Critical Riemannian metrics on cosympletic manifolds

J. Korean Math. Soc. 32 (1995), No. 3, pp. 553-562

CRITICAL RIEMANNIAN METRICS ON COSYMPLECTIC MANIFOLDS

BYUNG HAK KIM

1. Introd uction

In a Recent paper [3], D. Chinea, M. Delon and J. C. Marrero proved that a cosymplectic manifold is formal and constructed an example of compact cosymplectic manifold which is not a global product of a Kaehler manifold with the circle. In this paper we study the compact cosymplectic manifolds with critical Riemannian metrics.

Let M be an m-dimensional compact orientable Riemannian man- ifold and ^J..Io( M) be the set of Coo Riemannian metric G on M satis- fying fM dVa = 1, where dVa is the volume element of M measured by G. For an element G in J..Io(M), we assume that f(k) is a scalar field on M determined by G as the contraction of a tensor product of the curvature tensor. Then HM[G] = fM f(k)dVa defines a mapping H M : J..Io( M) -+ R. In this case, a critical point of H M is called a critical Riemannian metric with respect to the field f( k) and denoted by G ^H.

The following four kinds of critical Riemannian metrics have been studied by M. Berger [1] and Y. Muto [7] :

AM[G] = fM ^{K dVa,}

CM[G] = fM ^{S2 dVa,}

BM[G] = fM ^K 2 _dVa,

DM[G] = fM R ² dVa,

Received July 5, 1994.

1991 AMS Subject Classification: 53C20.

Key words: Critical Riemannian metrics, fibred Riemannian space, cosymplectic manifold.

This research was partially supported by TGRC-KOSEF and the Basic Science

Research Institute Programs, Ministry of Education, 1994.

where R, S and K are the Riemannian curvature tensor, Ricci curva- ture tensor and scalar curvature respectively. The equations of the critical Riemannian metrics are written as follows:

(1.1) Aji =CAGji,

Cji =CCGji,

Bji = CBGji, Dji = CDGji,

where CA, CB, Cc, CD are undetermined constants and Aji,Bji,Cji and D ji are given by

(1.2) (1.3)

(1.4)

(1.5)

Aji = ^-Sji + '2K Gji, 1

Bji = 2\ljViK - 2(b.K)Gji - 2K Sji + ~K2 ^Gji,

" 1

Cji =\lj\liK - \l"\l Sji - i(b.K)Gji - 2RjkhiS kh + %SkhSkhGji,

Dji =2VjViK - 4V"V ^k Sji + ^{4SjkSi k}

4R S kh 2R R HI 1 R RkhlmG

- jkhi - j/ohl i +"2 ^{khlm ji,}

where V is the Riemannian connection and b.K is the Laplacian of K with respect to G on M.

2. Cosymplectic Bochner curvature _t~nsor

Let M be an m-dimensional cosymplectic manifold with structure

(q,,~,.'1, G), that is, a manifold M which admits a I-form "l, a vector fields ~, a metric tensor G satisfying

"l(e) = 1, q,2 X = -X + "l(X)e, q,e = 0,

G(e,X) = "l(X), G( q,X, q,Y) = G(X, Y) - "l(X)"l(Y)

Critical Riemannian metrics on cosymplectic manifolds 555

and ~(X, Y) = G(cPX, Y) and TJ are both closed for arbitrary vector fields X and Y on .M. The relations

'VeK = 0 and R(X, _Y)~ = 0 are easily verified.

The cosymplectic Bochner curvature B is defined by B(X,Y,Z,U)

=R(X, Y, Z, U) + {G(X, U) - TJ(X)TJ(U)}L(Y, Z)

- {G(Y, U) -1J(Y)1J(U)}L(X, Z) + {G(Y, Z) - TJ(Y)1J(Z)}L(X, U) - {G(X, Z) - 1J(X)1J( Z)}L(Y, U) + q>(X, U)M(Y, Z)

- ~(Y, U)M(X, Z) + q>(Y, Z)M(X, U) - q>(X, Z)M(Y, U) - 2{q>(Z, U)M(X, Y) + q>(X, Y)M(Z, U)},

where we have put

L(X,Y) = (1 )[S(X,Y)+o:{G(X,Y)-TJ(X)7J(Y))], 2 m+2

M(X, Y) = -L(X, cPY) and

o:=-~-~

K 4(m + 1)

We assume that the cosymplectic Bochner curvature B vanishes iden- tically on M, then we get

(2.1)

R(X,Y,Z,U)

=~3 [S(Y, Z){G(X, U) - 1J(X)1J(U)}

m+

-SeX, Z){G(Y, U) - TJ(Y)1J(U)}

+S(X, U){G(Y, Z) - 1J(Y)7J(Z)} - S(Y, UH G(X, Z) - TJ(X)1J(Z)}]

- (m + 1)(m K + 3) [{G(Y, Z) - TJ(Y)TJ(Z)}{G(X, U) - TJ(X)7J(U)}

- {G(X,Z) - TJ(X)77(Z)}{G(Y,U) - TJ(Y)1J(U)}]

- q>(X, U)M(Y, Z) + q>(Y, U)M(X, Z) - q>(Y, Z)M(X, U)

+ q>(X, Z)M(Y, U) + 2{ q>(Z, U)M(X, Y) + q>(X, Y)M(Z, U)}.

LEMMA 2.1. litbe cosymplectic Bocbner curvature B oftbe cosym- plectic manifold M vanisbes identica.11y and M is Einstein, tben M is locally Euclidean.

Proof. From (2.1), we easily get

SeX, Y) = {-!..G(X, Y) - (4 ) T/(X)q(Y)}K.

m mm+3

This equation implies K = ^{0 and that S} = o. Then we get R = 0 by use of (2.1).

If the scalar curvature vanishes on M, then, from (2.1), we have (2.2)

R(X,Y,Z,U)

=~3 [S(Y; Z){G(X, U) - q(X)T/(Un

m+

-sex, Z){G(Y, U) - T/(Y)q(U)}

+S(X, U){G(Y, Z) - q(Y)q( Z)} - S(Y, U){ G( X, Z) - q( X)7](Z)} 1

- «p(X, U)M(Y, Z) + CP(Y, U)M(X, Z) - «p(Y, Z)M(X, U) + cP(X,Z)M(Y,U) + 2{cP(Z,U)M(X,Y) + cP(X,Y)M(Z,U)}.

Thus we can state

LEMMA 2.2. li tbe cosymplectic Bocbner curvature Band tbe scalar curvature vanisb on tbe cosymplectic manifold M, tben tbe curvature tensor on M is of tbe form (2.2).

3. Cosympleetic manifold with critical Riemannian metrics Let M be an m-dimensional cosymplectic manifold. If the Riemann- ian metric G is a critical Riemannian metric GB, Gc or G D, then, by use of (1.1)-(1.5), the undetermined constants CB, Cc and CD given by (1.1) are determined as follows :

1 2

(3.1) CB ="2K - 2~K,

1 .. 1

(3.2)· Cc =-S·-B 2 JZ ^Jz - -~K 2 '

1 kOOh

(3.3) CD ^{=-RkjihR JZ •}

2

Critical Riemannian metrics on cosymplectic manifolds 557

From (1.1), (1.3) and (3.1), we get

VxVyI( = K S(X,Y).

Then

fM ^K 2 _{dVGB =} fM _~K ^dVGB'

Thus, applying the Green's Theorem, we have

PROPOSITION 3.1. In a compact cosymplectic manifold M, G is a critical Riemannian metric GB on M if and only if tbe scalar curvature vanishes.

By use of (1.4) and (3.2), we have

PROPOSITION 3.2. In a cosymplectic manifold M, G is a critical Riemannian metric Gc on M ifand only iftbe Ricci curvature vanishes.

If the Riemannian metric G on M is the critical Riemannian metric G D, then we obtain

by use of (1.5) and (3.3). Thus we have

PROPOSITION 3.3. In a compact cosymplectic manifold M, G is a critical Riemannian metric G D on M if and only if M is locally Eu- clidean.

From (2.1), we easily see that the cosymplectic manifold with van- ishing cosymplectic Bochner curvature is locally Euclidean if the Ricci curvature vanishes. Moreover, it is well known that if G is a critical Riemannian metric GA, then G is an Einstein metric. From these facts and Lemma 2.1, Propositions 3.2 and 3.3, we have

PROPOSITION 3.4. The compact cosymplectic manifold with van-

islllng cosymplectic Bocbner curvature is locally Euclidean if and only

ifC is a critical Riemannian metrics CA or Gc or CD.

4. Critical Riemannian metrics in the fibred Riemannian space

Let {M,B,G,7r} be a fibred Riemannian space, that is, {M,G}

is an m-dimensional total space with projectable metric G,B an n- dimensional base space and 7r : M -+ B the projection with maximal rank n. The fibre passing through a point q E M becomes an p(= m - n )-dimensional submanifold of M, which is denoted by F q or generally F. Throughout this section, the ranges of the indices are as follows :

h,i,j,k,l = 1,2,··· ,m,

a,b,c,d,e = 1,2,··· ,n,

a,f3,~,6,€=n+1,··· ,n+p=m, unless stated otherwise.

Let {EO!, CP} be dual to the frame {Eb, COl} of M and denoting R _kii ^h components of the curvature in M, we have the structure equation as follows [2, 4, 5, 8] :

(4.3) RhP ⁼ RhP + _h6/h..,

h _r /h ₆ 0!e,

where R _de6a and R hP are the components of the curvature in B and F respectively, h = (hp:) and L = (LcbO!) are the components of the second fundamental tenSor and normal connection of each fibre F respectively, arid we have put

,

Rdcba =G(R(Ed' Ec)Eb, Ea),

R d..,';, =G( R( E d ; C.., )E b , COl), R6 ..,p =G(R(C6, C..,)CP, COl),

*V dhl\ =8dhlb - rdbhle + QdEO!hl b - QdphEO!b'

**V..,Lcb =8dhcbO! - rdcLebO! - r:bLceO! + _QdEO!Lcb

Qcl =PepO! - hPO!e,

Critical Riemannian metrics on cosymplectic manifolds 559

Pel' are local function related to Le", GP = P _d ! ^{E d and P3e is the}

Christoffel symbol induced by the metric in B.

Denoting by Sji components of the Ricci tensor in M, then we have [2,4,5, 8]

(4.4)

Seb = S(Ee,Eb) =Seb - 2Le/L b e

E -

h/eh/b + ~(*Veh/b ^{+ *Vbh/J,}

(4.5)

S",b = S(C"" Eb) =**Veh/b + _**Vthb\ + ·VeLb\ - 2h",E eLbeEl (4.6) S"ffJ =S(G""Cp) = S-yp - h-yph/e + *Veh"'fp - L a e

"'fLe a fJ , where Seb and S"'ffJ are the components of the Ricci tensor in Band F respectively and we have put

and f'pQ is the Christoffel symbol induced by the metric in F.

Let K, KB and KF be the scalar curvatures of M, B and each fibre F respectively, we have [2, 4, 5, 8]

(4.7) K = KB + ^{KF -} LebQUbQ - h"ffJeh"'ffJe - h-y\h/e + 2 ·VehE~' The present author [5] proved that

THEOREM 4.1. The almost contact metric structure (tP,~,7J,G) on M is cosymplectic, then

(1) B is Kaehlerian with a complex structure J,

(2) F is cosymplectic with a cosymplectic structure _(~, (, ij, g), (3) L = 0,

(4) each fibre is minimal in M.

It is well known that [4]

LEMMA 4.2. If the st11lcture tensor h and L vanish identically on M, then M is locally a Riemannian product space of the base space and a fibre.

Assume that the cosymplectic Bochner curvature vanishes on the

compact cosymplectic manifold M and the metric G on M is a critical

Riemannian metric G A or Gc or G D, then, from the Proposition 3. 4, M is locally Euclidean. So.that the Theorem 4. 1 and (4.1)-(4.3) imply Rdcba = ^{0, h} = ^{0 and.} R6-y; = O. Taking account of the Proposition 3.4 and Lemma 4. 2, we can state

THEOREM 4.3. Let the :fi.bred Riemannian space M be a compact cosymplectic manifold with vanishing cosymplectic Bochner curvature.

If the metric on M is a critical Riemannian metric GA or Gc or G D,

then M is locally the product of the two locally Euc1idean spaces.

Next, if the cosymplectic Bochner curvature vanishes on M and the scalar curvature vanishes, then the curvature of M is dete~d as (2.2). By use of the Theorem 4. 1 and (4.1). we get

(4.8)

(~ +3.)Rdcl t ^{=( Scb -} hPOte~P_0t~)§4~ (~Ilb ^- hpOtd.hP~)t5~

+ (Sl- hpOtdh'jOta)9cb - (Sea - hPOtehPOta)9db - (Sce - hPOtehP~)Jb ^e Jd a + ^{(Sde -} hPOtehP~)Jb ^e Jea - (Sde - hpOtdhP~)JaeJeb + (Sce - hPOtehP~)reJdb + 2(Sde - hpOtdhP~)J/)ba + 2(Sbe - hpOtbhP~).r·eJde, where geb is the metric components of B. From (4.8), we obtain

2Scb = - (m + l)hPOtehP~ - hPOlehPOle9cb (4.9) +3(Sde - hpOtdhP~)Jb ^e Jed + ^KBgeb

and that (4.10)

Moreover, by use of (4. 7), we get (4.11)

On the other hand, the Theorem 4.1 and (4. 2) imply

(4.12) (n + 4)hpabhfJOlb= -(p -l)KB - nKp.

(4.14)

Critical Riemannian metrics on cosymplectic manifolds 561

Taking "account of (4. 11) and (4. 12), we obtain

(4.13) }' 2(n + 2) h hftQb

\.B =1 -p + n ftOlb , K - -(m+3)h hftQb

F -1 _-p + _n PQb when p ::J n + 1.

If we substitute (4.13) into (4.10), then p(n + 3)hfta/lhPQb = 0 so that h = 0 and hence KB = 0 and KF = O.

In the case of p = n + 1, (4.12) is reformed to (4.15) (n +4)hpQbhPQb = -n(KB + KF).

The equations (4.11) and (4.15) give rise to 2(n + 2)hpQbhPQb = 0, that is h = O. Then (4.10) and (4.11) imply KB = 0 and KF = O.

Thus we have

LEMMA 4.4. Let M be the fibred Riemannian space with cosym- plectic structure. If tbe cosymplectic Bocbner curvature and tbe scalar curvature on M vanish identically, then M is locally tbe product of B and F, and the scalar curvatures of B and F vanish identically.

Considering the Theorem 4.1, Lemma 4.4 and (4.3), it is reduced to (m + 3)R 6oy;

=Soyp(6 6 - ^{fi6(Q) - S6P(6~ - fioy(Q} + S6 Q

(9OYP - fioyfip)

(4. 16) _ _ ,\ _ _ _ ,\ _ -

- 5oyQ(96P - fi6fip) - Soy <P13,\cP6

+ 56 <PP'"(cPoyQ

-

-

-

-

- 56,\<p

<poyp + S'"1>.<p

"cP6P + 2S6,\<P'"( <pl + 25p,\<p

<P6oy, where 9 is the induced metric on F. From (4. 16), we easily see that S'"(13 = 0 and that R6oy; = O.

Furthermore, if we consider (4.9) and Theorem 4.1, then Seb = 0 and that Rdet = 0 by use of (4.8). Thus taking account of the Theorem 4.3 and Lemma 4.4, we have

THEOREM 4.5. Let M be the fibred Riemannian space with cosym- plectic structure. If the cosymplectic Bochner curvature and the scalar curvature on M vanish, then M is locally product of the two locally Euclidean spaces.

The following fact can be directly reduced from Proposition 3. 1.

COROLLARY 4.6. H M is the compact cosymplectic manifold with vanishing cosymplectic Bochner curvature and the metric G on M is the critical Riemanman metric GB, then M is locally product of the two locally Euclidean spaces.

Finally, combining Theorem 4.3 and Corollary 4.6, we have

THEOREM 4.7. H M is the compact cosymplectic manifold with vanishing cosymplectic Bochner curvature and the metric G on M is one of the critical Riemanman metrics GA,GB,Gc and GD, then M is locally product of the two locally Euclidean spaces.

ReferenceS

1. M. Berger, Quelques formulas de variation pour une structure Riemannian, Ann. ScL Ecole Norm Sup. 4e series 3 (1970), 285-294.

2. A. Besse, Einstein manifolds, Springer-Verlag, Berlin, 1987.

3. D. Chinea, M. Deleon and J. C. Marrero, Topology of cosymplectic manifolds, J. Math. Pures Appl. 12 (1993),567-591.

4. S. Ishihara and M. Konishi, Differential Geometry of fibred spaces, Publ. Study Group of Diiferential Geometry 1; Tokyo (1973).

5. B. H. Kim, Fibre4 Riemannian spaces with quasi Sasakian structure, Hiroshima Math. J. 20 (1990), 477-513.

6. J. -H. Kwon, B. H. Kim and N.-G. Kim, Flbred Riemannian spaces with. critical Riemannian metrics, J. Korean Math. Soc. 31 (1994),205-211.

7. Y. Muto, Riemannian submersions anrl critical Riemannian metrics, J. Math.

Soc. Japan 29 (1977), 493-511.

8. Y. Tashiro and B. H. Kim, Almost complex and almost contact structures in fibred Riemannian spaces, Hiroshima Math. J. 18 (1988), 161-188.

Department of Mathematics

Kyung Hee University

Suwon 449-701, Korea