on hyper f -structures

Math. Ann. 306, 205-230 (1996) Mathematische Anna]en (~) Springer-Verlag 1996

On hyper f-structures

Gerardo Hernandez*

Department of Mathematics, University of New Mexico, Albuquerque, NM 87131, USA

Received: 10 January 1994 1 Revised version: 20 October 1995

Mathemat ics Subject Classification (1991): 32C17, 30G35, 32G07

1. Introduction

In 1963 Yano [Ya] introduced the notion off-structures, that are given by tensor fields f of type (1, 1) that satisfy the equation f3 + f = 0. Such a structure generalized the well-known almost complex structure defined in some manifolds of even dimension, and the (by then) recently introduced concept of almost contact structure defined by Gray [Gray] for odd dimensional manifolds, with its tensorial formulation due to Sasaki [Sa].

Although almost contact structures are naturally given on oriented hypersurfaces of almost complex manifolds, the motivation of such a notion came from the linear (tangent) approximation of the contact condition. However, contrary to the almost complex case, there is no integrability condition for the tangent approximation in the almost contact case [Blal]. Sasaki developed instead the notion of normality for these structures. Noticing that the product of an almost contact manifold with the real line inherits an almost complex structure, the almost contact structure is defined to be normal if the almost complex structure in the product is integrable [Sa]. It was also noticed that R 1 (S~) bundles over almost complex manifolds have an almost contact structure and that this one is normal if and only if the base manifold is complex [Moll.

A normal contact structure is usually called Sasakian and was intended to be the odd dimensional analog of Kiihler structures. There are some other relations between these two structures: every Sasakian manifold embeds as a hypersurface in a K ~ l e r manifold [Ta], and, under certain regularity conditions, fibers over a Kiihler manifold [Me2, Ha]. Hence, Sasakian structures appear naturally in the

* Present address: CINVESTAV-SMTC, Mexico (e-mail: [email protected])

206 G. Hernandez

K~hler reduction. Using the concept off-structure, the previous results show the existence of important relations between f-manifolds. However, the idea of f - structure is more general. For instance, hypersurfaces of almost contact manifolds are not in general almost complex manifolds, but they have f-structures associated to them. Moreover, it was also proved that every CR-manifold has an f-structure [Ia]. The great variety of manifolds with an f-structure motivated the study of properties parallel to those of K~ihler manifolds, extending the notion of normality for the so called globally framed f-manifolds, where the distribution annihilated by the tensor f admits global sections [Go, GoYa].

On the other hand, the quatemionic analog of almost complex structures was developed by defining (locally) 3 almost complex structures that satisfy the algebra of the imaginary quaternions. Special attention was paid to the quaternionic K~thler and hyper K~ihler, that are particular cases of almost quaternionic and almost hypercomplex structures. Naturally, hypersurfaces of almost hyper complex manifolds inherit three almost contact structures, but the tensor fields associated to them do not satisfy the algebra of the imaginary quaternions, as one might suspect based on their rank. This structure is called an almost contact 3-structure [Kuo] and immediately leads to the notion of 3-Sasakian structure. It was extensively studied by the Japanese school during the seventies; remark- ably, the only known examples were the spherical space forms. These manifolds received a systematic study in [BGM2], motivated by the discovery of a large family of examples. Following the case of a single Sasakian structure, it was proved that every 3-Sasakian manifold (with regularity conditions) fibers over a quaternionic K~ihler manifold [Ish], and embeds in a hyper K~ihler manifold [BGM2]. It was also noticed that 3-Sasakian manifolds play a natural role in the hyper K~ler-quaternionic K~ihler reduction [BGM1 ].

Since f-structures generalize those of almost complex and almost contact structures, an obvious extension of almost quaternionic and almost contact 3- structures could be given as quaternionic or hyper f-structures. This is the basic idea of the present article.

In Sect. 2 we give a precise definition of almost quaternionic and almost hyper f-structures and related concepts. In particular we study globally framed hyper f-structures and prove the equivalence of the notions of hyper f-structure of corank 3 and almost contact 3-structure (hyper f-structures of corank 0 are simply almost hyper complex structures). The former equivalence also shows that the almost contact 3-structure introduced by Kuo [Kuo] implies the almost 3-contact structure that is suggested by Lutz [Lu] work. However, the notions of 3-contact in these approaches are completely different. In the first case, each contact form is of maximal rank, whereas the second is defined by weaker conditions. Moreover, 3-Sasakian manifolds fiber (with fiber Sp(1) or SO(3)) over quaternionic K~hler orbifolds -or manifolds, in the regular case-, whereas regular, normal hyperf-manifolds with contact structure fiber (with fiber T 3) over hyper K~ihler manifolds. To prove this last statement we observe first that a notion of normality for hyper f-structures cannot simply be given by the condition that each f-structure is normal in the sense of Goldberg [Go] (Proposition 2.11).

On hyper f-structures 207

Therefore, a new definition of normality is given and we explore some of its consequences. Although clearly a hyperf-structure induces a quaternionic analog of a CR-structure, we do not develop this idea in the present work.

In Sect. 3 we begin by reviewing the technical conditions that guarantee the already mentioned relations between f-structures and hyper f-structures. Then we proceed to prove our main theorems:

Theorem A. Let M 4n+3 be a compact connected manifold with regular normal hyper f -structure. Then M 4n+3 iS the bundle space of a principal toroidal bundle over a hyper complex manifold M 4n. Moreover, if such a manifold has a 3-contact structure (it is a hyper PS-manifold), then M 4n is a hyper-Kiihler manifold. In this last case, the projection defines a riemannian submersion with totally geodesic fibers.

Theorem B. Let (M 4n , Js , ~s ) be a compact hyper Kgihler manifold whose Kgihler 2-forms associated to the complex structures .Is have integral period. Then there is a principal T3-bundle over M 4n whose total space is a compact manifold with a regular hyper PS-structure.

Hyper PS-structures are special cases of hyperf-structures whose importance is justified by the above mentioned theorems. The next two sections are therefore directed to the study of such structures.

In Sect. 4 we begin by studying general geometric properties of hyper PS- manifolds. In particular we prove that the metric compatible with the hyper f-structure is not Einstein. This last result contrasts with the 3-Sasakian, quaternionic Kfihler and hyper K~ihler cases. Then we turn our attention to the 3- Sasakian manifolds. It is proved in [BGM2] that 3-Sasakian manifolds embed as hypersurfaces in hyper K~ihler manifolds. Here we prove that it is not only a totally umbilical embedding, but is an extrinsic sphere as defined by Nomizu [No], with mean curvature vector parallel with respect to the normal connection. We prove then that there is a unique way of embedding a 3-Sasakian manifold as hypersurface in a hyper K~ihler manifold. Since every 3-Sasakian manifold looks (extrinsically) like a sphere, taking products with odd dimensional spheres we obtain new examples of generalized Hopf surfaces and Calabi-Eckmann manifolds.

Finally, in Sect. 5 we study the basic (but non-trivial) examples of hyper PS-manifolds. They are compact quotients of the quaternionic analog of the Heisenberg group and fiber over tori of dimension 4n. They also permit us to construct hypercomplex manifolds with strictly negative sectional curvature, as well as new examples of hypercomplex non hyper K~ihler manifolds. Similar generalizations of the Heisenberg group provide examples of hyper complex non hyper K~ihler, and hypercomplex, symplectic, non Kahler manifolds. They are presented in the addendum.

208 G. Hernandez

2. Hyper f-structures

Let M" be a C ~ manifold on which there is given a non vanishing tensor field f of type (1, 1) that satisfies f 3 + f = 0. I f f has constant rank r, it is called an f-structure and M an f-manifold. It is known that in this case r is even (r = 2k) and that the tensor fields defined by _ f2 and f2 + 1 are projection maps that split TM in two complementary distributions ~ and J/g. In a neighborhood of each point of M there exists a basis ~i, i = 1, 2 , . . . ,m = n - r = dim~//g, for the distribution ,//~ and, taking the 1-forms r/i E Ann ~)~, i = 1, 2, ...,m, dual to these vector fields, the f-structure satisfies

~2i(~j)=~5~, f~i=O, rlif=O, f 2 = - l + r l i |

(summation convention assumed). It is easily seen that the existence of such an f - structure is equivalent to the reduction of the structure group to U (~) x O (m). This structure has been extensively studied. One special case is when the distribution ,//~ is trivial, which implies the existence of global vector fields ~i with the described properties. Now the structure group can be reduced to U(~) x Ira and the f-structure is called a globally framed f-structure [GoYa]. Examples of these structures are almost complex (m = 0) and almost contact (m = 1) structures [Sa].

Consider now a manifold M provided with two bundles F and E of rank 3 and m respectively, F C End(TM) and E C TM. Assume that on any open set U there exist local sections]q, f2, f3 E F(F), and ( i , . . . ,~,, E F(E) that satisfy

(2.1) ~f) = -~J~ =A,

for ( i , j , k ) = (1,2, 3) and cyclic permutation of indices, and

(2.2) f ~ r = 0 , f ~ = - - l + ~ T r | r/~f~=0, s = l , 2, 3; r = l , . . . , m ,

where r/$ is a local base of E* dual to (~, ~s(~r) = (~sr, r,s = 1 , . . . ,m. In the intersection of two open sets U, U ' C M we have

and due to the relations (2.1) and (2.2) we conclude that (asr) E SO(3). A manifold M with this structure and such that the bundle E admit global sections ~s with the properties just described will be called an almost quaternionic f- structure of corank m. We conclude that a manifold admits a quaternionic f - structure if and only if. the structure group of the tangent bundle can be reduced to Sp(n). Sp(l) • We immediately see that the dimension of M is 4n + m. If the bundle F also admits global sections fs, the structure group can be reduced to Sp(n) • Ira, and we will call this a hyperf-structure ofcorank m. Examples of the first kind are the almost quaternionic (corank m = 0), and, of the second, hypercomplex (corank m = 0) and almost contact 3-structures (corank m = 3).


We will be particularly interested in this last case, where the rank of E is

three. From now on, we will assume this cyclic permutation convention of indices for i , j , k , and we also set r , s , t = 1,2,3.

Fixing an open set U C M, we define the local tensor fields

(2.3) 0i : f i "l- Tlj ~ k --T]k @ ~ j ,

and it is easy to check that they satisfy

Csfs = 0, //sr = 0, r = _ l + /Is @ ~s. $

Hence each r determines an almost contact structure [Sa]. Moreover, we have the relations

r162 viej=r/k, r162174162162162174162162

Therefore (r is , r/s)s=l.2,3 determines a local almost contact 3-structure as defined by Kuo [Kuo]. Then a hyper f-s t ructure of corank m = 3 is precisely an almost contact 3-structure. It is known that every almost contact 3-structure has an associated riemannian metric g that satisfies, for each s,

9(X, Y) = g(r esY) + ~Ts(X)~s(Y), g ( ( s ,X ) = r/s(X).

An easy calculation shows that this metric is also compatible with each f - structure, f i , i.e., it satisfies

9(X, Y) = g(fsX,fi Y)+ Z ~r(X)~r(Y).

Such a metric is not uniquely determined. For a fixed compatible riemannian metric such structure is called a metric hyperf-structure. From now on we will assume that we have chosen a compatible metric. We now define the following 2-forms

us(X, Y) = g(X,fsY), &s(X, Y) = g(X, CsY),

then ~i = w, - ~j A r/k, with our usual cyclic convention of indices. We observe that the two forms ~s are linearly independent. It is also clear that the two forms a2s are also linearly independent.

If &~ = dq?~ we have a contact 3-structure [Kuo]. The structure will be called

a 3-S-structure if us = d~/~. As was observed by Goldberg [Go], for a single globally framed f -manifo ld ,

�9 1 = +(1/]~kl)T/ l A ~2 A . . - h ~m A ~k, and ~ is of rank 2k (for a manifold of dimension 2k +m). Since this relation holds for each w~ in our case, we conclude that

(2.4) rh A r/2 A r/3 A ~sa~s 7~ 0,

with (A1, A2, A3) C S 2, each a~i of rank 4n. In the case of a 3-S-structure,

2 1 0 G . H e r n a n d e z

(sZ )2n (2.5) Ot A 7?2 A 773 A Asd77s # O.

This last equation defines a "p contact structure" in the sense of Lutz [Lu] (t9 = 3 in our case) who used it in an attempt to generalize the notion of contact structure to higher codimensions. Now let us recall Liebermann's definition of an almost contact structure (which Blair [Blal] proved to be equivalent to the structure of the same name introduced by Sasaki [SA]): A manifold M 2~+1 has an almost contact structure if there exist a 1-form ~7 and a two form 03 such that 77 A w n r 0. We would be tempted to define an "almost contact 3-structure" as a pair of triplets of 1-forms (rll , r12,773) and of 2-forms (031,032,033), each w~ of rank 4n, that satisfy (2.4). After all, notice that rh A 772 A r13 A (,ks~s) ~ =

- - - - - . 2 n + l - . 2 n + l 771A 772 A 773 A(AsWs) 2" r 0, and, in particular, z/1A~72/\r/3/\~s = 77~ A ~ S ~ 13 for each s. Therefore ~ are of maximal rank and each pair (rls, ~ ) determines an almost contact structure in the sense of Liebermann and, thus the almost contact structure of Sasaki. However, the triplets of almost contact structures do not necessarily satisfy the relations required by Kuo for an almost contact 3- structure. Nevertheless, given an almost contact 3-structure it is easy to recover, reversing the previous argument, the condition (2.4). Therefore we have:

Proposit ion 2.6. Let M 4n+3 be an almost contact 3-manifold (in the sense o f

Kuo). Then there exist on M 4n+3 globally defined one forms 771,772, ~3 and lineaHy

independent two forms wl , w2 , w3 o f rank 4n that satisfy (2.4). The converse of this proposition can be established under special circum-

stances, as we will see in theorem 3.5. The case of interest for us will be the one when

(2.7) d~?,(X, Y) = w~(X, Y) = 9 (X , f i Y).

Remark 2.8. This is what Lutz calls a "p-contact structure", d77s = 03.~. The "3 contact structure" is not equivalent to the "contact 3-structure" of Kuo [Kuo]. Notice that the first assume dr/s = ws, whereas the second d77s = ~s.

Another important concept associated to f-structures is that of normality. An f-structure is said to be normal if the Nijenhuis tensor o f f , Nf f ) , satisfies [GoYa]:

+ Z d O r | = O, (2.9) N( f ) r

whereas an almost contact structure is normal if it satisfies [SaHa]

N(40 + d r / | ~ = 0.

(that means that the almost complex structures J defined on the product manifold M z~+l • R by J ( X , h d / d t ) = (~X - h~, ~(X)dldt) are integrable, where h is any real function and t is the coordinate for R). Moreover, the normality o f f implies the normality of 4~, when they are related as in (2.3) [GoYa]. A normal f-structure that satisfies (2.7) is called by Blair [Bla2] an S-structure. It has

On hyper f-structures 21l

important properties. For instance, assuming regularity conditions, it fibers over a Kaehler manifold with toric fibers. However, this concept cannot be extended to the case of three f-structures that satisfy the quaternionic relations (2.1). For this normal S-structure, which is defined for a single f-structure, we have the following relations

5eC~,fr = 0, ~ s 9 = O, VX~s = - } f i X .

(In fact all of them are equivalent for this kind of structure). This implies that

(2,10) , V~,fr = 0.

If we assume that in our case all of the fs are normal S-structures satisfying (2,1), then

( & , f j ) X = [~ i , f jX] - f j [ ~ i , X ] = ( V { , f j ) X - V j j x ~ i + f j V x { i = ( V { , f j ) X ..t-fkX ,

which means that V{,j~ = - fk , contradicting (2.10). We have proved:

Proposi t ion 2.11. There are no triples fs o f normal S-structures that satisfy d~s = w~ and the quaternionie relations fifj = fk.

We can now establish the structure that properly generalizes Blair 's construction.

Definition 2.12 A hyper f-structure o f corank 3 is said to be normal i f the fs tensors satisfy

N (d~i)(X, Y) + drli(X , Y){i - dr~j(X, Y){j - d~Tk(X, Y){k = O.

And a normal hyper f-structure that satisfies drls = ~Os will be called a hype r PS-strueture.

We use PS for pseudo Sasakian. As we will see in Sect. 5, "sub Sasakian" or "hyper Sasakian" would be also an adequate term. We remark that in a normal hyper f-structure the individual tensors fi are not normal in the sense of (2.9).

Proposi t ion 2.13. For an 3, normal hyper f-structure the following relations hotd

) r ~9~ f r=O, ~rf;E, sf]r-----O , r , s = l , 2 , 3

d r l i ~ X , Y) + drli(X ,J) Y ) = 0

drls(fiX, Y ) = d r l s ( X , f i Y ) , s T~ i.

The proof is basically the same as that which appears in [Go] for single normal f-structures. Proof'. Sincefi satisfy (2.12), set Y = ~i and apply r/r to get drlr(X, ~s) = 0. But

Hence ~ 7/r = O.

212 G. Hernandez

Now set X = ~,, Y = ~r. We get3~[~.~,~r] = 0 which implies [~,,~r] = Cr~t.t Therefore ~i[~s,~r] -- c,.i However,

0 = ~sr/~(~,) = ( ~ , ~ i ) ~ s + ~ [~ , ~r].

Then c~i = 0 and [~ ,~r ] = 0. TO conclude the first set of equations, we notice that

f i ( ~ a S ) x + dni(X, r162 - ct nj(X , r )r - d ,k( X , ~ )r = O.

Thus j S ( ~ c 6 ) X = 0. Hence (5'J~J~)X = l_tt~r Now rtr(.~f~'~jS)X = #~. Using 0 = (~r = (~r + r/r(~-~JDX, we get

~tr~ = 0 and ~ J , - = 0.

Finally, r l~ iX , f iY] • drip(X, Y) = 0 , for all X, Y E ~r where we choose the positive sign for r = i and the negative if r r i. Then r/r[fiX,f~2Y]• Y) = 0. Now using drlr(fiX, Y) = -rlr[f,'X, Y] we get drlr([iiX, Y) • drl~(X,fi Y) = 0, from which the result follows. []

The following two results are immediate consequence of the equation @rf~ = 0.

Coro l l a ry 2.14. The vector fields r are Killing with respect to the metric 9, compatible with the hyper PS-structure, =Sif~r 9 = O.

Corol la ry 2.15. If (fs, 7/,, ~ ) is a hyper PS-structure, then XTx~ = -�89 for all X.

It also follows from this last corollary that

(2.16) V~r~$ = O,

which means, in particular, that the integral curves of the vector fields ~ are geodesics.

Other important consequences of LTC~r = 0 are given by the equations

(2.17) V{,./~ = - fk , V(,J} = 0.

The normality of the associated tensors r is also needed, and can be established for hyper PS-structures.

T h e o r e m 2.18. Let ~ , ~ , r/s),=1.=,3 be a hyper PS-structure. Then the associated almost contact 3-structure (r {s, r/s)s=~,2,3 is normal.

Proof'. For any quaternionic f-structure, we have

N(0D(X, Y ) = N (fi )(X , Y)+ rlj(X)(:~r ) r - rlk(X)( SL~r )Y - rl j(Y )(_~;~r )X

+o~(r ) (Yr + [ r ~ j ( r ) ( ~ o ~ ) x - ~ ( r ) ( ~ , r ~ k ) x + r ~ , ( x ) ( ~ , ~ ) r

- rlj ( X )( SC'~'r r/t, ) Y + d rlj ( X , Y ) ICj + [ rl~c ( Y )( ~r rlj )X - r/t ( X )( ~r r/j ) Y

+r~(X)(~,'~trtj)Y - rlj(Y)(5;C~rlj)X + drtk(X, Y)]r + [(~r v r/t)X -(9--);xr/k)Y]r


Using the proposition 2.13, we simplify

N(r Y) = - d ~ i ( X , Y)~i + 2d~j(X, Y)~j + 2dvik(X , Y)~k

Since ( ~ x ~ k ) Y =fiX~Tk(Y) - Ok[fiX, Y], a direct calculation shows that

( Y~f,x~Tk)Y - (5-'x2~ r~k)X = d r l k~X , Y) + drlk(X,f . Y) = 2d~?j(X, Y) ,

and similarly

( ~J~j;xq/)Y - (~'~t, Yrlj)X = d~j(f iX, Y) + dr l j (X , f iY ) = - 2 d ~ k ( X , Y )

Hence, N( r = - d ~ i | ~i"

The converse of this result is immediate.

Theorem 2.19. Let (0s, ~s, r/s)s=l~2,3 be a normal almost contact 3-structure. I f ~ r = 0 and there is a compatible metric such that d~i - ~ A ~ = wi, then the associated structure ~s, ~s, rls)s=~.~.3 is a hyper PS-structure.

3. Hyper PS-structures as fiber bundles

Let P ( M , G) be a smooth principal bundle with connection form ~ and curvature form (2. Since (2 is p valued, with p Lie algebra of G, we obtain a horizontal, scalar valued forms on P by taking compositions # • ~ with # E So*. If P admits a connection such that/20(2 is non degenerate for all # # 0, then P is called a fat bundle [We]. If the projection P ---+ M is a riemannian submersion with totally geodesic fibers, then the bundle P is fat if and only if the sectional curvatures of plane sections spanned by one horizontal and one vertical vector field (the so called vertizontal plane) is positive [We].

Regarding f-structures, it is known [Ko2] that the circle bundle over a manifold with certain pinching is a sphere and so it is a fat bundle [We]. More gen- erally, following Kobayashi 's results [Kol], over a symplectic manifold can be constructed a circle bundle with a contact structure [BoWa]. I f the base manifold is K~ihler, the circle bundle with totally geodesic fibers is necessarily Sasakian [Ha] (contact and normal) and since for Sasakian manifolds the sectional curvature spanned by the fundamental vector field (vertical) and any vector field orthogonal to it is I / 4 [Bla2], it follows that these circle bundles over a K/ihler manifold are fat. In this case the curvature form is the pull back of the K~ihler form by the projection. These kinds of manifolds appear naturally in the K~ihler reduction.

For 3-f-structures, it is known that the Hopf fibration S 3 ~ S 4n§ ---+ HP" provides an example of a fat bundle over a quaternionic K~ihler manifold [We]. It is possible to construct S3-bundles (or SO(3)-bundles) with totally geodesic

214 G. Hernandez

fibers over a quaternionic Kahler manifolds with positive scalar curvature [Kon]. In this case, the total space is 3-Sasakian and by the previous argument, these fi- brations provide also examples of fat bundles. Boyer, Galicki and Mann [BGMI] constructed several examples of this structure using hyper K~ihler-quaternionic Kahler reductions. As they observe [BGM1], the manifolds involved in this construction are Einstein manifolds.

Remark 3.1. In the 3-Sasakian case, [~, ~j] = ~k, which prevents the associated almost quaternion f-structure to be a hyper f-structure; the almost contact 3- structure is normal, though.

It is natural to ask whether there exists a 3-f-structure that fibers over a hyper K~ihler manifold with totally geodesic fibers and that is also a fat fibration. First let us recall the following theorem due to Blair, Ludden and Yano [BlaLuYa].

Theorem 3.2. Let M 2n+~ be a compact connected manifold'with regular normal f-structure. Then M is the bundle space of a principal toroidal bundle over a complex manifold N2n(= M~+S /~//g). Moreover if the associated ~'o-form on M is closed, then N is a Kdhler manifold.

The regularity of an f-s tructure refers to the regularity of the distribution spanned by the vector fields ~i, i.e., that every leaf of J / g intersects, in precisely one s-dimensional slice, the domain of some coordinate system with coordinates (Xl,... ,Xn) such that the set {O/Oxl, ...,O/Oxn} defines a basis for ~///~; an obvious condition to guarantee the manifold structure of the quotient M 2n+l/,j~. The following result is an easy consequence of the previous theorem:

T he o re m 3.3. Let M 4n+3 be a compact connected manifold with regular normal hyper f-structure. Then M 4n+3 is the bundle space of a principal toroidal T 3 bundle over a hyper complex manifold M 4n. Moreover, if M 4~+3 is a hyper PS-manifold, then M 4n is a hyper-Kahler manifold. In this last case, the projection defines a riemannian submersion with totally geodesic fibers.

The proof is basically a repetition of that given in [BlaLuYa]. We give a sketch here for completeness. Proof." From the normality off~ we know that [~r,~s] = 0 for r ,s = 1,2,3. The regularity of ~//g guarantees that we have a foliation of M 4n+3 by tori T 3, and we define M 4n = Man+3/T 3. The connection form 77 = (r/l, r/2, 7/3) defines a horizontal distribution and, with respect to the projection 7r : M 4n+3 ~ M 4n , a horizontal lift ~-. We define tensor fields Js on M 4~ by JsX = 7r.f~rX, which can be done since the tensors f~ are projectable, and compute (note that J f X = - X ) :

[Ji,Ji](X, Y)

= - [ X , Y] + [Tr.3]~'X, 7r.fi~-Y] - r.fi~[Tr.f,.~-X, Y] - 7r.fi~[X, ~r.fi~'r]

= "/T.(~/2[~'X, ~I'Y] - y ~ r/i([~X, ~Y])~i) + 7"f .[ f i~X,f i~Y]

- , r . j5 bS~'X, ~Y] - 7 r~ [~X, / ;~ r r ]

= 7r,([fi ,fi ](~-X, ~rY) + drli(~rX, ~rY)~i

-dr/j(~rX, ~rY)~j - drlk(~rX, ~rY)~k) = 0

On hyper f -structures 215

So M 4n J.s hyper complex. Now suppose M 4n+3 is a hyper PS-manifold. We define a metric ~ on M 4" by O(X, Y) = 9(frX, frY) where 9 is a metric compatible with the hyper PS-structure. Then {I(J~X, Js Y) = .~(X, Y)and we can define the Kiihler 2-forms ~2i on M 4n as usual: ~i (X , Y) = g ( J i X , Y ) . We get wi = 7r*~i and so 0 = d~i = dzc*s = 7r*dQi ~ d ~ i = 0 and M 4n is hyper-K~ihler. Finally, the submersion is totally geodesic since V~r(S = 0 and the vertical projection of V r X is zero (2.17). t3

Remark 3.4. We could also drop the condition of regularity and assume that T 3 acts locally freely on M instead. Since the vector fields ~.~ are Killing (2.14), the metric is bundle-like ([BGM2], lemma 1.7). The hyper PS-manifold fibers then over a hyper Kiihler orbifold. See [BGM2] for the details of this construction applied te 3-Sasakian manifolds fibering over quaternionic Kfihler orbifolds.

Theorem 3.5. Let (M4n,Js, .Qs) be a compact hyper Kiihler manifold whose K~ihler 2-forms associated to the complex structures Js have integral period. Then there is a principal T 3-bundle over M 4n whose total space is a compact manifold with a regular hyper PS-structure.

Proof" Since the 2-forms aQi are linearly independent and of maximal rank on M 4", they satisfy ( ~ Ai g2i) 2~ ~ 0 for ~ As = 1. Following Lutz [Lu, Proposition 2], there exists a principal bundle 7r : M 4n+3 ~ M 4n with structural group T 3,

such that the connection form r? = 0/1, r/2, zt3) with curvature co = 7r*O, defines a

regular 3-contact structure on M 4n+3. N o w w e see that rh A~/2Az/3 A ( ~ ) ~ w s ) 2" 0. We choose vector fields ~s, dual to the connection forms z/s, and a locally defined metric 9 o n M 4n+3 given as 9 = ~r l~ | r/s + 7r*~, with .~ the hyper Kiihler metric of M 4n . Now define Y:f = w~ - r/j A r/k. Then ~s are of maximal rank and satisfy the Liebermann condition and then each pair (7/~, ~ ) defines an almost contact structure. Since ~ is non degenerate on the distributions ker(z/s), it follows that the (1, 1) tensors ~s defined by ~:s(X, Y) = 9(X,r satisfy ~ = - I restricted to that distribution. We extend that tensor to the whole TM 4n§ by requiring ~bs~s = 0. Evidently, for each s we have the following relations

Also note that

g(~i~, X) = -~ , (~j , X) = ~: (~j)~(X) = ~k(X) = g ( ~ , X).

So 4~i~) = ~k. On the other hand,

o~Ojx = g(6, r = - 9 ( ~ j ~ , x ) = g ( ~ , x ) = ~ ( x ) ,

which proves that r/i~j = r/~. Finally, we can prove that, for all possible values of X, and Y,

ff;~((ojX, Y) - ~]i(X)~j(Y) = Cdk(X, Y ),

which proves that cki(~j -z/ j | = $~. We have constructed an almost contact 3- structure or, equivalently, a (necessarily regular) 3-S-structure (since d77i = wi).

216 G. Hernandez

It remains to prove that the associated hyper f-structure is normal. To prove it, we construct the complete structure on the total space explicitly. We define the tensors j~ as fi = ~rJiTr. (note that equation (2.3) is satisfied). Since ~i are vector fields dual to the connection forms we can immediately verify the conditions (2.1) and (2.2).

Now we compute

[f,, , f ] ( X , Y ) + d rh ( X , Y )~i - d oj ( X , }" )~j -- d ~lk ( X , Y )~k =

"~[Ji, Yi](']r. x , 7r, Y )+ ~-~ r]s[fiX , f i Y IG + d q i ( X , Y )~i - d r l j ( X , Y )~j - -dr lk(X , Y )~

= +drli(X, Y)~i - dr l j (X, Y)~j - drlk(X, r ) ~ + E r ls[ f ix , f i Y]~s

But r l s[ f iX , f ig] = - d r l s ~ X , f i Y ) . Using drb = a;, and (2.13) we get the result. [3

4. Geometry of hyper f-structures. Embeddings

In Sect. 3 we saw that, under suitable regularity assumptions, certain f-structures (Sasakian) fiber over others (K~ihler), and that the h y p e r f - structures which are 3-Sasakian and hyper PS-structures fiber over quaternionic K~ihler and hyper K~aler manifolds, respectively. Many more relations are known between the classical (quaternionic) (hyper)f -s t ructures K~ihler, hyper K~ihler, quaternionic K~hler and 3-Sasakian. In [BGM1] it is proved that for a quaternionic Kahler M manifold with positive scalar curvature there exists a commutative diagram

(4.1) H */Za .9 ~

M

where ~ is hyper K~ihler, - ~ is K~ihler-Einstein and . ~ is 3-Sasakian. A re- markable fact about this diagram is that all the geometries involved are Einstein.

On the other hand, we have seen that hyper PS-manifolds fiber over hyper K~ihler manifolds. However, the metric associated to the hyper PS-structure is not Einstein. We establish this fact as a theorem.

Theorem 4.2. Let M be a hyper P S - m a n i f o l d and g an associated metr ic to this

structure. Then the Ricci tensor r is given by

(--314n O ) r = 0 nI3 "


We prove this theorem through a sequence of easy lemmas.

L e m m a 4.3. Let M be a hyper PS-manifold and g an associated metric to the hyper PS-structure. Then the sectional curvature K satisfies K(~s, ~r) = 0 and, for

1 X c ~ , K(X,~s)= ~.

Proof" For any Y,

1 1 ~ f 2 Y 4 f 2 Y

And the result follows.

L e m m a 4.4. For a manifold M with PS-structure and dimension 4n+3, the Ricci curvature in the direction of (~ is equal to n, r(~s, ~,) = n9(~,, ~r).

L e m m a 4.5. Assume the same hypotheses as in the previous lemma and let X, Y be two vector fields in the distribution c~, then r(X, Y) = - 3 g(X, Y).

Proof Since M 4n+3 fibers over a hyper K~ihler manifold we can use the equation [Bes, 9.36cl

r(X, Y) = -29(Ax,Ay).

Recall that the fibration is totally geodesic and hyper K/ihler manifolds are Ricci flat. Now use the relations 9(Ax,Ay) = ~_,9(Ax(~,Ar~D and AxEs = -�89 to get the result. Finally, we show that

L e m m a 4.6. For X C 5ir r (X ,~ ) = O.

Proof" We choose an orthonormal local basis Xi for c~, and use the formula

3 4n

r(X,~i) = Zg(R(~. , ,X)~i ,~s)+ Zg (R(X j ,X )~ i ,X j ) . s=l j=l

Firstly we establish that 9(R(~s, X)~i, ~ ) = 0. We need to show that R(~, X)~i is horizontal. This is clearly a result of the following computation, which uses (2.17).

1 R(~,,X)~i = V~,Vx~i - V x V ~ I - Vt~,,xl~i = - � 8 9 + ~fif, X.

Now we prove that 9(R (Xj, X)~i, Xj) = O.

R(Xj, X)~ i = Vx, Vx~i - V x Vxj ~[ - Vtx ].Xl~i

= - 1 V x , ~ X ) + ~ V x ~ X j ) + ~f.[Xj,X]= - ~1 (Vx, fi)X + ~(Vxfi)Xj.

Since Xj and X are horizontal, (4.10) implies that (VxSi)x and (Vxfi)Xj are vertical. The theorem follows from 4.4, 4.5 and 4.6. []

Corol lary 4.7. Let M 4~+3 be a hyper PS-manifold with compatible metric g, then the scalar curvature is constant and equal to -3n.

218 G. Hernandez

Moreover, these manifolds are hopelessly non Einstein with respect to the submersion. Considering the canonical variation of the metric in the total space

9t Iv = t9 Iv , 9t Itt = 9 lu , 9t(V, H) = 0, we use proposition 9.72 of [Bes] to see that no metric 9r is Einstein.

We can also use the fibration result to compare the sectional curvatures of the hyper PS-manifold M with those of the hyper K~hler manifold N on which M fibers rr : M ---, N. Using standard techniques we get

Proposit ion 4.8. Let M be a hyper PS-manifold and N be a hyper K~ihler manifold and 7r : M ~ N a Riemannian submersion as in (3.3). Then the holomorphic sectional curvature K (X, Jif~) is related to the f-sectional curvature K (X, f i X ), X horizontal lift o f fC by the relation

3 /((X, J iX) = K (X , f iX) + -g.

The odd behavior of the PS-structure with respect to the classical hyper f - structures is also reflected in immersions. It is also proved in [BGM1] that every 3-Sasakian manifold embeds in a hyper K/ihler manifold in a "natural" way. We will prove that this is not the case for hyper PS-manifolds. But lets first explain the adjective "natural".

Let M and N be two (quaternionic) (hyper)f-manifolds, and L : M ~ N be a map that preserves the f-structure, we say then that e is an f -map , and if it preserves a fixed metric compatible with those f-structures, we call ~ an f- isometry or a isometric f -map. It is well known that every globally framed f - structure induces an almost contact structure for manifolds of odd dimension and an almost complex structure in the even-dimensional case [GoYa]. Similarly, a hyper f-structure induces an almost contact 3-structure or an almost hypercomplex structure depending whether the dimension of the manifold is 4n + 3 or 4n + 4. We extend the definition of an f - m a p to one that preserves either the (hyper)f-structure or the associated almost contact (3 -)structure or almost (hyper) complex structure. We observe that an f - isometry also preserves the hermitian metric or the metric associated to the almost contact structure. Con- sider, for example, a hyper K~ihler manifold N 4n+4 with complex structures I ~, a = 1,2, 3, and hyper hermitian metric g. Now take an embedded hypersurface t~ : M 4n+3 --+ N 4n+4. This embedding induces an almost contact 3-structure on

M. If M is a 3-Sasakian manifold and the almost contact 3-structure induced by the embedding is the one that defines the 3-Sasakian structure, we say that the embedding c is an f-embedding of a 3-Sasakian in a hyper K~ihler manifold, and clearly it is, in a sense, "natural".

Now we proceed to give an alternative characterization for a hyperf-structure to be a PS-structure. Let ~ , r/s, ~s, g) be a metric hyper PS-structure d~s (X, Y) = w~(X, Y ) = g (X , f sY ) . As in [Blal, p. 53-54] we simply use the identity

29(Vx Y, Z) = x g ( r , Z ) + Y 9(X, Z ) - Z9 (X , Y)

(4.1) +9([X, Y ] , Z ) + g(IZ,X] , Y) - 9([Y , Z I , X )


together with the well known formula

1 d~i (X , Y , Z ) = "~[Xcoi(Y,Z) + YCOi(Z~X) + ZCO,(X, Y)

-COl(IX, YI, Z) - c o ; ( [ Z , X ] , Y) - toe([Y, Z I , X ) ]

applied to the expression 29( (Vx f i )Y , Z) to obtain (see also proposition 2.13),

(4.9) (Vxj'})Z = l { - d r l i ( Z , f . X ) ( i + drl j (Z, f ,X){ j + dr~,(Z,f,.X){k

+)'li(Z)f.2X + ~j(Z)fkX - 1]k(Z)fjX } + oj(X)fkZ - 7"lk(X)fjZ.

Or, equivalently,

~7xwi(Y ,Z ) = • { - d ~ ( Z , f ~ X ) ~ ( Y ) + d~j(Z,f~X)~j(r ) + dwk(Z,f~X)~k(Y )

+d~i(r,f,X)~i(Z) - d ~ ( r ,aSX)rlj(Z) - d ~ ( r , ~ X ) ~ k ( Z ) }

(4.10) +drlk(Y ,Z)o)(X) - dr~j(Y ,Z)rlk(X).

Conversely, suppose that M has a hyper f-structure ~ , rt~, (s ,9) such that equation (4.9) holds, and assume also that the vector fields ~s are Killing. Then we get Vx~i = - � 89 and

d~i(X, Y) = ((•xrli)Y - ( ~ y ~ i ) X ) = 2g(Vx~i , Y) = coi(X, Y),

thus dr/, = COs. Finally we use the formula

N ~ ) ( X , Y) = (fsVvfs - V f # f D X - (fsVxfs - V/,xf+)Y,

to get

N(fii) + d~i | {i - dr~j | {j - d r l k | {k = O.

We have then established

Theorem 4.11. A metric hyper f-structure ~ , *Is, (s, 9) is a hyper PS-structure if and only if the tensors fs satisfy (4.9), and the vector fields (s are Killing.

Suppose that a hyper PS-manifold M embeds in an almost hypercomplex N manifold, c : M -+ N, with L an f-embedding such that ~*g~, = &~, where ~b, are the K~ihler forms associated to the almost hypercomplex structure in N. Using the Gauss equation fg~.x~.Y = ~ , V x Y +h(X, Y )C , where C is a normal vector, the covariant derivatives of these K~ihler forms satisfy

(4.12) ( V X & s ) ( Y , Z ) = (XTX~s)(Y,Z) + h ( X , Y)rb(Z) - h(X,Z)r ls (Y) ,

where we have identified t , X with X, since the context is clear. We use the equations (4.9), &i =COi - ~j A~7k and drl~(X, Y) = - 2 ( V r r b ) ( X ) to get

(VxYoD(Y ,Z ) = �89 - d ~ , ~ Y ,X)rl~(Z)

+dTlk(Y , Z)rIj(X) - drlj(Y , Z)pk(X)

(4.13)

220 G. Hernandez

Substituting (4.13) in (4.12) we get the conditions

(4.14) Vx~i (~ i ,Z ) - h (X ,Z ) = g ~ X , f Y ) , Z ~ ~i

XTx#i(Y,Z)=d~Tt(Y,Z)~Tj(X)-dzlj(Y,Z)~7k(X), Z # {i, r (:~,.

Suppose that xT#i = 0, then, using the second equation of (4.14) and setting X = ~j, we obtain d~Ti(Y,Z) = 0 for all Y # ~i, Z r (i, which is impossible. Hence N cannot be hyper K~.hler. We have proved that

Proposit ion 4.15. There does not exist an isometric f-embedding t : M 4n+3 --~ N 4n+4 with M hyper PS-manifold and N a hyper Kgthler manifold.

It might be possible to embed a PS-manifold in a hyper Kiihler manifold in other ways - for example, the quatemionic analog of the Heisenberg group, which will be studied in detail in the next section, embeds in H ~ as a quadratic submanifold [BaSa] (see also [Bog, pp. 113-114] for this quadratic structure of the Heisenberg group), but the embedding does not preserve the metric f - structures.

We already mentioned that Boyer, Galicki and Mann [BGM2] proved that every 3-Sasakian manifold M embeds (as an f -embedding) in a hyper K~ihler manifold N. For the particular embedding that they construct, they also proved that M is a totally umbilic submanifold of N. We prove that this condition has to be true independently of the embedding used.

L e m m a 4.16. Let M *n+3 be a 3-Sasakian manifold, N 4n+4 a hyper Kiihler manifold and t : M --~ N an isometric f-embedding. Then M is a totally umbilic submanifold of N.

Proof" Blair [Blal] proved that a hypersurface of a K~ihler manifold admit a Sasakian structure if and only if the second fundamental form h satisfies h = -9+c~-r/| a a function. Since this must be true for each contact structure of the 3-Sasakian manifold, and clearly h is independent of the contact structure used, h = - 9 + a - f i x Q~'ts for each s = 1, 2, 3. But then a-7)i(X)l" l i (Y ) = a . r l j (X)17 j (Y ). Set X = ~i, Y = (i to obtain a = 0. Hence M is totally umbilically embedded.

Recall that an almost contact structure is Sasakian if and only if

(4.17) (Vx ~b) Y = 9(X, Y )~ - ~I(Y )X.

Hence if a hypersurface of a hyper K~ihler manifold is totally umbilic, h = - 9 , (4.12) and (4.17) imply that the hypersurface is a 3-Sasakian manifold. Therefore

Theorem 4.18. An oriented hypersurface of a hyper Kiihler manifold has a 3- Sasakian structure if and only if it is a totally umbilic submanifoM with h = -9 .

There is another condition that is trivially satisfied for this kind of embedding, which is worth mentioning since it clarifies the geometry involved in the case of Sasakian structures. Chen [Chen] used the concept of an extrinsic sphere introduced by Nomizu [No] to study the geometry of K~ihler manifolds. An extrinsic sphere of a riemannian manifold is a totally umbilic submanifold that


has parallel mean curvature vector H ~ 0. Here parallel means parallel with respect to the normal connection. It is easily seen that an f -embedding of a 3-Sasakian manifold in a hyper Kiihler manifold has parallel mean curvature vector (which is simply the normal unit vector of the embedding). The converse is trivially true and we can state:

Theorem 4.19. Let M 4n+3 be a 3-Sasaklan manifold, N 4n+4 a hyper KLihler man-

ifold and ~ : M --+ N an f-embedding. Then M is an extrinsic hypersphere o f N. Conversely any extrinsic hyper sphere o f a hyper Kgihler manifold, with h = -g , admits a 3-Sasakian structure.

We conclude that, at least from the extrinsic point of view, every 3-Sasakian manifold looks like a sphere. If they are compact, taking the Cartesian product of a 3-Sasakian manifold with S 1 we construct a large class of examples (large after [BGM2]) of compact non hyper K~hler hypercomplex manifolds that generalize the Hopf surfaces. Using the theorem 1 of Chen [Chen], we immediately get

Corol lary 4.20. A non flat hyper Kdhler manifold as in the previous theorem has no sectional curvature o f constant sign.

In [BGM1], the authors proved that if N is a hyper K~ihler manifold that admits a locally free action of Sp(1) or SO(3) permuting the complex structures on the manifold and such that the obstruction section ~b of the fourth order symmetric product of the spin bundle S4H is constant on N, then each level set of the moment map admits a 3-Sasakian structure. We can use the converse of that result [BGM2, Theorem 3.2] to establish:

Theorem 4.21. A hyper Kgihler manifold N that admits a locally free action o f Sp(1) or SO(3) has a constant obstruction ~ if and only if N admits an extrinsic hypersphere.

Let (M, ~,77,~) and (M r, ~br,r? t, ~r) be two almost contact manifolds. Mori- moto [Mol] defined an almost complex structure on the product M x M r by

J ( X , Y) = (OX - rlt(Y)~, ~ 'Y + rl(X)~')

where ( X , X r) denotes the decomposition of a tangent vector to M • M ' in elements X, Y, tangent to M and M r respectively, and he proved that J is integrable if and only if the almost contact structures are normal. Noticing that the standard spheres are Sasakian manifolds, he reconstructed the complex non K~ihler manifolds of Calabi-Eckmann, S 2n+l x S 2m+1 .

Once again, using the fact that 3-Sasakian manifolds look like spheres (extrinsically), we generalize the Calabi-Eckmann construction by taking products of 3-Sasakian manifolds. We can also see that there is a S z x S 2 worth of complex structures in these products. Observe first that every 3-Sasakian manifold has a two-sphere's worth of almost contact structures, i.e., every structure ( ~ a i O i , ~ a i o i , Y ' ~ a i ~ i ) is almost contact when )--]~a 2 = 1. Equation (4.17) implies that for every a = (ai ,az,a3) C S 2, the almost contact structure ( c~a , rl, , ~ ) = (F, ai r ~ ai rli , ~ ai ~i ) is Sasakian. Consider now the 3-Sasakian

222 G. Hernandez

manifolds M 4~+3 and M ~4~+3 with Sasakian structures (0a, ~a, ~a) and (0~, r/~, (/,), then the almost complex structure defined by

Jab(X, Y) = (OaX ' t - ~b(Y)~o, ~bY + ~a(X)~;))

is integrable for each a and b (since the almost contact structures are normal). We emphasize that there are no complex structures defined in this way that satisfy the quaternionic relations, which is an obvious consequence of the dimension of the manifold. We have then the following result.

Proposition 4,22. Let M and M' be 3-Sasakian manifolds. Then M • M ~ has an

S 2 • S 2 worth o f complex structures.

There is no obvious obstruction for these manifolds to have Kahler structures. However we can prove that for a special family of exampleg such Kahler structures do not exist.

Consider the examples of non-homogeneous 3-Sasakian manifolds constructed in [BGM2], called .5~(p). It is a family of homotopically inequivalent 3-Sasakian manifolds with cohomology groups [BGM2] (dim.5~(p) = 4n - 5):

Z i = 1 , 2 . . . . , 2 n - 3 , 2 n , 2 n + 2 , . . . , 4 n - 6 H'6.fff(p), Z ) = i = 0, 2 . . . . , 2n - 4 , 2n - 1, 2n + 1 , . . . , 4n - 5

Za,_~(p) i = 2n - 2

n

where O ' n _ l ( p ) = ~-'~Pl "Pz'" "Pj "" "P, and p = (P l , . . . Pn ) E Z~. is any n-tuple j = l

of pairwise relatively prime positive integers. Since S 3 is a 3-Sasakian manifold, the previous proposition guarantees the

existence of complex structures on S(p) • S 3. These manifolds are not K~ihler: a simple computation shows that H2n(S~(p) • S 3, Z) = 0. In general, using the Sasakian manifolds S 2k+1 , a similar computation shows that

Theorem 4.23. S(p) x S 2J:+l are generalized Calabi-Eckmann manifolds.

Similar constructions of complex manifold can be made by taking products of hyper PS-manifolds, or products of hyper PS-manifolds with 3-Sasakian. We emphasize, however, that on hyper PS-manifolds there is no a sphere worth of PS-structures.

5. The basic example: Lie groups of type

Let us recall first a general construction of a 2-step nilpotent Lie algebra due to Kaplan [Kal]. Let U and V be two real inner product vector spaces of dimension m and n respectively. The symbol < , > will denote the inner product for both spaces. Suppose there exists a linear map 3 : U ~ End(V) that satisfies

13(a)pl = Ipl laf , p ~ v , a ~ u

( 3 ( a ) ) Z = - [ a ] Z l , a C U.


Polarizing, we get (j(a)p,3(b)p) = (a, b} [pl 2

(5.1) (3(a)p,j(a)q) = la[ 2 (p,q)

for all p , q 6 V, a,b c U. Now consider the space u = U | with its natural inner product and bracket defined as

[a+p ,b+q]=[p ,q ] E U

(5.2) < [p,q],c > = < 3(c)p,q >

for all a, c C U. Then u is called a Lie algebra of type ~.Nr This is a 2-step nilpotent Lie algebra with center U. The connected, simply connected Lie group N with Lie algebra u is called a Lie group of type ,-~i~ or a generalized Heisenberg group. N can be identified with its a Lie algebra through the exponential map; the multiplication on N being defined as

(p,a) . (q ,b)= (p + q,a + b + [p,q])

We recover the Heisenberg group as special case when U is the space of imaginary numbers and V is C ". We define now the quaternionic analog as the group with Lie algebra of type ~ with U the space of imaginary quaternions and V = H". The following result due to Kaplan [Ka3] says that these two examples share an important property.

Proposition 5.3. The homogeneous manifold (N, <, >) is naturally reductive if and only if N is the Heisenberg group or a quaternionic analog.

The Heisenberg group is a Sasakian manifold, as it is easily verified, and has been used to model (locally) contact manifolds [Goz]. Since, as we have observed, the PS-structure generalizes in a certain sense the contact structure and the quaternionic analog has a natural hyper PS-structure which can be used also to model these kinds of manifolds, we will study this group in some detail in this section.

Now consider h = H n G Im H with its natural inner product and define a Lie bracket on/z by (5.2).

If we choose the orthonormal basis {1~, it, jr, kl }, l = 1 , 2 , . . . , n for H ~, and el, ej, ek, the standard basis for Im H then

[lt, it] = [jr, kl] = ei

[ l t , j l ] = [kl, it] = ey

[11, kt] = [i/,jl] = ek

and the rest of the brackets are zero. We define now the connected, simply connected Lie group H with Lie algebra

ft which can be realized (cf. [BaSa]) as the set of points in H n | H with product defined by

224 G, Hernandez

(Pl, .-.Pn, a ) . (ql , ---, q , , b) = (Pl + ql, ..., Pn + qn, a + b + Z Im(q,O,)) r"

for Pr,qr E H; a ,b E ImH. If we introduce the coordinates (Xr ~ + xl i + x~j + xr3k) = pr E H, a = a l e i + a2ej + a3ek , then the left-invariant vector fields

_ 0 U, - ~-~g,

0 x ~ o x o

0 - - - - x , ' U , - x ,~u2 - xr~U3, X,' = ~ + x ~ + x ~ u ~ - x ,~w~

0 + x~ ul x~ u2 + x ~ o ~ x~ +X'r U~, X~ x :, - Ox~ xr u, + = -

and the left-invariant l-forms

] 0 2 3 C~l =clal + X r ~ r -xOrdx 1, 3 2 - - X r d x r ) + x r dxr

(5.4) 2 0 3 1 0 2 1 3 o<2 = da2 + x r d.)c r - x r dr. r , -- x r d r r Xr dx r +

3 0 2 1 1 2 0 3 _ _ X r d x r o~3 = da3 + x r dx r + x r d x r X r d x r

satisfy

~ i ( U j ) = ~ , o4(Xrk)=O, ~ , j = l , 2, 3; k=O, 1, 2, 3; r = l , . . . , n .

Moreover, the tensors j5 induced on H by left translation from 3(ei) and the metric 9 defined by gx = L~ ( . , �9 ), define a hyper f-structure with associated metric g. In this particular example ./ /g = s p a n { U l , U 2 , U 3 } and

= s p a n { X ~ r = 1, ...,n}. Now g defines a connection that satisfies

9 ( V x Y , Z ) = �89 Y ] , Z ) + g ( [ Z , X ] , Y ) - 9 ( [ Y , Z I , X ) )

for all X, Y, Z in the Lie algebra ,~. Using (5.2) we obtain [Ka2]

Vpq = l i p , q ]

V~p = Vpa = - �89

V a b = 0

where a, b E ImH, p , q E H n, and so, the sectional curvature function is given by [Ka2]:

(5.5) K(p,q)=-31[p,q]l 2, g(p,a)=�88 2, g(a,b)=O. Since the globally framed hyper f-structure is left-invariant, the local geome-

try of H passes to the quotient H / H z , where Hz is the subgroup of H consisting of elements with integer coordinates. H / H z is a compact hyperf-manifold and, as an easy calculation shows, each f-structure satisfies (2.12). Therefore it fibers over a hyper-complex manifold M with fiber T 3 (in our case we already knew that [Ui , Uj] = 0). Also, (5.2) and (5.4) show that the two-forms


wi(a +p , b + q) = 9(](ei)p, q) = 9([P, q], el)

for all a , b E I m H , and all p , q E H ~, are given by 02i ---- 2dOLl, and so the manifold H / H z has a hyper PS-structure. Hence M is hyper-K~ihler. In addition, the R3-valued connection 1-form c~ = (cq, c~2, a3) has curvature w = (o21, ca2, w3), which clearly defines a fat fibration, We see that , / /g coincides with the vertical distribution of the fibration and ~ with the horizontal distribution for the defined connection.

Using the fundamental equations for a riemannian submersion, we observe that the fbers are totally geodesically embedded in the total space H / H z . (The

fatness of this bundle is again easily seen using the last observation and the fact that K ( p , a ) = �88 for p and a unitary vectors.) We will also use the fact that Aap = 0 and that Apq = �89 q].

If K denotes the sectional curvature function of the metric 9, k the sectional curvature of the induced metric on the fibers ~, and K the corresponding to the metric ~ defined on the base manifold, we obtain for riemannian submersions [O'Ne]: (assuming lal = Ibl = Ipl = lq[-- 1 and [a Ab] = [p Aql = 1)

K ( a , b ) = K ( a , b ) = 0

K ( p , a ) = IApal 2 1 1 = ~ Im(a)pl 2 =

/~(p, q) = K(p, q) + 3 IApql 2 = O.

Then both the fiber and the base manifold are fiat manifolds. From the homotopy exact sequence of the fibration we conclude that the base manifold has the homotopy type of a torus T 4n and we simply apply Bieberbach classification for fiat manifolds to conclude that the base manifold is in fact a torus. Another way to see this is noticing that the hyper K/J.hler metric is a homogeneous Ricci-flat metric on this manifold, so by a result due to Aleksevskii and Kimeldfeld ([Bes], p. 191) the metric is fiat. We have then an example of a fat fibration of a torus bundle over a torus.

Remark 5.6. Recall that the Ricci tensor computed by [Ka2] for these manifolds is [3 1 --~14n 0

r = 0 n13 "

The fundamental group of H / H z , is isomorphic to Hz. We use the following i ' k notation. Let p ) , p r , ~ , p r denote the real and imaginary quaternion unit vectors

in the r-th component of an element of Hz, i. e., p) = (1 + Oi + Oj + Ok)r, p~ = (0 + i + Oj + Ok),, ~ = (0 + Oi + j + Ok)r, pr k = (0 + Oi + Oj + k)r, then

1 i - - i . - l - - i ) - I i k , i - - l - - k ) - I s PrPs~Pr) ~Ps = lYrPs ~P~r) [Ps = 6rei

l i ~ i . - I r _ i ) - I k i - - k ) - l - - i ) - I ~ e j Prl~s~Pr ) ~Ps = PrPs~Pr [Ps =

1 i - - k . - l ~ k ) - I i i - - i ) - l , i ) -1 ~ e k Pr Ps ~Pr ) ~Ps = Pr[Ys ~ r ~ s =

226 G. Hernandez

and the rest of the commutators are zero, the unit element of the group. Then HI(H/Hz) ~ Hz/[Hz, Hz] is isomorphic to the free Abelian group of 4n gen- erators, Hz/[Hz,Hz] ~ ~]~4n z , hence bl(H /Hz )= 4n.

Let us now define the compact manifold M 4"+4 = H / H z • S i. This manifold cannot be a hyper K~ihler manifold since bl (M 4n+4) = 4n + 1. On the other hand, since the tensors f~ satisfy (2.12), the associated almost contact 3-structure is normal (2.15). But the normality of ~bi implies that the almost complex structures Ji on M 4n+4 defined by J i ( X , h d / d t ) = (~iX - h U i , a i ( X ) d / d t ) are integrable, where X is tangent to H / H z and h is any real function. Hence we have proved the next proposition.

Proposit ion 5.7. The manifolds M 4n§ = H /Hz • S 1 are hyper complex but not hyper Kgihler.

As noticed by Baker and Salamon [BaSa], the quotient of h by the ideal generated by Uz, U3 is the Heisenberg algebra of dimension 4n + 1. Thus we have the following diagram off-structures

M 4n+3

(5.8) T 3 M 4n+l

M 4~

where M 4n+3 is a hyper PS-structure, M 4"+j is a contact-normal (Sasakian) structure and M 4n is a hyper Kahler manifold.

In the case of a general regular hyper PS-structure (~i = d77i) a diagram like this always exists:

Theorem 5.9. Let M 4n+3 be a compact, connected manifold with regular hyper PS-structure and such that each ~.~ is a regular vector field. Then there exist a compact regular Sasakian manifold M 4~+j and a compact hyper Kgihler manifold M 4n such that the diagram (4.8) commutes.

Proof" Choose one almost contact structure, say El, 77l, ~l. Obviously, M 4n+3 fibers over a principal circle bundle M 4n+j over M an. Since ~'~r 77,~ = 0, [~r, ~s] = 0, _5~,fs = 0 for all r and s, in particular it is true for s = 1. Then ~l, 01, 4~1 are projectable by the projection 7r : M 4~§ ~ M 4n§ and we can define the tensor fields 6, (, 7/on M 4n+l by

where ~" denotes the horizontal lift with respect to the connection (772,773). An easy computation shows that these tensors define a normal almost contact structure.


Now we define a metric ~ on M a"+l by 9ffrX, frY) = ~(X, Y). This is clearly compatible with the almost contact structure. Moreover, defining w(X, Y) = ~I(X, OY), we see that

a2(X, Y) = ~(X, 4)Y) = 9(~rX, ~rTr.f~rY) = w[(SX, "TRY)

and oJl = d~h. So ~ = dr /and the manifold M 4"+1 is a contact manifold. []

Since the quaternionic analog Lie group has a natural hyper PS-structure and it is nilpotent, there is a set of linearly independent vector fields that span the whole tangent bundle, we could use the techniques of subriemannian geometry [Str] to study some properties of this group. This has been done by Kaplan [Kap] (see also [Str], especially Sect. 8). This would justify the name "hyper subSasakian" or "hyper Sasakian" for this kind of structure, since the Heisenberg group has the same generating property and is Sasakian.

Consider now S = H �9 R § the semidirect product of H and R +, R + acting as dialations

a . (x, b) = (ax,a2b).

Identifying S with U x V • R § with product

(p, a, t ) . (q, b, t ') = (p + tq, a + t2b + �89 q], tt')

S has a lie algebra ~ = h | ? with bracket

[p + a + th ,q + b + trh] = tq - t 'p + 2tb - 2t' a - [p ,q]

h E ? such that [h,p] =p. In g we choose the inner product

< p +a + th ,q +b + J h > = < p , q > + < a ,b > +4tt t

which induces a left invariant metric on S. Inducing almost complex structures on S as we have previously done using the almost contact structures defined on H, it follows that S is a hyper complex manifold. Damek [Da] has proved that such manifolds have strictly negative sectional curvature. We have then examples of hyper complex manifolds with strictly negative curvature. If we use the Heisenberg group instead of its quaternionic analog, we have complex manifolds with negative sectional curvature. In particular we have proved, noticing that we can identify H with H x 1 through the embedding H ~ (H, 1):

Proposition 5.10. The Heisenberg group (the quaternionic analog) embeds in a (hyper) complex manifold with strictly negative sectional curvature.

228 G. Hernandez

Addendum

5.1. Hypercomplex not hyper Kiihler and complex, symplectic non Kdihler manifolds

In the last section, we have found a family of hypercomplex but not byper K/ihler manifolds. On the other hand, Thurston [Th] provided the first example of a compact symplectic manifold that is not K/ihler. It is a T 2 bundle over T 2. This is a fat bundle [We]. This example was generalized by Cordero, Fernandez and de Leon [CoFeLe]. We can easily construct a family of manifolds with hyper complex but not hyper K~hler structures different from the one given, and, in a particular case where our family of examples coincides with the one studied in [CoFeLe], an example of a complex, symplectic not K~ihler structure.

Consider the group G of real matrices of the form

L• Q 0 1

where P is a (3 x r) matrix, r is a (3 x 1) matrix and Q is a (r x 1) matrix. This is a connected and simply-connected two-step nilpotent Lie group (as it is our previous example), of dimension 4n + 3. The Lie algebra of this group is generated by the elements

/i~ /i ai]= 0 , b j = 0 I , c i = 0 0 , 0 0 0 0

where lijl denotes the (3 x r) matrix with 1 in the i - j t h entry and zero every where else. The same convention (with the obvious modification) defines Ii[ and [/' I- The Lie bracket is defined by

[aij , bj ] = ci

and the other brackets are zero. Once again, we construct the compact manifold M 4m+3 = G / G z , where Gz is the set of matrices in G with integer entries. The fundamental group is isomorphic to Gz and, again it is easy to see that bl(M 4'~+3) = 4n and so M 4(n+l) = M 4n+3 x S 1 has first Betti number 4n + 1, which prevents M 4n§ from having a K~ihler structure.

Now we introduce coordinates for A E G:

xj(A) = Qj, yij(A) = Pij, zi(A) = Ti

for i = 1,2,3, j = 1 , 2 , - - - , r . Then the following l-forms are left-invariant (and so they descend to the quotient M 4n+3)

O~j = dxj , ~ij = dyi j , ")/i = dz i - E Yi jdxj

J

as well as the vector fields


x / = OlOxs + ~-~yoOlOz~, i

the tensor fields of type (1,1)

ft = Z cej | Ylj

and the metric

9

Then ~ . , "7:, Zs, 9) defines a

Yo = 0/03,0, Z~ = O/Oz~

-- [Jlj ~ Xj + ~2j (~ Y3j - ~33j (~ Y2j

- ~2j (~ Xj + ~3j | YIj - 3lj (~ Y3j

- 33j | x~ + 3,s | Y2s - 32j | YIj

~) +3o

metric hyperf-structure. We note that d% (X, Y) g(X,fs Y); however, the almost contact 3-structure is normal and hence the compact manifolds M 4n§ = G/Gz x S 1 have hypercomplex structures; they are

obviously non hyper Kfihler. In the special case r = 1 we can construct a symplectic structure on M 8 = G/G• S I. We define the symplectic form as

F = c ~ A 6 + Z ~ i A yl

w h e r e a = c~i, 3i = 3il and t5 = dr, t is the c o o r d i n a t e fo r S 1 . W e h a v e t h e n

p r o v e d

P r o p o s i t i o n The manifolds M 4n+4 = G / G z • S 1 have hypercomplex structures but do not admit hyper K~ihler structures. In the case r = l, the manifold M 8 = G / Gz • S 1 is hyper complex and symplectic but not Kahler.

I want to thank Charles P. Boyer, my dissertation advisor, for his acute comments and guidelines.

I also thank K. Galicki for helpful discussions about 3-Sasakian and quateruionic Kahler manifolds.

The referee made so many important observations that he dangerously came close to being a co-

author. I gratefully incorporated his suggestions. This research was partially supported by CONACYT,

Mexico.

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on hyper f -structures

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