DOUBLY WARPED PRODUCTS WITH HARMONIC WEYL CONFORMAL CURVATURE TENSOR

Transcription

1 C O L L O Q U I U M M A T H E M A T I C U M VOL. LXVII 994 FASC. DOUBLY WARPED PRODUCTS WITH HARMONIC WEYL CONFORMAL CURVATURE TENSOR BY ANDRZEJ G Ȩ B A R O W S K I (RZESZÓW). Introduction. An n-dimensional (n 4) Riemannian maniold (whose metric g ij need not be deinite) is called conormally symmetric [4] i its Weyl conormal curvature tensor () C hijk = R hijk n 2 (g ijr hk g ik R hj + g hk R ij g hj R ik ) R + (n )(n 2) (g ijg hk g ik g hj ) is parallel, i.e., i C hijk,l = 0. Here and in the sequel we denote by R hijk, R ij and R the curvature tensor, Ricci tensor and scalar curvature, respectively, while the comma stands or covariant dierentiation with respect to g. Clearly, the class o conormally symmetric maniolds contains all locally symmetric as well as all conormally lat maniolds o dimension n 4. It is easy to check that or every conormally symmetric maniold the condition (2) R ij,k R ik,j = 2(n ) (R,kg ij R,j g ik ) holds. A Riemannian maniold is said to have harmonic curvature respectively harmonic Weyl (conormal curvature) tensor i the divergence o its curvature tensor (respectively Weyl tensor) is identically zero, i.e. R r ijk,r = 0, respectively C r ijk,r = 0 ([2], p. 440). The second Bianchi identity implies that R r ijk,r = R ij,k R ik,j and C r ijk,r = n 3 n 2 [ R ij,k R ik,j ] 2(n ) (R,kg ij R,j g ik ). 99 Mathematics Subject Classiication: Primary 53B83; Secondary 53C43. [73]

2 74 A. GȨBAROWSKI Including the case n = 3 in our considerations we say that M has harmonic conormal curvature i the tensor R ij R 2(n ) g ij is Codazzi on M, i.e. satisies condition (2). Now it is clear that any conormally symmetric maniold as well as maniolds with harmonic curvature have harmonic conormal curvature. Moreover, a Riemannian maniold has harmonic conormal curvature i and only i C r ijk,r = 0 in case n 4 or i it is conormally lat in case n = 3. In the paper [2] an n-dimensional (n 3) Riemannian maniold satisying condition (2) was called nearly conormally symmetric. (M n, g) is Einstein i R ij is proportional to g ij. It is well known that every surace is Einstein and that an Einstein space satisying C h ijk = 0 is a space o constant curvature. The aim o this paper is to give a description o the local structure o doubly warped products with harmonic conormal curvature. 2. Deinition and basic ormulas. Let (M, g) and (M, g ) (dim M = r, dim M = n r, r < n) be Riemannian maniolds and let : M R + and h : M R + be positive C -unctions. The Riemannian maniold M := M M urnished with the metric tensor g := (h σ) 2 π (g) + ( π) 2 σ (g ), where π : M M M and σ : M M M are the natural projections, will be called a doubly warped product and denoted by M h M [6]. I in particular h or is constant, then the doubly warped product reduces to a (singly) warped product [3] (or semi-decomposable space [0]). The doubly warped product is the special case o the so-called conormal product (g = hg kg, where h and k are unctions deined on M M ) investigated by Yano [4] and Wong [3]. Some theorems on conormally symmetric, Ricci-symmetric and Ricci-pseudosymmetric doubly warped products can be ound in [8] and [9]. Lorentzian doubly warped products have been studied by Beem and Powell []. By similar considerations to [], 2, we obtain Theorem. An n-dimensional doubly warped product M = M h M (dim M = r, r < n) ibers into two mutually orthogonal (n r)- and r-codimensional Riemannian oliations, the leaves o each being totally umbilical in M and pairwise (locally) homothetic.

3 DOUBLY WARPED PRODUCTS 75 Let {U V : x,..., x r, x r+ = y,..., x n = y nr } be a product chart or M M. The local components o the metric g = g h g with respect to this chart are h 2 g ab i i = a and j = b, (3) g ij = 2 g αβ i i = α and j = β, 0 otherwise, where a, b, c, d, e {,..., r}, α, β, γ, δ, ε {r +,..., n} and i, j, k, l {,..., n}. We assume that each object marked with a dash is ormed rom g ab, and those marked with a star come rom g αβ. Moreover, the period and semicolon in subscripts denote covariant dierentiation in M and M, respectively. The local components Γij h o the Levi-Civita connection on M h M are Γ a bc = Γ a bc, Γ a αb = h h αδ a b, Γ α βa = aδ α β, (4) Γ α βγ = Γ α βγ, Γ α ab = h hα g ab, Γ a αβ = a g αβ, where a = ( / x a ), h α = ( / x α )h, a = g ae e and h α = g αε h ε. The local components R l ijk = k Γ l ij j Γ l ik + Γ r ijγ l rk Γ r ikγ l rj, where j = / x j, o the curvature tensor o M h M, which in general do not vanish identically, are (5) R a bcd = R a bcd ( h) h 2 (δ a dg bc δ a c g bd ), R α bcd = h (hα d g bc h α c g bd ), R a βγδ = h (h δ a g βγ h γ a g βδ ), R α bcβ = δα β bc h hα βg bc, R a βγd = h δa dh βγ a d g βγ, R α βγδ = R α βγδ ( ) 2 (δ α δ g βγ δ α γ g βδ ), where ( h) = g αβ h α h β, ( ) = g ab a b, h ij = h i,j and ij = i,j.

4 76 A. GȨBAROWSKI (6) where The local components R ij = R r ijr o the Ricci tensor o M h M R ab = R ab n r ab (r ) ( h) h 2 g ab h (hε.ε)g ab, R αd = n 2 h h α d, R αβ = R αβ r h h αβ (n r ) ( ) 2 g αβ ( ẹ e)g αβ (h ε.ε) = g αβ h αβ and ( ẹ e) = g ab ab. The scalar curvature R o the metric g h g satisies are (7) R = R h 2 + R r(r ) 2 h 2 ( h) 2r h (hε.ε) 2(n r) ( ẹ e). (n r)(n r ) 2 ( ) By straightorward computations we establish the ollowing three lemmas. Lemma. In a doubly warped product M h M, h ab = h ( h)g ab, h aα = ah α, h αβ = h α;β, ab = a.b, aα = h ah α, αβ = ( )g αβ. Lemma 2. Let M = M h M. Then c ( h) = 2 ( h) c, γ ( h) = 2h α h αγ, α ( ) = 2 h ( )h α, c ( ) = 2 e ec, c h αβ = 0, c (h ε.ε) = 2 (hε.ε) c, γ ab = 0, γ ( ẹ e) = 2 h ( ẹ e)h γ. Lemma 3. In M h M, h αcβ = 2 ch αβ + h ( h) c g αβ + h ch α h β, aγb = 2 h h γ ab + h ( )h γ g ab + h a b h γ,

5 DOUBLY WARPED PRODUCTS 77 g ab abc = c ( ẹ e) 2 h 2 ( h) c, g αβ h αβγ = γ (h ε.ε) 2 2 ( )h γ, where ijk = i,jk and h ijk = h i,jk. (8) On the other hand, the Ricci ormula gives g ab abc = g ab acb h 2 er e c + r h 2 ( h) c, g αβ h αβγ = g αβ h αγβ 2 h εr ε γ + n r 2 ( )h γ. Dierentiating the local components o the Ricci tensor covariantly and taking into account (4), (6) and Lemmas and 2, we get R ab,c = R ab.c n r + abc + n r 2 c ab 2(r ) h 2 ( h) c g ab + 2 h (hε.ε) c g ab + r 2 h 2 ( h)( b g ac + a g bc ), R αb,c = h h αr bc + n 2 h h α bc + h hε R αε g bc, h 2 h α b c (9) R ab,γ = 2 h h γr ab + + 2(n r) h γ ab h 2(r ) h 3 ( h)h γ g ab + h 2 (hε.ε)h γ g ab h 2 a b h γ r h 2 g ab γ ( h) h g ab γ (h ε.ε), R αβ,c = 2 cr αβ + 2r h ch αβ ( ẹ e) c g αβ 2(n r ) 3 ( ) c g αβ h 2 h α h β c n r 2 g αβ c ( ) g αβ c ( ẹ e),

6 78 A. GȨBAROWSKI (9) [cont.] R aβ,γ = ar βγ + n 2 h ah βγ + e R ae g βγ, R αβ,γ = R αβ;γ r h h αβγ + r h 2 h γh αβ + Moreover, rom (7) we ind (0) h 2 a h β h γ 2(n r ) h 2 ( )h γ g αβ + 2 h ( ẹ e)h γ g αβ + n r 2 h 2 ( )(h β g αγ + h α g βγ ). R,c = h 2 R.c 2 3 R c + + 2r(r ) h 2 ( h) c 2(n r)(n r ) (n r)(n r ) 3 ( ) c 2 c ( ) + 4r 2(n r) h (hε.ε) c + 2 ( ẹ e) c R,γ = 2 h 3 Rh γ + 2 R ;γ + r(r ) h 2 γ ( h) + + 2r h 2 (hε.ε)h γ 2r h γ(h ε.ε) + 2r(r ) h 3 ( h)h γ 2(n r) c ( ẹ e), 2(n r)(n r ) h 2 ( )h γ 4(n r) ( ẹ h e)h γ. 3. Doubly warped products with harmonic Weyl tensor and an r-dimensional actor ( < r < n ). Let M = M h M with n = dim M 4. We can now prove the ollowing results. Lemma 4. I M h M has harmonic Weyl tensor and r = dim M 2, n r = dim M 2, then () R ab + r 2 ab R r g ab r 2 r ( ẹ e)g ab = 0 on M, where ( ẹ e) = g ab a.b, and ( ) R αβ + n r 2 h αβ R h n r g αβ n r 2 (n r)h (h ε.ε )g αβ = 0 on M, where (h ε.ε ) = g αβ h α;β.

7 DOUBLY WARPED PRODUCTS 79 P r o o. From (2), taking all the subscripts i, j, k rom the same range,..., r or r +,..., n and substituting suitable ormulae (9), we get (2) R ab.c R ac.b + n r e R e abc + n r 2 ( c ab b ac ) + 2 h (hε.ε)( c g ab b g ac ) n 2r h 2 ( h)( c g ab b g ac ) = 2(n ) (R,cg ab R,b g ac ) and (2 ) R αβ;γ R αγ;β + r h h εr ε αβγ + r h 2 (h γh αβ h β h αγ ) = + 2 h ( ẹ e)(h γ g αβ h β g αγ ) + n 2r h 2 ( )(h γ g αβ h β g αγ ) 2(n ) (R,γg αβ R,β g αγ ). Applying contractions to (2) and (2 ), and making use o (0) and Lemma 2, we obtain (3) and (3 ) h 2 R 2(r ) R.c = n r 3 c 2(n ) h 2 e R e c + 2n nr + r2 2 2 c ( ) 2(n r) 2 ( ẹ e) c + 4(r )(n r ) (h ε (n r)h.ε) c + 2(n r ) R 2 R ;γ = r h 3 h γ + 2n nr + r2 2 h 2 γ ( h) 2r h 2 (hε.ε)h γ + 2(r ) c ( ẹ e) 2(r )(n r ) h 2 ( h) c 2(n ) h 2 h ε R ε γ 2(r )(n r ) 3 ( ) c 2(n r ) γ (h ε h.ε) 2(r )(n r ) h 2 ( )h γ 4(r )(n r ) ( ẹ 2(r )(n r ) rh e)h γ + h 3 ( h)h γ. From (2), irst taking i = α, j = β, k = c and remembering that the mixed

8 80 A. GȨBAROWSKI components o the metric tensor are zero, we have R αβ,c R αc,β = 2(n ) R,cg αβ, which, in view o (9), (0) and (3), yields [ c R αβ + n r 2 h αβ R h n r g αβ n r 2 ] (n r)h (h ε.ε )g αβ = 0. The last result together with the non-constancy o implies ( ). In the same way, putting i = a, j = b, k = γ in (2), we get R ab,γ R aγ,b = 2(n ) R,γg ab, which, in view o (9), (0) and (3 ) yields similarly [ h γ h R ab + r 2 ab R r g ab r 2 r ( ẹ e)g ab and consequently () by the assumption about h. proo. ] = 0, This completes the R e m a r k. We observe that or M (resp. M ) two-dimensional, the condition () (resp. ( )) is identically satisied. For higher dimensions Lemma 4 states that in a doubly warped product M h M with harmonic Weyl tensor, the Hessian o (resp. h) is proportional to the metric tensor g (resp. g ) i and only i M (resp. M ) is an Einstein maniold. R e m a r k 2. I in particular h = const, then the doubly warped product reduces to a warped product and ( ) shows that M is an Einstein maniold (see also [7], Lemma 3). We are now in a position to prove Theorem 2. Let M h M be a doubly warped product with harmonic Weyl conormal curvature tensor. Then (i) M has harmonic conormal curvature i and only i its curvature tensor and the unction satisy (4) e R e abc + ( c ab b ac ) = { e (R e cg r ab R e bg ac ) where a = g ae e, and + [( ẹ e) c e ec ]g ab [( ẹ e) b e eb ]g ac },

9 DOUBLY WARPED PRODUCTS 8 (ii) M has harmonic conormal curvature i and only i its curvature tensor and the unction h satisy (4 ) h ε R ε αβγ + h (h γh αβ h β h αγ ) { = h ε (R ε γ g αβ R ε β g αγ) n r where h α = g αε h ε. + h [(h ε.ε )h γ h ε h εγ ]g αβ h [(h ε.ε )h β h ε h εβ ]g αγ P r o o. To prove (i), by calculating R rom (3) and substituting the result to (0) one obtains (5) R,c = n (r )h 2 R.c + + which, in view o (2), yields R ab.c R ac.b + n r Thereore 2(n )(n r) (r )h 2 e R e c (n )(n r) (r ) 2 c ( ) 2(n )(n 2r) h 2 ( h) c 4(n ) (h ε 2(n )(n r) h.ε) c + (r ) 2 ( ẹ e) c, [ e R e abc + ( c ab b ac ) = 2(r ) (R.cg ab R.b g ac ) + n r { e (R e cg r ab R e bg ac ) + [( ẹ e) c e ec ]g ab [( ẹ e) b e eb ]g ac }. R ab.c R ac.b = 2(r ) (R.cg ab R.b g ac ) is equivalent to (4). Next, by the same argument as above, we conclude (ii) rom the star ormulas. This completes the proo. R e m a r k 3. It is easy to veriy that i the Hessian o (resp. h) is proportional to the metric tensor g (resp. g ), then the condition (4) (resp. (4 )) is satisied. ] },

10 82 A. GȨBAROWSKI R e m a r k 4. By making use o Lemma 4 it is not diicult to show that or a doubly warped product with harmonic Weyl conormal curvature tensor the conditions (4) and (4 ) are equivalent to (6) e R e abc + ( c ab b ac ) and R = r(r ) ( cg ab b g ac ) + [ ] 2 r ( ẹ e) b e eb g ac (6 ) h ε R ε αβγ + h (h γh αβ h β h αγ ) [ ] 2 r ( ẹ e) c e ec g ab R = (n r)(n r ) (h γg αβ h β g αγ) + [ ] 2 h n r (h ε.ε )h γ h ε h εγ g αβ [ ] 2 h n r (h ε.ε )h β h ε h εβ g αγ, respectively. Theorem 3. A doubly warped product M h M has harmonic Weyl conormal curvature tensor i and only i the conditions (2), (), (2 ) and ( ) are satisied. P r o o. The necessity ollows rom the harmonicity o the Weyl tensor and Lemma 4. To prove the suiciency it suices to check the condition (2) in the cases when i, j, k are (a) a, b, c, (b) α, β, γ, (c) a, b, γ, (d) α, β, c. From (2), (2 ), (9) and the Ricci ormula, the cases (a) and (b) are evident. To veriy the case (c), besides () we need the condition (3), which ollows (as shown in the proo o Lemma 4) rom (2). Thereore, relations (), (3), (9) and Lemmas 3 give (c). Analogously, we check the case (d). This completes the proo. Dierentiating () covariantly in M and taking into account the Ricci ormula, we get (7) R ab.c R ac.b r 2 e R e abc r 2 2 ( c ab b ac )

11 DOUBLY WARPED PRODUCTS 83 = r (R.cg ab R.b g ac ) r 2 r 2 ( ẹ e)( c g ab b g ac ) + r 2 r [g ab c ( ẹ e) g ac b ( ẹ e)]. It is worth remarking that we may use (7) and the analogous ormula (7 ) to obtain some relations equivalent to (2) and (2 ). Lemma 5. Suppose that M h M has harmonic Weyl conormal curvature tensor. Then (8) (8 ) (i) or r = dim M > 2, 2 R.c + R c = r(r ) 2 (ii) or n r = dim M > 2, 2 R ;γ + R h h γ = [ e ec r ( ẹe) c ] (n r)(n r ) h 2 r c ( ẹ e), [ h ε h εγ ] n r (h ε.ε )h γ n r γ (h ε h.ε ). P r o o. Contracting (7) with g ab, we get, or r > 2, (9) 2r R.c = er e c + 2 e ec r 2 ( ẹ e) c r r c( ẹ e). On the other hand, transvecting () with a, we obtain easily (20) e R e c = R r c r 2 e ec + r 2 r ( ẹ e) c. Together with (9), this yields (i). The assertion (ii) ollows rom the analogous argument via the star way instead o the dash way. This completes the proo. Corollary. Suppose that M h M has harmonic Weyl conormal curvature tensor. I the Hessian o (resp. h) is proportional to the metric tensor g (resp. g ) then (i) or r = dim M > 2, (ii) or n r = dim M > 2, R c + (r ) c ( ẹ e) = 0, R h γ + (n r ) γ (h ε.ε ) = 0.

12 84 A. GȨBAROWSKI Lemma 6. Suppose that M h M has harmonic Weyl conormal curvature tensor. Then or r > 2, n 2 r 2 (R ab.c R ac.b ) = + 2nr r2 2n 2r(r )(r 2) (R.cg ab R.b g ac ) (n r)r r(r ) ( cg ab b g ac ) + n r r 2 ( ẹ e)( c g ab b g ac ) n r 2 ( e ec g ab e eb g ac ) + n r r (g ab c ( ẹ e) g ac b ( ẹ e)) on M, and the corresponding relation holds on M. P r o o (or the dash ormula). Calculating e R e abc + ( c ab b ac ) rom (7), and substituting the result to (2), together with (5) and (20), we get our assertion. Since on a two-dimensional maniold the condition (2) is identically satisied, we conclude rom Lemmas 5 and 6 the ollowing Theorem 4. I a doubly warped product M h M has harmonic conormal curvature, then so do the actor maniolds M and M. 4. Doubly warped products with harmonic Weyl tensor and a -dimensional actor. Let dim M =. In a suitable product chart t = x, x 2,..., x n or M M, putting g = g h g, we have g = h 2, g α = 0, g αβ = 2 g αβ, Γ = Γ = 0, Γ α = h h α, Γ α β = δ α β, Γ α = hh α, Γ αβ = h 2 g αβ, Γ α βγ = Γ α βγ. Here and in the sequel α, β, γ run through {2,..., n}, the prime stands or d/dt, while the Γ s (resp. Γ s) are Christoel symbols o g (resp. o g with respect to the chart x 2,..., x n o M ). Furthermore, (2) R = n R αβ = R αβ h h αβ h 2 R = R 2 2 h (hε.ε) n h 2 h(h ε.ε), R α = n 2 h h α, [ + (n 2) ( ) 2 [2 + (n 2) ( ) 2 ] g αβ, ].

13 DOUBLY WARPED PRODUCTS 85 Hence in this case (22) R,γ = R γ, = 2(n ) h γ + (h ε h.ε)h γ h γ (h ε.ε), h γ + (h ε h.ε)h γ h 2 ( ) 2 h γ h 2 ( ) 2 h γ + h 2 h εr ε γ 2 γ( h) n 2 h 2 ( ) 2 h γ, R αβ, = 2 R αβ + 2 h h αβ + h 2 3 ( ) 3 g αβ h 2 h α h β 2n 5 h 2 2 g αβ h 2 g αβ, R α,β = R αβ + n h h αβ + n 2 h 2 3 ( ) 3 g αβ h (hε.ε) g αβ R, = 2 3 R + R,γ = 2 γr + h 2 h α h β n 2 h 2 2 g αβ, 2(n )(n 2) h 2 3 ( ) 3 2(n )(n 3) h h (hε.ε) 2 h γ(h ε.ε) + 2(n ) h 2, 2(n )(n 2) h 3 2 ( ) 2 h γ + 2 h 2 (hε.ε)h γ 4(n ) h 3 h γ. Lemma 7. I M h M has harmonic conormal curvature and r = dim M =, n = dim M 2, then (23) R αβ + n 3 h h αβ R n g αβ n 3 (n )h (h ε.ε )g αβ = 0 on M. Moreover, or n > 3, (24) 2 R ;γ + R h h γ = (n )(n 2) h 2 [ h ε h εγ ] n (h ε.ε )h γ n 2 h γ(h ε.ε ). P r o o. Taking i = j = and k = γ in (2), we get R,γ R γ, = 2(n ) R,γg,

14 86 A. GȨBAROWSKI which, in view o (22), yields (25) 2 h εr ε h γ = 2(n ) 2 γr + 2h γ( h) n 2 n γ(h ε.ε) (n )h (hε.ε)h γ. On the other hand, taking i = α, j = β and k = in (2), we have R αβ, R α,β = which, in view o (22), yields [ R αβ + n 3 h h αβ R 2(n ) R,g αβ, n g αβ n 3 ] (n )h (h ε.ε )g αβ = 0. Together with the non-constancy o, this implies (23). Now, rom (23), (25) and Lemma 2, we get (24) by the same argument as in the proo o Lemma 5. This completes the proo. By proceeding as in Section 3, in the case dim M =, rom Lemma 7 one can easily obtain the ollowing theorems. M Theorem 5. Let dim M = and n 3. A doubly warped product M h has harmonic conormal curvature i and only i the conditions (23) and (2 ) are satisied. Theorem 6. Let dim M = and n 3. I a doubly warped product M h M has harmonic conormal curvature, then so does M. 5. Examples o doubly warped products with harmonic Weyl tensor. In this section we construct our examples using some ideas rom the paper [5] (see Examples and 3). Example. Assume that M (dim M 2) is an open subset o R r {(0,..., 0)}, g ab = δ ab, a, b {,..., r}, = (x,..., x r ) = B ( r (x a ) 2), 2 and M (dim M 2) is an open subset o R nr {(0,..., 0)}, g αβ = δ αβ, α, β {r +,..., n}, and h = h(x r+,..., x n ) = A ( n (x α ) 2), 2 a= α=r+

15 DOUBLY WARPED PRODUCTS 87 with positive constants A and B. It is then easy to veriy that a = Bx a, h α = Ax α, ( ) = 2B h 2, ( h) = 2Ah 2, ab = Bg ab, h αβ = Ag αβ, ( ẹ e) = rb, (h ε.ε ) = (n r)a and 2(n )r 2(n )(n r) R = h 2 A h 2 B const. Thereore, by Theorem 3, we conclude that the doubly warped product M h M has harmonic Weyl conormal curvature tensor but it does not have harmonic curvature (since R const). Example 2. Let M = {(x, y) R 2 : y > 0} with the metric tensor g deined by ( ) ( g x, = uv, g x y, ) = ( y v, g x, ) = 0, y where u = u(x), v = v(y) are smooth unctions not vanishing at any point on M. Next let = (y) = y, and M = {(z, t) R 2 : t > 0} with g deined by ( g z, ) ( = pq, g z t, ) = ( t q, g z, ) = 0, t where p = p(z), q = q(t) are smooth unctions not vanishing at any point on M, and h = h(t) = t. Then the only non-zero components o the Christoel symbols Γ a bc and curvature tensor R abcd are Moreover, Γ = 2u xu, Γ 2 = 2v yv, Γ 2 22 = 2v yv, Γ 2 = 2 uv yv, R 22 = 2 u yyv. = 0, 2 =, ( ) = v t 2, R = yyv, ab = 2 ( yv)g ab, ( ẹ e) = y v and we have analogous ormulae on M. The conditions (2) and (2 ) take the orm and y 3 yyy v + y 2 yy v 2y y v + 2v = t 2 tt q + 2t t q 2q t 3 ttt q + t 2 tt q 2t t q + 2q = y 2 yy v + 2y y v 2v,

16 88 A. GȨBAROWSKI respectively. Thereore, i we take the solutions v(y) = c y + c 2 y 2 ϕ, q(t) = γ t + γ 2 t 2 + ϕ, where c, c 2, γ, γ 2, ϕ R, then, by Theorem 3, the doubly warped product M h M has harmonic Weyl conormal curvature. It is easily checked that ( c2 R = 2 y 2 + γ 2 t 2 + c 2 yt 2 + ) γ 2 y 2 const, t so M h M does not have harmonic curvature. Example 3. Let M (dim M = 2) and be as in Example 2, and M (dim M = n 2 2) and h be as in Example. Then (2) and (2 ) take the orm yyy v + n 3 2(n 3) 2(n 3) yy v y y 2 y v + y 3 v = 0 and h γ ( h 3 yy v 2y yv + 2y ) 2 v = 0, respectively. Taking a solution v(y) = c y+c 2 y 2, c, c 2 R, we get a doubly warped product M h M which has harmonic Weyl conormal curvature. REFERENCES [] J. K. B e e m and T. G. P o w e l l, Geodesic completeness and maximality in Lorentzian warped products, Tensor (N.S.) 39 (982), [2] A. L. Besse, Einstein Maniolds, Springer, 987. [3] R. L. B i s h o p and B. O N e i l l, Maniolds o negative curvature, Trans. Amer. Math. Soc. 45 (969), 49. [4] M. C. Chaki and B. Gupta, On conormally symmetric spaces, Indian J. Math. 5 (963), [5] R. Deszcz, L. Verstraelen and L. Vrancken, On the symmetry o warped product space-times, Gen. Relativity Gravitation 23 (99), [6] P. E. Ehrlich, Metric deormations o Ricci and sectional curvature on compact Riemannian maniolds, Ph.D. dissertation, SUNY, Stony Brook, N.Y., 974. [7] A. Gȩbarowski, Nearly conormally symmetric warped product maniolds, Bull. Inst. Math. Acad. Sinica 20 (992), [8] M. Hotloś, Some theorems on doubly warped products, Demonstratio Math. 23 (990), [9], Some theorems on Ricci-pseudosymmetric doubly warped products, to appear. [0] G. I. Kruchkovich, On semi-reducible Riemannian spaces, Dokl. Akad. Nauk SSSR 5 (957), (in Russian). [], On some class o Riemannian spaces, Trudy Sem. Vektor. Tenzor. Anal. (96), (in Russian). [2] W. R o t e r, On a generalization o conormally symmetric metrics, Tensor (N.S.) 46 (987),

17 DOUBLY WARPED PRODUCTS 89 [3] Y. C. W o n g, Some Einstein spaces with conormally separable undamental tensors, Trans. Amer. Math. Soc. 53 (943), [4] K. Y a n o, Conormally separable quadratic dierential orms, Proc. Imper. Acad. Tokyo 6 (940), MATHEMATICAL INSTITUTE PEDAGOGICAL UNIVERSITY OF RZESZÓW REJTANA 6A RZESZÓW, POLAND Reçu par la Rédaction le