ON PSEUDOSYMMETRIC PARA-K ¨ AHLER MANIFOLDS

VOL. 74 1997 NO. 2

ON PSEUDOSYMMETRIC PARA-K ¨ AHLER MANIFOLDS

BY

FILIP D E F E V E R (LEUVEN), RYSZARD D E S Z C Z (WROC LAW)

AND

LEOPOLD V E R S T R A E L E N (LEUVEN)

1. Introduction. Let (M, g, J ) be a connected, n = 2m-dimensional, m ≥ 2, semi-Riemannian manifold of class C ^∞ with a not necessarily definite metric g and an almost complex structure J such that

(1) g(J X, J Y ) = g(X, Y ), X, Y ∈ Ξ(M ),

where Ξ(M ) is the Lie algebra of C ^∞ vector fields on M . A manifold (M, g, J ) satisfying (1) is called almost Hermitian. The almost Hermitian manifold (M, g, J ) is said to be a para-K¨ ahler manifold ([12], [13], see also [10], p. 69) if its Riemann–Christoffel curvature tensor R satisfies the K¨ ahler identity

(2) R(X, Y, Z, W ) = R(X, Y, J Z, J W ), X, Y, Z, W ∈ Ξ(M ).

Evidently, every K¨ ahler manifold satisfies (2). The converse statement is not true; see e.g. [13] or [17].

In this paper we consider para-K¨ ahler manifolds which satisfy curvature conditions of pseudosymmetric type. In Section 2 we give precise defini- tions. Pseudosymmetric manifolds constitute a generalization of spaces of constant (sectional) curvature, along the line of locally symmetric (∇R = 0) and semisymmetric (R · R = 0, cf. [14]) spaces, consecutively. Profound in- vestigation of several properties of semisymmetric manifolds gave rise to their next generalization: the pseudosymmetric manifolds. Both the study of the intrinsic aspect and the study of the extrinsic aspect led to this con- cept. We have e.g. the following two theorems. Every manifold M which can be mapped geodesically onto a semisymmetric manifold is pseudosym- metric. Every totally umbilical submanifold, with parallel mean curvature

1991 Mathematics Subject Classification: 53B20, 53C25.

The first author is a postdoctoral researcher of N.F.W.O., Belgium.

Research of the second author supported by a research-grant of the Research Council of the Katholieke Universiteit Leuven.

Research of the third author supported by the grant OT/TBA/95/9 of the Research Council of the Katholieke Universiteit Leuven.

[253]

vector field, of a semisymmetric manifold is pseudosymmetric. This concept of pseudosymmetry in the proper sense belongs to a larger class of curva- ture conditions of pseudosymmetric type. For more detailed information on the geometric motivation for the introduction of pseudosymmetry, and for a review of results on different aspects of pseudosymmetric spaces, see e.g.

[6] and [16]. We just mention here the following application. Curvature conditions of pseudosymmetric type often appear in the theory of general relativity, which is rather surprising in view of the purely geometrical origin of the concept. For more information on this aspect see e.g. [3].

2. Preliminaries. Let (M, g) be an n-dimensional, n ≥ 3, semi- Riemannian connected manifold of class C ^∞ . We denote by ∇, S and κ the Levi-Civita connection, the Ricci tensor and the scalar curvature of (M, g), respectively. We define on M the endomorphisms e R(X, Y ), X ∧ Y and e C(X, Y ) by

R(X, Y )Z = [∇ e X , ∇ Y ]Z − ∇ [X,Y ] Z, (X ∧ Y )Z = g(Y, Z)X − g(X, Z)Y, C(X, Y ) = e e R(X, Y ) + 1

n − 2

 κ

n − 1 X ∧ Y − (X ∧ e SY + e SX ∧ Y )

 , respectively, where X, Y, Z ∈ Ξ(M ), Ξ(M ) being the Lie algebra of vector fields on M , and the Ricci operator e S of (M, g) is defined by S(X, Y ) = g(X, e SY ). The (0, 4)-tensor G is defined by

G(X 1 , . . . , X 4 ) = g((X 1 ∧ X ₂ )X 3 , X 4 ).

The Riemann curvature tensor R and the Weyl curvature tensor C of (M, g) are defined by

R(X 1 , X 2 , X 3 , X 4 ) = g( e R(X 1 , X 2 )X 3 , X 4 ), C(X 1 , X 2 , X 3 , X 4 ) = g( e C(X 1 , X 2 )X 3 , X 4 ).

Further, for a symmetric (0, 2)-tensor field A on M , we define the endomor- phism X ∧ A Y of Ξ(M ) by (X ∧ A Y )Z = A(Z, Y )X − A(Z, X)Y , where X, Y, Z ∈ Ξ(M ). Evidently, we have X ∧ g Y = X ∧ Y . For a (0, k)-tensor field T on M , k ≥ 1, and a symmetric (0, 2)-tensor field A on M , we define the (0, k + 2)-tensor fields R · T and Q(A, T ) by

(R · T )(X 1 , . . . , X k ; X, Y ) = − T ( e R(X, Y )X ₁ , X 2 , . . . , X k )

− . . . − T (X 1 , . . . , X k−1 , e R(X, Y )X k ), Q(A, T )(X 1 , . . . , X k ; X, Y ) = − T ((X ∧ A Y )X 1 , X 2 , . . . , X k )

− . . . − T (X ₁ , . . . , X k−1 , (X ∧ A Y )X k ).

Curvature conditions involving tensors of the form R · T and Q(A, T ) are

called curvature conditions of pseudosymmetric type. E.g. manifolds satis-

fying the condition

(3) R · R = L R Q(g, R)

on the set U R = {x ∈ M | R − _n(n−1) ^κ G 6= 0 at x} are called pseudosym- metric; in particular, for L R = 0 they contain the semisymmetric spaces (R · R = 0). Manifolds satisfying the condition

(4) R · S = L S Q(g, S)

on the set U S = {x ∈ M | S − ^κ _n g 6= 0 at x} are called Ricci-pseudo- symmetric; in particular, for L S = 0 they contain the Ricci-semisymmetric spaces (R · S = 0). Manifolds satisfying the condition

(5) R · C = L C Q(g, C)

on the set U C = {x ∈ M | C 6= 0 at x} are called Weyl-pseudosymmetric;

in particular, for L C = 0 they contain the Weyl-semisymmetric spaces (R · C= 0).

The inclusions among the above mentioned classes of manifolds can be summarized in the following table; in general, for manifolds with dimension

≥ 4, all inclusions are strict [6].

R · S = L S Q(g, S) ⊃ R · R = L R Q(g, R) ⊂ R · C = L C Q(g, C)

∪ ∪ ∪

R · S = 0 ⊃ R · R = 0 ⊂ R · C = 0

In the present paper, we prove that on para-K¨ ahler manifolds the cur- vature conditions of pseudosymmetric type R · T = LQ(g, T ) for T = R, S and C reduce to the corresponding curvature conditions of semisymmetric type, e.g. R · T = 0 for T = R, S and C, respectively. This question for Ricci- generalized pseudosymmetric para-K¨ ahler manifolds (i.e. manifolds realizing a curvature condition of the form R · R = LQ(S, R)) was treated in [2].

Let L denote the class of all almost Hermitian manifolds (M, g, J ). Fur- ther, the class of all para-K¨ ahler manifolds will be denoted by L 1 . According to [10], we denote by L 2 and L 3 the classes of all almost Hermitian manifolds realizing the relations

R(X, Y, Z, W ) = R(J X, J Y, Z, W ) + R(J X, Y, J Z, W ) + R(J X, Y, Z, J W ) and

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

respectively, where X, Y, Z, W ∈ Ξ(M ). As shown in [10] (Lemma 5.1, p. 68), we have the following inclusions:

L ₁ ⊂ L ₂ ⊂ L ₃ ⊂ L.

Some results on the above classes of manifolds are presented in Chapter II

of [10].

3. Para-K¨ ahler manifolds of pseudosymmetric type. Let (M, g, J ), n = dim M ≥ 4, be a para-K¨ ahler manifold covered by a system of coordi- nate neighbourhoods {U ; u ^h }. We denote by J _h ^k = J _h ^k the local components of the almost complex structure J . Morever, let C hijk , R hijk , S ij and g ij

be the local components of the Weyl conformal curvature tensor C, the Riemann–Christoffel curvature tensor R, the Ricci tensor S and the metric tensor g, respectively. Thus, by (1), we have

(6) J _k ^s J _s ^l = −δ _k ^l , J _k ^r J _l ^s g rs = g kl , J ^kl = g ^ks J _s ^l . Further, (2) takes the form

(7) J _h ^r J _i ^s R rsjk = R hijk , whence, by (6), we obtain

(8) J _h ^s R sljk − J _l ^s R shjk = 0.

Further, using the above relations, we find

(9) J ^rs R rsjk = 2A jk , J ^rs R rijs = −A ij , where A jk = −J _j ^s S sk .

3.1. Ricci-semisymmetric para-K¨ ahler manifolds

Proposition 3.1. Every semi-Riemannian Ricci-pseudosymmetric para- K¨ ahler manifold (M, g, J ), dim M ≥ 4, is Ricci-semisymmetric.

P r o o f. Let x be a point of the set U S and let ˜ U ⊂ U S be a coordinate neighbourhood of x. Then the equality

(R · S) hirs = L S Q(g, S) hirs

holds on e U . Transvecting this with J _j ^r J _k ^s and using the Ricci identity and (7) we obtain

(R · S) hirs = L S Q(g, S) hirs J _s ^r J _k ^s . Thus by (4) we have

L S Q(g, S) hirs J _j ^r J _k ^s = L S Q(g, S) hijk .

Suppose that the function L S is non-zero at x. Then the last equality gives Q(g, S) hirs J _j ^r J _k ^s = Q(g, S) hijk ,

which, by contraction with g ^hk , yields S ij = ^κ _n g ij , a contradiction. Thus the function L S vanishes identically on U S , which completes the proof.

The above proposition generalizes Theorem 1 of [11].

3.2. Pseudosymmetric para-K¨ ahler manifolds

Proposition 3.2. Every semi-Riemannian pseudosymmetric para-

K¨ ahler manifold (M, g, J ), dim M ≥ 4, is semisymmetric.

P r o o f. Let x be a point of the set U R and let e U ⊂ U R be a coordinate neighbourhood of x. Then the equality

(R · R) hijkrs = L R Q(g, R) hijkrs

holds on e U . Now, in the same way as in the proof of Proposition 3.1 we obtain

L R Q(g, R) hijkrs J _l ^r J _m ^s = L R Q(g, R) hijklm .

Suppose that the function L R is non-zero at x. Then the last equality gives (10) Q(g, R) hijkrs J _l ^r J _m ^s = Q(g, R) hijklm .

Since every pseudosymmetric manifold is Ricci-pseudosymmetric, from Proposition 3.1 it follows that the equality

(11) S ij = κ

n g ij

holds at x. Contracting (10) with g ^hm and using (11), (7), and (9), we get (12) J _l ^r J _j ^s R rkis − J _l ^r J _k ^s R rjis + 2κ

n J li J jk − κ

n J lj J ki + κ n J lk J ji

= (n − 3)R lijk − κ n G lijk , where J li = J _l ^s g si and G lijk = g lk g ij − g _lj g ik are the local components of the tensor G. Transvecting (12) with J _h ^j and and using (6) and (7), we find

−J _l ^r R rkih − J _k ^s R sihl − 2κ

n J li g hk − κ

n J ki g hl − κ n J lk g hi

= (n − 3)J _h ^s R skli − κ

n g kl J hi + κ n g ik J hl . Contracting this with g ^hi and applying (6), (7), (8) and (11), we find κJ kl

= 0, whence κ = 0. Thus (11) yields S ij = 0. Now (9) reduces to (13) J ^rs R rsjk = 0, J ^rs R rijs = 0,

respectively. Further, from (7), by (8), we have

J _h ^r J _i ^s (R · R) rsjkab = (R · R) hijkab ,

which, by virtue of (3), (7) and the assumption that L R is non-zero at x, turns into

g ia R bhjk + g hb R aijk − g ha R bijk − g ib R ahjk

= J ha J _i ^s R sbjk + J ib J _h ^s R sajk − J _ia J _h ^s R sbjk − J _hb J _i ^s R sajk . This yields

g ia (R · R) bhjklm + g hb (R · R) aijklm − g _ha (R · R) bijklm − g _ib (R · R) ahjklm

= J ha J _i ^s (R · R) sbjklm + J ib J _h ^s (R · R) sajklm

− J _ia J _h ^s (R · R) sbjklm − J _hb J _i ^s (R · R) sajklm .

Finally, applying (3) and contracting the resulting equality with g ^am and g ^hb we get R lijk = 0, a contradiction. Thus L R vanishes at x. But this completes the proof.

The above proposition generalizes Theorem 4 of [4].

3.3. Weyl-pseudosymmetric para-K¨ ahler manifolds

Proposition 3.3. Every semi-Riemannian Weyl-pseudosymmetric para- K¨ ahler manifold (M, g, J ), dim M ≥ 4, is Weyl-semisymmetric.

P r o o f. Let x be a point of U C . First assume that dim M ≥ 5. Thus, in view of Theorem 1 of [7] and our Proposition 3.2, the tensor R · R vanishes at x. Thus R · C = 0 holds at x, completing the proof of this case. Now assume that dim M = 4. Transvecting the equality

(R · C) hijklm = L C Q(g, C) hijklm

with J _a ^l J _b ^m and using the Ricci identity and (7) we obtain (R · C) hijkab = L C J _a ^l J _b ^m Q(g, C) hijklm , whence

L C Q(g, C) hijkab = L C J _a ^l J _b ^m Q(g, C) hijklm .

Suppose that the function L C is non-zero at x. Thus the last equality reduces to

Q(g, C) hijkab = J _a ^l J _b ^m Q(g, C) hijklm . This, by contraction with g ^hb , gives

2C aijk = J _a ^s J _i ^r C srjk + J _a ^s J _j ^r C skir − J _a ^s J _k ^r C sjir

(14)

− J _ai J ^sr C srjk − J _aj J ^sr C skir + J ak J ^sr C sjir , which, by transvection with J _h ^a , turns into

2J _h ^s C sijk = J _i ^r C rhjk − J _j ^r C hkir + J _k ^r C hjir

(15)

+ g hj J ^sr C skir − g _hk J ^sr C sjir + g hi J ^sr C srjk . Next, contracting (15) with g ^hi we obtain

(16) J ^sr C srjk = 0,

which reduces (15) to

(17) 2J _h ^s C sijk = J _i ^s C shjk − J _j ^s C sikh + J _k ^s C sijh . Next, summing (17) cyclically in h, j, k we get

J _h ^s C sijk + J _j ^s C sikh + J _k ^s C sihj = 0.

Now (17), by making use of the above relation, yields

J _k ^s C sijk = J _i ^s C shjk .

From this, by (8) and (5), it follows that

J ha C bijk + g ia J _h ^s C sbjk − J hb C aijk − g ib J _h ^s C sajk

= J ia C bhjk + g hl J _i ^s C sbjk − J _ib C ahjk − g _hb J _i ^s C sajk

and

J ha Q(g, C) bijklm − J _hb Q(g, C) aijklm

+ g ia J _h ^s Q(g, C) sbjklm − g ib J _h ^s Q(g, C) sajklm

= J ia Q(g, C) bhjklm − J _ib Q(g, C) ahjklm

+ g hl J _i ^s Q(g, C) sbjklm − g _hb J _i ^s Q(g, C) sajklm . Contracting the last equality with g ^am and g ^hb , after some calculations, we obtain

J _i ^s C sljk = 0.

Applying this and (15) in (14) we get at x the relation C = 0, a contradic- tion. Our theorem is thus proved.

R e m a r k 3.1. An example of a non-conformally flat and non-semisym- metric Weyl-semisymmetric manifold (M, g), dim M = 4, which is a K¨ ahler manifold was described in [5] (Lemme 1.1).

R e m a r k 3.2. Let B be the Bochner curvature tensor ([1], [18], [15]) of a para-K¨ ahler semi-Riemannian manifold (M, g, J ), n = 2m ≥ 4. In [8] K¨ ahler Riemannian manifolds with semisymmetric Bochner tensor (R · B = 0) were considered. From the main results of [8] (Theorem) we can conclude that if the tensor B of a para-K¨ ahler Riemannian manifold (M, g, J ), n = 2m ≥ 4, is semisymmetric then the Riemann–Christoffel curvature tensor R of (M, g, J ) is semisymmetric on the subset U B ⊂ M consisting of all points of M at which B is non-zero. Most recently this statement was generalized as follows [9]: if the Bochner tensor B of a para-K¨ ahler semi-Riemannian manifold (M, g, J ), n = 2m ≥ 4, is pseudosymmetric, i.e. R·B = L B Q(g, B) holds on U B , then the tensor R of (M, g, J ) is semisymmetric on the set U B . Acknowledgements. The first author (F.D.) acknowledges support from the project “Gauge theories, applied supersymmetry and quantum gravity”, contract SC1-CT92-0789 of the European Economic Community.

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Filip Defever Ryszard Deszcz

Institute of Theoretical Physics Department of Mathematics Katholieke Universiteit Leuven Agricultural University of Wroc law

Celestijnenlaan 200 D Grunwaldzka 53

B-3001 Leuven, Belgium PL-50-375 Wroc law, Poland

E-mail: Filip.Defever@wis.kuleuven.ac.be E-mail: rysz@ozi.ar.wroc.pl Leopold Verstraelen

Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 200 B

B-3001 Leuven, Belgium

E-mail: Leopold.Verstraelen@wis.kuleuven.ac.be

Received 13 November 1996;

revised 27 January 1997