We also have the following relations

(3.2.2) φm=φ− 1

1 +η(σm)φξm⊗η =φ−f(εm)φξm⊗η.

This implies that φmξm = 0 and the almost complex structure J on D remains unchanged. However, the induced metrics become

gm=f(εm)gD⊕f(εm)²η⊗η

=g−η(ξm)(gD+ 2η⊗η) +o(ε²_m).

Forεmsmall enough,gmare well-defined Rimannian metrics onM which can easily be seen to satisfy the compatiblility condition

(3.2.3) gm(φmX, φmY) =gm(X, Y)−ηm(X)ηm(Y).

Moreover, sinceξm∈t⊂a(M, η), it follows that the functions f(εm)∈C^∞(M)^T, where C^∞(M)^T denotes the subalgebra of C^∞(M) invariant under the action of the torusT. Thus, by (2.2.2), we have

(3.2.4) Lξmφm= 0.

Hence, (M, φm, ξm, ηm, gm) is a sequence of quasi-regular K-contact structure onM whose limit with respect to the compact-openC^∞topology is the original K-contact Einstein structure (M, φ, ξ, η, g). By taking account of the fundamental formulas, we have established by O’Neill. We see that the Ricci tensorρm ofgmis given by (3.2.5) ρm=λmgm+A(εm, ξm, g),

whereA(εm, ξm, g) is a traceless symmetric (0,2)-tensor field depending onεm,ξm, gthat tends to 0 withεm, andλm∈C^∞(M) satisfies lim

m→∞λm= 2n. Now, there is a sequence of orbifold Riemannian submersionsπm:M → Zm, where (Zm, hm) are

a sequence of compact almost K¨ahler orbifolds satisfyingπ_m^∗h_j =f(ε_j)g_D. Further, by (2.2.5), we see that the scalar curvature of hm are all positive. From (1.2.2), and making necessary adjustments, we find that there are non-negative numbers δmand non-negative smooth functionsFmsuch that

(3.2.6)

Z

Zm

µ

Fm+˜τm

n k∇˜mJ˜mk²_Zm+ 1

2nk∇˜mJ˜mk⁴_Zm

¶

dvm5δm,

where ˜∇m, ˜Jm, ˜τm, dvm, k · kZm are the Levi-Civita connection, almost complex structure, scalar curvature, volume element and Riemannian norm, respectively on (Zm, hm). Now, since the metrics g, gm are bundle-like, the leaves of the char- acteristic foliations are geodesics and the O’Neill tensors T and N vanish. More- over, for any K-contact manifold of dimension 2n+ 1, O’Neill tensor A satisfies kAk² =g(Aξ, Aξ) = 2n. Thus, we have the relation between the functions λm on M and the scalar curvatures ˜τmonZm

(3.2.7) τ˜m= (2n+ 1)λm+ 2n,

and therefore, lim

m→∞τ˜m = 2n+ (2n+ 1) lim

m→∞λm= 4n(n+ 1) ((3.2.7) can be also derived from (3.1.5) and (3.2.5)). Thus, sinceFmand ˜τmare non-negative for each m, the estimate (2.2.6) implies the estimate

(3.2.8) k∇˜mJ˜mkZm 5δ_m⁰ , where δ_m⁰ are non-negative numbers satisfying lim

m→∞δ_m⁰ = 0. For each m, the horizontal lift of ˜∇mJ˜mis the horizontal projection (∇mJm)^h = (∇mϕm)^h, where

∇m, Jm, ϕm are the corresponding Levi-Civita connection and tensor fields with respect to the metrics onM. SinceJm=J for allm, we have

(3.2.9) k(∇J)^hk= lim

m→∞k(∇mJ)^hkm5 lim

m→∞δ⁰_m= 0,

where k · km is the Riemannian norm with respect togm. Thus, we see that the almost CR-structure on D is integrable which implies that (M, η, g) is Sasakian- Einstein.

In [12], S. Morimoto has obtained the foliation version of Theorem A, namely, proved the following by adjusting the proof of Theorem A suitably.

Theorem I. LetF be a harmonic almost K¨ahler foliation on a compact orientable Riemannian manifold (M, g) whose transverse scalar curvature is non-negative.

ThenF is a K¨ahler foliation.

We can see that Theorem H follows from the above Theorem I immediately. In fact, since the integral curves of ξare geodesics, we may observe that the characteristic foliationFξassociated with a K-contact manifoldM = (M, φ, ξ, η, g) is a harmonic, almost K¨ahler foliation by virtue of the arguments in§3.1. Further, we assume that the Ricci tensorρofM satisfies the conditionρ=ag+ (2n−a)η⊗η for a constant a(i.e.,M isη-Einstein). Then, by (3.1.5), we have

(3.2.10) ρD(X, Y) = (a+ 2)g(X, Y),

for X, Y ∈ X(D), and hence Fξ is a harmonic Einstein almost K¨ahler foliation on M. Therefore, from (3.2.10), by applying Theorem I to our situation, we have immediately

Theorem J. Let M = (M, φ, ξ, η, g)be a (2n+ 1)-dimensional compact K-contact manifold. If the Ricci tensorρsatisfiesρ=ag+(2n−a)η⊗η for a constanta=−2, then M is a Sasakian manifold.

We remark that Boyer and Galicki has discussed compactη-Einstein manifolds and obtained the global structure theorem for such manifolds (cf. [4], Theorem 7.2).

In the proof of the structure theorem, they made use of the result of Theorem H.

The simplest example of a compact simply connected Sasakian-Einstein manifold is the unit round sphere S²ⁿ⁺¹ whose associated cone defined in§3.1 is flat. One of the first examples for which the associated cone is not flat was constructed by Tanno [19].

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