Almost Hermitian and Kähler structures on product manifolds

Almost Hermitian and K¨ ahler structures on product manifolds

Yoshiyuki Watanabe

Department of Mathematics, University of Toyama, Toyama, 930-8555, Japan e-mail : ywatnabe@sci.u-toyama.ac.jp

(2000 Mathematics Subject Classification : 53C40, 53C15.)

Abstract. In this paper, we survey almost Hermitian, K¨ahler, almost quaternionic Hermi- tian and quaternionic K¨ahler structures, naturally constructed on products of manifolds with almost contact metric and Sasakian structures and open intervals. As an application of these constructions, the notion of a unitary-symmetric K¨ahler manifold was introduced in [46]. Then we guide its geometric properties in [27], [47]. Next, we investigate almost Hermitian structures, naturally defined on the product manifolds of two almost contact metric and Sasakian manifolds, and ask some problems related to these topics.

1 Introduction

By means of a natural change of the product metric, one can construct widely almost Hemitian structures on product manifolds of two almost contact metric manifolds. Since many almost contact metric and Sasakian structures are now found (for example, see [5], [7], [8]), this method enables us to provide various kinds of almost Hermitian structures.

In the later half of 1950’s, Japanese geometers initiated to study certain types of almost Hermitian geometry, motivated by finding out the nearly K¨ahler structure in the 6-dimensional sphereS⁶and the almost K¨ahler structure on the tangent bundle of a Riemannian manifold (see [15], [35], [37] and so on). Kotˆo [25] established inclusion relations between various classes. Unfortunately, in this early period, it was a weak point for almost Hermitian geometry to hold two kinds of examples only, mentioned above.

In 1971, Gray pointed out that nearly K¨ahler geometry corresponds to weak holonomy U(n). Consequently many interesting theorems about the topology and geometry of nearly K¨ahler manifolds were proved (see [16]). Then from view point of weak holonomy, Gray-Hervella [17] gave a classification of almost Hermitian man- ifolds in terms of the covariant derivatives of the fundamental forms. Falcitelli- Farinola-Salamon [14] combined the theory of Gray-Hervella [17] and Tricceri- Vanhecke [43] into a representation-theoretic framework. In this direction, Cabrera- Swann [9] systematically have studied the interaction between these classes when one has an almost hyper-Hermitian structure and in general dimension found at most 167 different almost hyper-Hermitian structures. However, these classification

1

are rigorous but algebraic. Therefore it needs that explicit examples of various type support now the bases of their theories.

On the other hand, by deforming the canonical almost Hermitian structure with some functions of the norm of vectors, one can construct many geometric structures of new type (almost Hermitian and K¨ahler structures, almost quaternionic Hermi- tian and K¨ahler structures, etc.) on the tangent bundles of Riemannian and almost Hermitian manifolds (see [30], [32], [34], [39], [40]).

By means of warped product there is a one-to-one correspondence between Sasakian and K¨ahler structures (see [42], [31]). By a similar tecnique, B¨ar pointed out that there is a one-to-one correspondence between Sasakian 3-structures and hyper-K¨ahler structures. But it seems, from view points of construction of examples, that this technique is not natural. In fact, a simple but natural one enables us to construct many examples of almost Hermitian structures on the productM ×Iof a manifoldM with an almost contact metric structure and an open intervalI(see [29]) and also to do many model spaces of quaternionic geometry on the product of a 3-Sasakian manifold and an open interval (see [48]).

In [11] Capursi showed that for the product of two almost contact metric man- ifolds, the product metric is K¨ahlerian if and only if both factors are cosymplectic.

Then Blair-Oubi˜na [6] asked the open question: What kind of change of the product metric will make both factors Sasakian?

Here by generalizing the cases ofM×I, we investigate almost Hermitian struc- tures on the product of two almost contact metric manifolds. Thus for these ex- amples of product manifolds together with almost Hermitian structures on tangent bundles mentioned above, we suggest to notice again the theories of classifications of Gray-Hervella [17] and Falcitelli-Farinola-Salamon [14].

2 Preliminaries

2.1 Almost contact metric structures and Sasakian structures

LetM be an odd-dimensional differentiable manifold. An almost contact struc- ture onM is by definition a pair (Σ, g) of an almost contact structure Σ = (φ, ξ, η) and a Riemannian metricg, whereφis a tensor field of type (1,1),ξis a vector field andη is a 1-form, satisfying the following conditions (see [4], [5]):

(2.1) φξ= 0, η(φX) = 0, η(ξ) = 1, φ²X=−X+η(X)ξ

for any vector field X on M. A Riemannian metric g is compatible with these structure tensors if

(2.2) g(X, Y) =g(φX, φY) +η(X)η(Y)

for any vector fieldsX and Y on M and (Σ, g) is called an almost contact metric structure. Note also thatg(X, ξ) =η(X). If it satisfies

(2.3) dη(X, Y) =g(φX, Y)

for any vector fields X, Y on M, then (M,Σ, g) is called a contact Riemannian manifold.

LetM be an almost contact manifold and define an almost complex structure J onM×Rby

(2.4) J(X+f d

dt) =φX−f ξ(X) +η(X)d dt

for any vector fieldX onM, wherefis aC^∞function onM×R. An almost contact structure is said to be normal ifJ is integrable (see (3.4)). An almost contact metric structure (Σ, g) is called Sasakian if furthermore

(2.5) (∇Xφ)(Y) =η(Y)X−g(X, Y)ξ for any vector fields X, Y onM.

On the other hand, Blair defined a cosymplectic structure to be a normal almost contact metric structure (Σ, g) with both η and Φ closed, given by Φ(X, Y) = g(φX, Y) for any vector fieldsX, Y onM (see [4], [5]).

Suppose that a differentiable manifold admits three almost contact structures (φ_(p), ξ_(p), η_(p)), p=1,2,3 satisfying

(2.6) η_(p)(ξ_(q)) =δ_pq,

φ_(p)ξ_(q)=−φ_(q)ξ_(p)=ξ_(r), η_(p)◦φ_(q)=−η_(q)◦φ_(p)=η_(r), φ_(p)φ_(q)−ξ_(p)⊗η_(q)=−φ_(q)φ_(p)+ξ_(q)⊗η_(p)=φ_(r).

for ε(p, q, r) = 1 where ε(p, q, r) = 1 means that (p, q, r) is a cyclic permutation of (1,2,3).Then (φ_(p), ξ_(p), η_(p)), p= 1,2,3 is called an almost contact 3-structure. It is well known (see [26]) that the dimension of a manifold with an almost contact 3-structure is 4m+ 3 for some non-negative integer m. A Riemannian metricg is said to be associated to the 3-structure if it satisfies

(2.7) g(φ(p)X, φ(p)Y) =g(X, Y)−η(p)(X)η(p)(Y), p= 1,2,3

for any vector fields X, Y on M. In such a manifold with an almost con- tact 3-structure there always exists a Riemannian metric g satisfying (2.7), and (φ_(p), ξ_(p), η_(p), g), p= 1,2,3 is called an almost contact metric 3-structure.

An almost contact metric 3-structure (φ(p), ξ(p), η(p), g), p = 1,2,3 is called a Sasakian 3-structure if each (φ(p), ξ(p), η(p), g) is a Sasakian structure. Then ξ(1), ξ(2), ξ(3) are orthonormal vector fields, satisfying

(2.8) [ξ_(p), ξ_(q)] = 2ξ_(r)

for ε(p, q, r) = 1 (see [38], [41]). Such a manifold with a Sasakian 3-structure is called a 3-Sasakian manifold. Remark that a 3-Sasakian manifold is an Einstein manifold.

3 Almost Hermitian and K¨ahler structures on products of manifolds and open intervals, and unitary-symmetric K¨ahler manifolds

Let M be an even-dimensional differentiable manifold. An almost Hermitian structure onM is by definition a pair (J, g) of an almost complex structureJ and a Riemannian metricg satisfying

(3.1) J²X=−X, g(J X, J Y) =g(X, Y)

for any vector fieldsX, Y onM. A manifold with such a structure (J, g) is called an almost Hermitian manifold. The fundamental form Ω of an almost Hermitian structure is defined by

Ω(X, Y) =g(J X, Y)

for any vector fieldsX, Y and is skew-symmetric. An almost Hermitian manifold is called an almost K¨ahler manifold if its fundamental form Ω is closed, that is, dΩ = 0 is equivalent to

(3.2) (∇XΩ)(Y, Z) + (∇YΩ)(Z, X) + (∇ZΩ)(Z, X) = 0

for any vector fieldsX, Y, ZonM, where∇ is the Levi-Civita connection ofg. On the other hand, an almost Hermitian manifold is a nearly K¨ahler manifold if (J, g) satisfies

(3.3) (∇_XJ)(Y) + (∇_YJ)(X) = 0 for any vector fieldsX, Y onM.

The Nijenhuis (or the torsion) tensor of an almost complex structureJ is defined by

(3.4) N(X, Y) = [X, Y]−[J X, J Y] +J[X, J Y] +J[J X, Y]

for any vector fieldsX, Y onM. An almost complex structure is said to be integrable if it has no torsion. It is well known (see [24]) that an almost complex structure is a complex structure if and only if it has no torsion. A complex manifold with a Hermitian structure (J, g) is said to be K¨ahlerian if its fundamental form is closed.

It is also well known (see [24]) that an almost Hermitian manifold (M, J, g) is K¨ahlerian if and only if its almost complex structureJ is parallel with respect to the Levi-Civita connection, that is,

(3.5) ∇J= 0.

3.1 Almost Hermitian and K¨ahler structures on the product of an almost contact metric manifold and an open interval

Let (M, φ, ξ, η, g) be an almost contact manifold of dimension 2p+ 1 andI be Ror an open interval.

First, we introduce an almost complex structure J on the product manifold M ×I, defined in such a way that

(3.6) J(X+T) =φX+1

λdt(T)ξ−λη(X)d dt

for any vector field X of M and any vector field T of I, where λ is a positive (or negative) function onM×I. Next, we define a Riemannian metric onM×I, defined by

(3.7) G(X+T, Y +T⁰) =αg(X, Y) +βη(X)η(Y) +dt²(T, T⁰)

for any vector fields X, Y of M and any vector fields T, T⁰ of I, where α, β are functions on I, satisfying

(3.8) α >0, α+β >0.

Then it is easily seen that (J, G) is an almost Hermitian structure on the product M ×Iif and only if their functions satisfy

(3.9) λ=±p

α+β.

Then by summing up these results, we have the following propositions (see [29], [46], [47]) .

Proposition 3.1. Let (M, φ, ξ, η, g) be an almost contact metric manifold. Let α=α(t)andβ =β(t)be functions onI, satisfying (3.8). Then,(J, G)is an almost Hermitian structure on M ×I. Moreover, (J, G) is a Hermitian structure if and only if(φ, ξ, η, g)is a normal almost contact metric structure.

Proposition3.2. Let (M, φ, ξ, η, g)be a contact Riemannian manifold. Then the almost Hermitian structure constructed such as Proposition 3.1 is almost K¨ahlerian if and only if the functions αandβ satisfy

(3.10) β =1

4(α⁰)²−α, denoting by α⁰= ^dα_dt.

Proposition3.3 (Ejiri [13], Watanabe [46]). Let(M, φ, ξ, η, g)be a Sasakian mani- fold. Then the Hermitian structure constructed such as Proposition 3.2 is K¨ahlerian if and only if the function α is an increasing(or decreasing) function, satisfying (3.10).

3.2 Application: Unitary-symmetric K¨ahler manifolds

A K¨ahler manifold (M, J, g) of complex dimension n is said to be unitary- symmetric at a point mof M if the linear isotropy group at m of automorphisms (that is, holomorphic isometries) ofM is the unitary groupU(n).

As is seen from Kaup’s result [22] on complex manifolds, a connected, unitary- symmetric K¨ahler manifold is biholomorphic to one of the complex space forms.

Then, from a view point of Riemannian geometry, Watanabe [46] initiated to study such K¨ahler metrics which are not homothetic to the complex space forms. Thus it seems to us that unitary-symmetric K¨ahler manifolds may be regarded as model spaces in K¨ahler geometry, corresponding to the concept of rotationally symmetric spaces in Riemannian geometry (see [12], [18]).

Let us fix some notations for explaining some results. Let m ∈ M. We de- fine δ to be the distance from the origin O of the tangent space Tm(M) at m to the fist tangential conjugate locus Qm. Define ˜Bδ = {X ∈ Tm(M)||X| < δ}, where|X|=p

gm(X, X). Then it is clear that ˜Bδ becomes a Riemannian manifold equipped with metric exp^∗_mg since the exponential map at m expm : ˜Bδ → M is non-singular. Then we have the following ([47]).

Theorem 3.4. Let (M, J, g) be a complete, connected, simply-connected K¨ahler manifold of complex dimensionn≥2 andm be a point of M. Then the following conditions (I), (II), (III) and (IV) are equivalent each other:

(I) (M, J, g)is unitary-symmetric atm.

(II)The metricexp^∗_mg and the fundamental formexp^∗_mΩ, pulled back under the exponential mapping expm, are given by

(3.11)

exp^∗_mg = dr²+f(r)²dΘ²+f(r)²(f⁰(r)²−1)η⊗η, exp^∗_mΩ = 2f(r)f⁰(r)η∧dr+f(r)²Ψ,

on the punctured ballB˜_δ−{O}of radiusδinT_m(M), wheref is aC^∞odd function on (−δ, δ) such that f⁰(0) = ^df_dt(0) = 1. Here we assume that δ is infinite when M is non-compact, and we denote by (r,Θ) the usual polar coordinate system of Cⁿ ≡Tm(M), by(φ, ξ, η, dΘ²) the standard Sasakian structure on the unit sphere S²ⁿ⁻¹ inTm(M), and setΨ(X, Y) =dΘ²(φX, Y).

(III)The Riemannian curvature tensorR˜ in terms of exp^∗_mg satisfies R(J γ˜ ⁰, γ⁰)γ⁰=h(r)J γ⁰, R(E(r), γ˜ ⁰)γ⁰=k(r)E(r),

whereγ⁰ is the tangent vector field of a radial geodesicγstarting fromO,h(r), k(r) are functions depending only onr, being the distance from the origin O, and E(r) is a parallel vector field alongγ which is perpendicular to bothγ⁰ andJ γ⁰.

(IV) The exponenential image of any complex linear subspace (resp. real sub- space spanned by u, v) of T_m(M)is a closed, totally geodesic, complex (resp. real) submanifold ofM, whereu, J u andware orthogonal each other.

In the proof of Theorem 3.4, by making use of the result in [45], we need to show the following.

Theorem 3.5. Let (M, J, g) be a connected, simply-connected, compact K¨ahler manifold of complex dimension n ≥ 2. If there exists a point m in M such that exp^∗_mg, exp^∗_mΩsatisfy the condition (II), then(M, J, g)is a Blaschke manifold atm and the first conjugate locus Q(m)of mis a totally geodesic, complex hypersurface of (M, J, g).

Example 3.1. Let (M, J, g) be a complex space form, endowed with the canoni- cal K¨ahler metric of constant holomorphic curvature 4k. Then, sinceM is locally Fubinian, from Tachibana’s result [36], the K¨ahler metric is represented by the fol- lowing functions f(r) in (II) of Theorem 3.4:

(1) fork= 0, M =Cⁿ, f(r) =r.

(2) fork >0, M =CPⁿ, f(r) = ^√1

ksin√ kr.

(3) fork <0, M =CHⁿ, f(r) = ^√1

−ksinh√

−kr.

Example 3.2. Let us consider a K¨ahler metric g = (g_αβ¯) in Cⁿ, given by the potential function

h(t) = Z t

0

1

slog(1 +s)ds

for t = Σz^α¯z^α. Klembeck [23] showed that (Cⁿ, g) is complete and has positive curvature. By using Itoh’s result [21], we have

g = dr²+ 2log(coshr)2dΘ²+ (tanh²r−2log(cosh r))η⊗η, Ω = 2(tanh r)η∧dr+ 2log(cosh r)Ψ,

with respect to the polar coordinate (r,Θ) about the originO ofCⁿ.

As a good application of Theorem 3.4, Mori-Watanabe [27] gave the following.

Theorem3.6. Let (ds²_can, Jcan) be the canonical K¨ahler structure on the complex projective n-space CPⁿ. Then, for a sufficient small positive numberεthere exists a one-parameter family of K¨ahler structures(ds²_a, Jcan)(−ε < a < ε)on CPⁿ, sat- isfying the following properties

(1)Fora= 0,ds²_a=ds²_can.

(2)For different valuesa, b∈(−ε, ε)and for each positive λ,ds²_a6=λds²_b. (3)For anya(−ε < a < ε), there is a pointmofCPⁿ and a constant`(a)such that (CP<sup>n</sup>, ds<sup>2</sup><sub>a</sub>) is a SC<sup>m</sup> manifold (see [3]), that is, all geodesics issuing fromm are simple closed ones with length `(a).

3.3 Almost quaternionic Hermitian and quaternionic K¨ahler structures on products of 3-Ssasakian manifolds and open intervals

Following Alekseevsky-Marchiafava [1] and Ishihara [20], we recall the defi- nitions of almost quaternionic Hermitian, quaternionic K¨ahler and hyper-K¨ahler structures.

An almost hypercomplex structure on a manifold M of dimension 4m is by definition a tripleH = (J_(p)), p= 1,2,3 of almost complex structure, satisfying

(3.12) J_(p)J_(q)=J_(r)

forε(p, q, r) = 1. By T M we denote the tangent bundle ofM. Then it generates a subbundle Q =< H > of the bundle End(T M) of endomorphisms whose fiber Qx=RJ₍₁₎|x+RJ₍₂₎|x+RJ₍₃₎|x in a pointx∈M is isomorphic to the Lie algebra sp1of the symplecticSp(1). More generally, an almost quaternionic structure on a manifoldM is defiend as a subbundleQ⊂(T M) of the bundle of endomorphisms which is locally generated by an almost hypercomplex structureH. We shall refer to suchH as an almost hypercomplex structure compatible with Q. Let Qbe an almost quaternionic structure on M with a Riemannian metric. Then M can be equipped with a Q-Hermitian metricg, that is, all endomorphisms fromQare skew- symmetric with respect to g. An almost quaternionic structure Qtogether with a Q-Hermitian metric g is called an almost quaternionic Hermitian structure and a manifold with such a structure is called an almost quaternionic Hermitian manifold.

An almost quaternionic Hermitian manifold is called a quaternionic K¨ahler man- ifold if an almost hypercomplex structure J_(p), p = 1,2,3 in any local coodinate neighborhoodU satisfies

(3.13)

∇_XJ₍₁₎= ω₃(X)J₍₂₎ −ω₂(X)J₍₃₎,

∇XJ₍₂₎=−ω3(X)J₍₁₎ +ω1(X)J₍₃₎,

∇XJ(3)= ω2(X)J(1)−ω1(X)J(2)

for any vector fieldX onU, where∇ is the Levi-Civita connection of the Rieman- nian metric, andωi are certain local 1-forms defined in U. In particular, if all ωi

for eachU are vanishing, then the structure is called hyper-K¨ahler. Remark that if m >1, a quaternionic K¨ahler manifold is an Einstein manifold.

Let (φ(p), ξ(p), η(p), g), p = 1,2,3 be an almost contact metric 3-structure on a manifold M of dimension 4m+ 3. By I we denote R or some open interval in R. For a positive function λ on I, we define an almost hypercomplex structure J˜_(p), p= 1,2,3 onM ×Iby (3.6), that is,

(3.14) J˜(p)=





φ(p) −λη(p)

λ⁻¹ξ(p) 0



. Letα, β be real valued functions onI, satisfying

(3.15) α >0, α+β >0.

Then, we give a Riemannian metric onM ×Iby (3.16) ˜g=α(t)g+β(t)

3

X

p=1

η_(p)⊗η_(p)+dt²

where dt² is the usual metric onI. Then by (2.1) and (2.2) Nakashima-Watanabe [29] showed the following.

Proposition3.7. Let (φ_(p), ξ_(p), η_(p), g), p= 1,2,3 be an almost contact metric 3- structure on a manifold M of dimension4m+ 3andλa function onI. Letα, βbe real valued functions on I, satisfying (3.15). Then( ˜J_(p),g), p˜ = 1,2,3 is an almost quaternionic Hermitian structure onM ×Iif and only ifλ=±√

α+β.

Next, let (M, φ_(p), ξ_(p), η_(p), g), p= 1,2,3 be a 3-Sasakian manifold of dimension 4m+ 3 andα, β be real valued functions on I, satisfying (3.15) and λ=√

α+β.

By using the equalities from (2.1) to (2.7), we compute directly ˜∇J˜_(p). Then, if the functions αandβ satisfies

(3.17) 2p

α+β= dα

dt, αdβ

dt = 4βp α+β,

then it is easily seen that ( ˜J(p),g), p˜ = 1,2,3 is a quaternionic K¨ahler structure.

Moreover, putting α = f² and hence β = f²(f⁰−1). Thus the metric (3.16) reduces to

(3.18) g˜=f(t)²g+f²(f⁰²−1)

3

X

p=1

η_(p)⊗η_(p)+dt²

Proposition 3.8. Let (M, φ_(p), ξ_(p), η_(p), g), p = 1,2,3 be a 3-Sasakian manifold.

An almost quaternionic Hermitian structure constructed on M×Isuch as Propo- sition 3.7 is quaternionic K¨ahler if and only if the function f satisfies the ODE (3.19) f f⁰⁰−(f⁰)²+ 1 = 0.

with the conditions f >0 andf⁰>0 onI.

Now, puttingp=^df_dt =f⁰, we havef⁰⁰=p^dp_df, from which (3.19) reduces to p²−1 =kf²,

where k is constant. Recall that p =f⁰(t) and that f(t) >0 and f⁰(t)> 0. We may, up to a motion of parametert, have that a solutionf(t) of the OED (3.19) is of the form:

Case 1. k= 0, f(t) =t, 0< t <∞.

Case 2. k <0, f(t) =^√1

−ksinh(√

−kt)), 0< t <∞.

Case 3. k >0, f(t) =^√1

ksin(√

kt), 0< t <^√π

k.

Thus we have now a generalization of Therem B in Boyer-Galicki-Mann [7], since f⁰= 1 in the casek= 0.

Theorem3.9. Let (M, φ_(p), ξ_(p), η_(p), g), p= 1,2,3 be a 3-Sasakian manifold. Let f be a real valued function on I, satisfying the ODE (3.19). Then we have

(1) If f(t) =t, then the product M×R⁺ with the cone metric in (3.18) is hyper- K¨ahlerian,

(2) If f(t) = ^√1

−ksinh(√

−kt), then the product M ×R⁺ with the cone metric in (3.18) is quaternionic K¨ahler, wherek is a negative constant.

(3)If f(t) =^√1

ksin(√

kt), the the product M×(0,^√π

k)with the metric in (3.18) is quaternionic K¨ahler, wherek is a positive constant.

4 Almost Hermitian structures on the product of two almost contact metric manifolds

Let us consider the product manifold M ×M⁰ of two almost contact metric manifolds (M,Σ, g) and (M⁰,Σ⁰, g⁰).

First, we introduce a class of almost complex structure J_(ρ,σ) on the product manifoldM×M⁰ :

(4.1)

J(ρ,σ)(X+X⁰) =φX− {ρ

ση(X) +ρ²+σ²

σ η⁰(X⁰)}ξ+φ⁰X⁰+{1

ση(X) +ρ

ση⁰(X⁰)}ξ⁰, for any vector fieldsX, Y ofM and any vector fieldsX⁰, Y⁰ of M⁰, whereρ, σ are functions onM ×M⁰ and further σis a positive (or negative) function. WhenM and M⁰ are odd-dimensional spheres and ρ = constant, σ = constant(6= 0), the almost complex structureJ(ρ,σ)is integrable (see [44]).

Next, we define a Riemannian metric onM ×M⁰, by

(4.2) G(α,β,γ,δ,µ)=αg+βη⊗η+µ(η⊗η⁰+η⁰⊗η) +γg⁰+δη⁰⊗η⁰ whereα,β,γ,δandµare functions onM ×M⁰, satisfying

(4.3) α >0, α+β >0, γ >0, γ+δ >0, (α+β)(γ+δ)> µ².

Then it is easily seen from (2.1) and (2.2) that (J_(ρ,σ), G(α,β,γ,δ,µ)) is an almost Hermitian structure on the productM×M⁰ if we defineρandσby

(4.4) ρ= µ

α+β, σ=±

p(α+β)(γ+δ)−µ²

α+β .

In this section we denote the almost Hermitian structure (J_(ρ,σ), G(α,β,γ,δ,µ)) by (J, G) for simplicity.

By a direct computation using (2.1) and (2.2) again, one can check that the fundamental 2-form Ω of (J, G) is very simply as follows:

(4.5) Ω =αΨ +γΨ⁰+ 2p

(α+β)(γ+δ)−µ²(η∧η⁰),

where we denote by Ψ(X, Y) = g(φX, Y) and Ψ⁰(X⁰, Y⁰) = g⁰(φ⁰X⁰, Y⁰) for any vector fieldsX, Y of M and any vector fieldsX⁰, Y⁰ of M⁰, although J andGare so complicatedly defined. In particular, when α= 1, β= 0, γ= 1, δ= 0 andµ= 0, the almost Hermitian metric (J, G) is a Hermitian structure which J is given by Morimoto [28] andGis the product metric.

4.1 Open question of Blair and Oubi˜na

It is natural to research for some conditions in order that almost Hermitian structures defined on the product of two Sasakian manifolds are K¨ahlerian. This is the open question due to Blair-Oubi˜na (see [5], [6]). But such a case, in general, is incorrect as is seen from Calabi-Eckmann manifold, which is not able to admit any K¨ahler metric (see [10]).

Our motivation, that introducesJ andGgiven in (4.1) and (4.2), is to consider the open question of Blair and Oubi˜na and to construct almost Hermitian structures of various type as many as possible.

4.2 Almost Hermitian structures on the products of 3-Sasakian mani- folds and Sasakian manifolds

Let (M, φ_(p), ξ_(p), η_(p), g), p= 1,2,3 be a 3-Sasakian manifold M of dimension 4m+ 3 and (M, φ⁰, ξ⁰, η⁰, g⁰) be a Sasakian manifold. Then we define three kind of almost complex structuresJ_(p), p= 1,2,3 on the product manifoldM×M⁰ :

(4.6)

J(p)(X+X⁰) = φ(p)X− {ρ

ση(p)(X) +ρ²+σ²

σ η⁰(X⁰)}ξ(p)

+φ⁰X⁰+{1

ση_(p)(X) + ρ

ση⁰(X⁰)}ξ⁰, p= 1,2,3 for any vector fieldsX, Y of M and any vector fieldsX⁰, Y⁰ of M⁰, where ρ, σare functions on M×M⁰ and further σis a positive (or negative) function.

Next, we define a Riemannian metric onM×M⁰, by (4.7)

G=αg+β

3

X

p=1

(η_(p)⊗η_(p)) +µ

3

X

p=1

(η_(p)⊗η⁰+η⁰⊗η_(p)) +γg⁰+δη⁰⊗η⁰, p= 1,2,3, where α, β, γ, δ andµ are functions on M ×M⁰, satisfying the conditions (4.3).

Then one can see that for each p=1,2,3, (J_(p), G) given by (4.4) is an almost Her- mitian structure on the productM×M⁰. Then we ask the following two problems.

Problem 4.1. How almost Hermitian geometry can one extend onM ×M⁰ with three kinds of almost Hermitian structures (J_(p), G), p = 1,2,3 in contrast with quaternionic geometry ?

Problem 4.2. Let (φ_(p), ξ_(p), η_(p), g), p = 1,2,3 be a Sasakian 3-structure on a manifoldM of dimension 4m+ 3 and (φ⁰, ξ⁰, η⁰, g⁰) a Sasakian structure on a man- ifoldM⁰of dimension 2q+ 1. Can one construct quaternionic K¨ahler structures on M×M⁰ whenqis an even number ?

4.3 Almost Hermitian structures on the product of a 3-Sasakian mani- fold and a 3-Sasakian manifold

Further, by extending Problem 4.2, we ask the following.

Problem 4.3. Let (M, φ_(p), ξ_(p), η_(p), g), p = 1,2,3 be a 3-Sasakian manifold M and (M⁰, φ⁰_(q), ξ_(q)⁰ , η⁰_(q), g⁰), q= 1,2,3 be another 3-Sasakian manifold. How almost Hermitian geometry can one extend onM×M⁰ with nine kinds of almost complex structures ?

On the other hand, it is known (see [19]) that there exists a homogeneous nearly K¨ahler structure onS³×S³. This structure onS³×S³is related to a general con- struction due to Ledger and Obata: If G is any compact simple Lie group, then G×Gis a Riemannian 3-symmetric. AsS³admits a Sasakian 3-structure (see [33]), we ask the following.

Problem 4.4. Can one realize the homogeneous nearly K¨ahler structure onS³×S³ in terms of Sasakian the 3-structure onS³ ?

Problem 4.5. Can one construct the other non-homogeneous nearly K¨ahler struc- ture onS³×S³ ?

In particular, from Riemannian geometric point of view , we are very interested in the following.

Problem 4.6. The Hopf conjecture is as follows (for example, see [49], [50]): Does S²×S² admit a metric with positive sectional curvature ? Can one, by making use of a similar change of metric to (4.7), construct a metric with positive sectional curvature onS³×S³?

Acknowledgment

The author would like to thank Professor H. Mori for his valuable suggestions and kind check of some computations.

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