CURVATURE OF MULTIPLY WARPED PRODUCTS. Contents

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1 CURVATURE OF MULTIPLY WARPED PRODUCTS FERNANDO DOBARRO AND BÜLENT ÜNAL arxv:math/ v3 [math.dg] 11 Dec 004 Abstract. In ths paper, we study Rcc-flat and Ensten Lorentzan multply warped products. We also consder the case of havng constant scalar curvatures for ths class of warped products. Fnally, after we ntroduce a new class of spacetmes called as generalzed Kasner space-tmes, we apply our results to ths knd of space-tmes as well as other relatvstc space-tmes,.e., Ressner-Nordström, Kasner space-tmes, Bañados-Tetelbom-Zanell and de Stter Black hole solutons. Contents 1. Introducton. Prelmnares 3. Specal Multply Warped Products 3.1. Ensten Rcc Tensor 3.. Constant Scalar Curvature 4. Generalzed Kasner Space-tme 5. 4-Dmensonal Space-tme Models 5.1. Type (I) 5.. Type (II) 5.3. Type (III) 6. BTZ ( 1)-Black Hole Solutons 7. Conclusons References Date: October 6, Mathematcs Subject Classfcaton. 53C5, 53C50. Key words and phrases. Warped products, Rcc tensor, scalar curvature, Ensten manfolds. Research of F. D. partally supported by funds of the Natonal Group Anals Reale of the Italan Mnstry of Unversty and Scentfc Research at the Unversty of Treste. 1

2 FERNANDO DOBARRO AND BÜLENT ÜNAL 1. Introducton The concept of warped products was frst ntroduced by Bshop and O Nell (see [0]) to construct examples of Remannan manfolds wth negatve curvature. In Remannan geometry, warped product manfolds and ther generc forms have been used to construct new examples wth nterestng curvature propertes snce then (see [19, 0, 4, 7, 3, 40, 41, 5, 53, 54, 57, 60]). In Lorentzan geometry, t was frst notced that some well known solutons to Ensten s feld equatons can be expressed n terms of warped products n [1] and after that Lorentzan warped products have been used to obtan more solutons to Ensten s feld equatons (see [1, 13, 19, 0, 44, 59, 65]). Moreover, geometrc propertes such as geodesc structure or curvature of Lorentzan warped products have been studed by many authors because of ther relatvstc applcatons (see [, 3, 4, 5, 10, 11, 14, 17,, 3, 8, 30, 3, 33, 34, 36, 37, 45, 50, 55, 56, 63, 67, 68, 69, 70, 71, 74, 75]). We recall the defnton of a warped product of two pseudo-remannan manfolds (B,g B ) and (F,g F ) wth a smooth functon b: B (0, ) (see also [13, 65]). Suppose that (B,g B ) and (F,g F ) are pseudo-remannan manfolds and also suppose that b: B (0, ) s a smooth functon. Then the (sngly) warped product, B b F s the product manfold B F equpped wth the metrc tensor g = g B b g F defned by g = π (g B ) (b π) σ (g F ) where π: B F B and σ: B F F are the usual projecton maps and denotes the pull-back operator on tensors. Here, (B,g F ) s called as the base manfold and (F,g F ) s called as the fber manfold and also b s called as the warpng functon. Generalzed Robertson-Walker space-tme models (see [, 11, 36, 68, 70, 71]) and standard statc space-tme models (see [3, 4, 5, 55, 56]) that are two well known solutons to Ensten s feld equatons can be expressed as Lorentzan warped products. Clearly, the former s a natural generalzaton of Robertson-Walker space-tme and the latter s a generalzaton of Ensten statc unverse. One way to generalze warped products s to consder the case of mult fbers to obtan more general space-tme models (see examples gven n Secton ) and n ths case the correspondng product s so called multply warped product. In [75], covarant dervatve formulas for multply warped products are gven and the geodesc equaton for these spaces are also consdered. The causal structure, Cauchy surfaces and global hyperbolcty of multply Lorentzan warped products are also studed. Moreover, necessary and suffcent condtons are obtaned for null, tme-lke and space-lke geodesc completeness of Lorentzan multply products and also geodesc completeness of Remannnan multply warped products. In [, 3], the author studes manfolds wth C 0 -metrcs and propertes of Lorentzan multply warped products and then he shows a representaton of the nteror Schwarzschld space-tme as a multply warped product space-tme wth certan warpng functons. He also gves the Rcc curvature n terms of b 1,b for a multply warped product of the form M = (0,m) b1 R 1 b S. In [45], physcal propertes (1) charged Bañados- Tetelbom-Zanell (BTZ) black holes and (1) charged de Stter (ds) black holes are studed by expressng these metrc as multply warped product space-tmes, more explctly, Rcc and Ensten tensors are obtaned nsde the event horzons (see also [9]). In [69], the exstence, multplcty and causal character of geodescs jonng two ponts of a wde class of non-statc Lorentz manfolds such as ntermedate Ressner-Nordström or nner Schwarzschld and generalzed Robertson-Walker space-tmes are studed. In [37], geodesc connectedness and also causal geodesc connectedness of mult-warped space-tmes are studed by usng the method of Brouwer s topologcal degree for the soluton of functonal equatons. There are also dfferent types of warped products such as a knd of warped product wth two warpng functons actng symmetrcally on the fber and base manfolds, called as a doubly warped product (see [74]) or another knd

3 CURVATURE OF MULTIPLY WARPED PRODUCTS 3 of warped product called as a twsted product when the warpng functon defned on the product of the base and fber manfolds (see [35]). Moreover, Easley studed Local Exstence Warped Product Structures and also defned and consdered another form of a warped product n hs thess (see [31]). In ths paper, we answer some questons about the exstence of nontrval warpng functons for whch the multply warped product s Ensten or has a constant scalar curvature. Ths problem was consdered especally for Ensten Remannan warped products wth compact base and some partal answers were also provded (see [41, 5, 53, 54]). In [53], t s proved that an Ensten Remannan warped product wth a non-postve scalar curvature and compact base s just a trval Remannan product. Constant scalar curvature of warped products was studed n [5, 7, 3, 33] when the base s compact and of generalzed Robertson-Walker space-tmes n [3]. Furthermore, partal results for warped products wth non-compact base were obtaned n [7] and [1]. The physcal motvaton of exstence of a postve scalar curvature comes from the postve mass problem. More explctly, n general relatvty the postve mass problem s closely related to the exstence of a postve scalar curvature (see [78]). As a more general related reference, one can consder [51] to see a survey on scalar curvature of Remannan manfolds. The problem of exstence of a warpng functon whch makes the warped product Ensten was already studed for specal cases such as generalzed Robertson-Walker space-tmes and a table gven the dfferent cases of Ensten generalzed Robertson-Walker when the Rcc tensor of the fber s Ensten n [] (see also references theren). Ensten Rcc tensor and constant scalar curvature of standard statc space-tmes wth perfect flud were already consdered n [55, 63]. Moreover, n [56], the conformal tensor on standard statc space-tmes wth perfect flud s studed and t s shown that a standard statc space-tme wth perfect flud s conformally flat f and only f ts fber s Ensten and hence of constant curvature. In [8], ths problem s consdered for arbtrary standard statc space-tmes, more explctly, an essental nvestgaton of condtons for the fber and warpng functon for a standard statc space-tme (not necessarly wth perfect flud) s carred out so that there exsts no nontrval functon on the fber guaranteng that the standard statc space-tme s Ensten. Duggal studed the scalar curvature of 4-dmensonal trple Lorentzan products of the form L B f F and obtaned explct solutons for the warpng functon f to have a constant scalar curvature for ths class of products (see [30]). Moreover, n the present paper, we ntroduce an orgnal form to generalze Kasner space-tmes and then we obtan necessary and suffcent condtons as well as explct solutons, for some specal cases, for a generalzed Kasner space-tme to be Ensten or to have constant scalar curvature. Besdes than the form mentoned here, there are also other generalzatons n the lterature (see [46, 58]). In [46], an extenson for Kasner space-tmes s ntroduced n the vew of generalzng 5-dmensonal Randall-Sundrum model to hgher dmensons and n [58], another mult-dmensonal generalzaton of Kasner metrc s descrbed and essental solutons are also obtaned for ths class of extenson. One can also consder [6, 39, 47, 48, 61, 66, 76] for recent applcatons of Kasner metrcs and ts generalzatons. We organze the paper as follows. In Secton, we gve several basc geometrc facts related to the concept of curvatures (see [73, 75]). Moreover, we recall two well known examples of relatvstc space-tmes whch can be consdered as generalzed multply Robertson-Walker space-tmes. In Secton 3, we obtan two results n whch, under several assumptons on the fbers and warpng functons, multply generalzed Robertson-Walker space-tmes are Ensten or have constant scalar curvature. In Secton 4, after we ntroduce generalzed Kasner space-tmes, we state condtons for ths class of space-tmes to be Ensten or to have constant scalar curvature. In Secton 5, we gve an explct classfcaton of 4-dmensonal multply generalzed Robertson-Walker

4 4 FERNANDO DOBARRO AND BÜLENT ÜNAL space-tmes and 4-dmensonal generalzed Kasner space-tmes whch are Ensten. In the last secton, we focus on BTZ (1)- Black Hole solutons and classfy (BTZ) black hole solutons gven n Secton by usng a more formal approach (see [8, 9, 45, 6]) and then we also prove necessary and suffcent condtons for the lapse functon of a BTZ (1) Black Hole soluton to have a constant scalar curvature or to be Ensten. Our man results are obtaned n Sectons 3,4 and 5, especally see Theorem 3.3, Propostons 4.3 and 4.11 as well as Tables 1, and 3.. Prelmnares Throughout ths work any manfold M s assumed to be connected, Hausdorff, paracompact and smooth. Moreover, I denotes for an open nterval n R of the form I = (t 1,t ) where t 1 < t and we wll furnsh I wth a negatve metrc dt. A pseudo-remannan manfold (M,g) s a smooth manfold wth a metrc tensor g and a Lorentzan manfold (M,g) s a pseudo-remannan manfold wth sgnature (,,,,). Moreover, we use the defnton and the sgn conventon for the curvature as n [13]. For an arbtrary n-dmensonal pseudo-remannan manfold (M,g) and a smooth functon f : M R, we have that H f and (f) denote the Hessan (0,) tensor and the Laplace-Beltram operator of f, respectvely ([65]). Here, we use the sgn conventon for the Laplacan n [65],.e., defned by = tr g (H), (see page 86 of [65]) where H s the Hessan form (see page 86 of [65]) and tr g denotes for the trace, or equvalently, = dv(grad), where dv s the dvergence and grad s the gradent (see page 85 of [65]). Furthermore, we wll frequently use the notaton grad f = g(grad f,grad f). When there s a possblty any msunderstandng, we wll explctly state the manfold or the metrc for whch the operator s consdered. We begn our dscusson by gvng the formal defnton of a multply warped product (see [75]). Defnton.1. Let (B,g B ) and (F,g F ) be pseudo-remannan manfolds and also let b : B (0, ) be smooth functons for any {1,,,m}. The multply warped product s the product manfold M = B F 1 F F m furnshed wth the metrc tensor g = g B b 1 g F 1 b g F b m g F m defned by (.1) g = π (g B ) (b 1 π) σ 1(g F1 ) (b m π) σ m(g Fm ) Each functon b : B (0, ) s called a warpng functons and also each manfold (F,g F ) s called a fber manfold for any {1,,,m}. The manfold (B,g B ) s the base manfold of the multply warped product. If m = 1, then we obtan a sngly warped product. If all b 1, then we have a (trval) product manfold. If (B,g B ) and (F,g F ) are all Remannan manfolds for any {1,,,m}, then (M,g) s also a Remannan manfold. The multply warped product (M, g) s a Lorentzan multply warped product f (F,g F ) are all Remannan for any {1,,,m} and ether (B,g B ) s Lorentzan or else (B,g B ) s a one-dmensonal manfold wth a negatve defnte metrc dt. If B s an open connected nterval I of the form I = (t 1,t ) equpped wth the negatve defnte metrc g B = dt, where t 1 < t, and (F,g F ) s Remannan for any {1,,,m}, then the Lorentzan multply warped product (M, g) s called a multply generalzed Robertson-Walker space-tme or a mult-warped space-tme. In partcular, a multply generalzed Robertson- Walker space-tme s called a generalzed Ressner-Nordström space-tme when m =. We wll state the covarant dervatve formulas for multply warped products (see [, 73, 75]).

5 CURVATURE OF MULTIPLY WARPED PRODUCTS 5 Proposton.. Let M = B b1 F 1 bm F m be a pseudo-remannan multply warped product wth metrc g = g B b 1 g F 1 b mg Fm also let X,Y L(B) and V L(F ), W L(F j ). Then (1) X Y = B X Y () X V = V X = X(b ) V b 0 f j, (3) V W = F V W g(v,w) grad b B b f = j One can compute the gradent and the Laplace-Beltram operator on M n terms of the gradent and the Laplace-Beltram operator on B and F, respectvely. From now on, we assume that = M and grad = grad M to smplfy the notaton. Proposton.3. Let M = B b1 F 1 bm F m be a pseudo-remannan multply warped product wth metrc g = g B b 1 g F 1 b m g F m and φ: B R and ψ : F R be smooth functons for any {1,,m}. Then (1) grad(φ π) = grad B φ () grad(ψ σ ) = grad F ψ (3) (φ π) = B φ (4) (ψ σ ) = F ψ b b s g B (grad B φ,grad B b ) b Now, we wll state Remannan curvature and Rcc curvature formulas from [73]. Proposton.4. Let M = B b1 F 1 bm F m be a pseudo-remannan multply warped product wth metrc g = g B b 1 g F 1 b mg Fm also let X,Y,Z L(B) and V L(F ),W L(F j ) and U L(F k ). Then (1) R(X,Y )Z = R B (X,Y )Z () R(V,X)Y = Hb B (X,Y ) V b (3) R(X,V )W = R(V,W)X = R(V,X)W = 0 f j. (4) R(X,Y )V = 0 (5) R(V,W)X = 0 f = j. (6) R(V,W)U = 0 f = j and,j k. (7) R(U,V )W = g(v,w) g B(grad B b,grad B b k ) b b k U f = j and,j k. (8) R(X,V )W = g(v,w) b B X(grad B b ) f = j. (9) R(V,W)U = R F (V,W)U grad B b B b (g(v,u)w g(w,u)v ) f,j = k. Proposton.5. Let M = B b1 F 1 bm F m be a pseudo-remannan multply warped product wth metrc g = g B b 1 g F 1 b mg Fm, also let X,Y,Z L(B) and V L(F ) and W L(F j ). Then (1) Rc(X,Y ) = Rc B (X,Y ) () Rc(X,V ) = 0 (3) Rc(V,W) = 0 f j. s b H b B (X,Y )

6 6 FERNANDO DOBARRO AND BÜLENT ÜNAL ( B b (4) Rc(V,W) = Rc F (V,W) k=1,k b (s 1) grad B b B b g B (grad s B b,grad B b k ) ) k g(v,w) f = j. b b k Now, we wll compute the scalar curvature of a multply warped product. In order to do that, { one can use the } followng { orthonormal } frame on M constructed as follows: Let x 1,, x r and y 1,, y s be orthonormal frames on open sets U B and V F, respectvely for any {1,,m}. Then { } x 1,, x r, b 1 y1 1,, b 1 y s,, 1 1 b m ym 1,, b m ym sm s an orthonormal frame on an open set W B F contaned n U V B F, where F = F 1 F m. Proposton.6. Let M = B b1 F 1 bm F m be a pseudo-remannan multply warped product wth metrc g = g B b 1 g F 1 b m g F m. Then, τ admts the followng expressons (1) () τ = τ B τ k=1,k = τ B s B b b τ F b s k s g B (grad B b,grad B b k ) b b k, g B ( m s B b b dv s grad B b b, s (s 1) grad B b B b grad s B b b s grad B b b The followng formula can be drectly obtaned from the prevous result and notng that on a multply ( generalzed Robertson-Walker space-tme grad B b = b, grad B b B = (b ), g B t, ) ( = 1, H b B t t, ) = b and B b = b, we denote the usual t dervatve on the real nterval I by the prme notaton (.e., ) from now on. Corollary.7. Let M = I b1 F 1 bm F m be a multply generalzed Robertson- Walker space-tme wth the metrc g = dt b 1 g F 1 b m g F m. Then, τ admts the followng expressons (1) () τ = τ = s b b τ F b s (s 1) (b ) b ) ( b m ) ( b m ) b s s s b b b k=1,k τ F b. τ F b. s k s b b k b b k,

7 CURVATURE OF MULTIPLY WARPED PRODUCTS 7 We now gve some physcal examples of relatvstc space-tmes and state some of ther geometrc propertes to stress the physcal motvaton and mportance of Lorentzan multply warped products. The frst example s Schwarzschld black hole soluton or known as nner Ressner-Nordström space-tme and the second one s Kasner spacetme. Our last two examples are closely related to each other, more explctly, the thrd example s Bañados-Tetelbom-Zanell (BTZ) black hole soluton and the fnal example s de Stter (ds) black hole soluton. Schwarzschld Space-tme We wll brefly dscuss the nteror Schwarzschld soluton. We show how the nteror soluton can be wrtten as a multply warped product. The lne element of the Schwarzschld black hole space-tme model for the regon r < m s gven as (see [44]) ( ) m 1 ( ) m ds = r 1 dr r 1 dt r dω, where dω = dθ sn θd on S. In [], t s shown that ths space-tme model can be expressed as a multply generalzed Robertson-Walker space-tme,.e., where ds = dt b 1(t)dr b (t)dω, m b 1 (t) = F 1 (t) 1 and b (t) = F 1 (t) also ( ) m r t = F(r) = m arccos r(m r) such that m lm r m F(r) = mπ and lm r 0 F(r) = 0. Moreover, we also need to mpose the above multply generalzed Robertson-Walker space-tme model for the Schwarzschld black hole to be Rcc-flat due to the fact that the Schwarzschld black hole s Rcc-flat (see also the revew of Mguel Sánchez n AMS for []). Kasner Space-tme We consder the Kasner space-tme as a Lorentzan multply warped product (see [64]). A Lorentzan multply warped product (M,g) of the form M = (0, ) t p 1R t p R t p 3R wth the metrc g = dt t p 1 dx t p dy t p 3 dz s sad to be the Kasner space-tme f p 1 p p 3 = (p 1 ) (p ) (p 3 ) = 1 (see [49]). It s known by [43] that 1/3 p 1,p,p 3 < 1. It s also known that, excludng the case of two p s zero, then one p s negatve and the other two are postve. Thus we may assume that 1/3 p 1 < 0 < p p 3 < 1 by excludng the case of two p s zero and one p equal to 1. Furthermore, the only soluton n whch p = p 3 s gven by p 1 = 1/3 and p = p 3 = /3. Note also that snce 1/3 p 1,p,p 3 < 1, we have to assume B to be (0, ). Clearly, the Kasner space-tme s globally hyperbolc (see [75]). By makng use of the results n [75], t can be easly seen that the Kasner spacetme s future-drected tme-lke and future-drected null geodesc complete but t s past-drected tme-lke and past-drected null geodesc ncomplete. Moreover, t s also space-lke geodesc ncomplete. Notce that the Kasner space-tme s Ensten wth λ = 0 (.e., Rcc-flat) (see [49] and page 135 of [59]) and hence has constant scalar curvature as zero. Ths fact can be proved as a partcular consequence of our results n the next secton, namely by usng Theorem 3.3. Statc Bañados-Tetelbom-Zanell (BTZ) Space-tme

8 8 FERNANDO DOBARRO AND BÜLENT ÜNAL In [45], authors classfy (BTZ) black hole solutons n three dfferent classes as statc, rotatng and charged. Here, we wll only gve a bref descrpton of a statc BTZ space-tme n terms of Lorentzan multply warped products,.e., multply generalzed Robertson-Walker space-tmes (see also [8, 9, 6]). The lne element of a statc BTZ black hole soluton can be expressed as ds = N dr N dt r dω, where dω = dθ sn θd on S. The lne element of the Statc BTZ black hole space-tme model for the regon r < r H can be obtaned by takng N = m r l. In ths case, the space-tme model can be expressed as a multply generalzed Robertson- Walker space-tme,.e., where r H = l m b 1 (t) = ds = dt b 1(t)dr b (t)dω, m (F 1 (t)) l and b (t) = F 1 (t) also ( ) r t = F(r) = l arcsn such that lm F(r) = lπ and lm F(r) = 0. r r H r 0 Here, note that the constant scalar curvature τ of the multply generalzed Robertson- Walker space-tme ntroduced above s τ = 6/l (see [45]) or apply Corollary.7. Note that, n [45], they also classfy (ds) black hole soluton n three classes as statc, rotatng and charged, smlar to (BTZ) black hole solutons (see [8, 9, 6]). We now state a couple of results whch we wll frequently be appled along ths artcle. The frst one s an easy computaton whch we wll show explctly below. Let (M,g) be an n-dmensonal pseudo-remannan manfold. For any t R and v C>0 (B) = {v C (B) : v > 0}, (.) grad g v t = tv t 1 grad g v r H g v t = t[(t 1)v t grad g v g v t 1 g v] [ g v t v t = t (t 1) grad ] g v g v gv. v The second one s a lemma that follows (for a proof and some extensons as well as other useful applcatons, see Secton of [9]). Lemma.8. Let (M,g) be an n-dmensonal pseudo-remannan manfold. Let L g be a dfferental operator on C>0 (M) defned by k g v a (.3) L g v = r v a, where r,a R and ζ := () k r a, η := k r a. Then, (.4) L g v = (η ζ) grad g v g v ζ gv v.

9 CURVATURE OF MULTIPLY WARPED PRODUCTS 9 () If ζ 0 and η 0, for α = ζ η ζ and β =, then we have η (.5) L g v = β gv 1 α. v 1 α 3. Specal Multply Warped Products 3.1. Ensten Rcc Tensor. In ths secton, we state some condton to guarantee that a multply generalzed Robertson-Walker space-tme s Rcc-flat or Ensten. Now, we recall some elementary facts about Ensten manfolds startng from ts defnton. Recall that an n-dmensonal pseudo-remannan manfold (M,g) s sad to be Ensten f there exsts a smooth real-valued functon λ on M such that Rc = λg, and λ s called the Rcc curvature of (M,g) (see also page 7 of [6]). Remark 3.1. Concernng to ths noton, t should be ponted out: (1) If (M,g) s Ensten and n 3, then λ s constant and λ = τ/n, where τ s the constant scalar curvature of (M,g). () If (M,g) s Ensten and n =, then λ s not necessarly constant. (3) If (M,g) has constant sectonal curvature k, then (M,g) s Ensten wth λ = k(n 1) and has constant scalar curvature τ = n(n 1)k. (4) (M,g) s Ensten wth Rcc curvature λ and n = 3, then (M,g) s a space of constant (sectonal) curvature K = λ/. (5) If (M,g) s a Lorentzan manfold then (M,g) s Ensten f and only f Rc(v,v) = 0, for any null vector feld v on M. By usng Proposton.5, we easly obtan the Rcc curvature of Lorentzan multply warped products, (M,g) of the above form. Proposton 3.. Let M = I b1 F 1 bm F m be a multply generalzed Robertson- Walker space-tme wth the metrc g = dt b 1 g F 1 b mg Fm also let t X(I) and v X(F ), for any {1,,m}. If v = m v X(F), then ( Rc t v, ) t v = b b ( Rc F (v,v ) k=1,k ( b b (s 1)(b ) b ) k b ) s k g F (v,v ) s b k b Proof. By substtutng X = m t v and Y = m t v and by notng that grad B b = b, g B ( t, ) t ( B b = b and Rc B t, t result. ( = 1, g B (grad B b,grad B b ) = (b ), H b B t, ) = b, ) t = 0 and by usng Proposton.5, we obtan the The followng result can be easly proved by substtutng v j = 0 for any j {1,,m} {} and v 0, n Proposton 3. along wth the method of separaton of varables. Theorem 3.3. Let M = I b1 F 1 bm F m be a multply generalzed Robertson- Walker space-tme wth the metrc g = dt b 1 g F 1 b m g F m. The space-tme (M,g) s Ensten wth Rcc curvature λ f and only f the followng condtons are satsfed for any {1,,m}

10 10 FERNANDO DOBARRO AND BÜLENT ÜNAL (1) each fber (F,g F ) s Ensten wth Rcc curvature λ F for any {1,,m}, b () s = λ and b (3) λ F b b (s 1)(b ) b b b k s k = λb b k k=1,k Remark 3.4. In Theorem 3.3, Equaton (3) can be expressed n dfferent forms and here we want to present some of them. By applyng Equaton., we can have (E grw ) or equvalently, (E grw ) λ F b λ F b 1 (b s ) s b s b b b (b ) b b b b k=1,k k=1 s k b k b k = λ, s k b k b k = λ. 3.. Constant Scalar Curvature. It s possble to obtan equvalent expressons for the scalar curvature n Corollary.7, namely the followng just follows from Equaton., (sc grw ) τ = [s b b (bs ) b s ] τ F b k=1,k s k s b b b k b k. Snce s 0 and s s = s (s 1) 0, by Lemma.8, there results (sc grw ) τ = 4s s 1 s 1 Thus, defnng ψ = b, results (sc grw ) τ = 4s ψ s 1 ψ s 1 (b ) b s 1 ψ τ F 4 s 1 τ F b k=1,k k=1,k s k s (ψ s k s b b b k b k. s 1 ) s 1 ψ (ψ s k 1 k ) s k 1 k Note that when m = 1 ths relaton s exactly that obtaned n [7] and [9] when the base has dmenson 1. The followng result just follows from the method of separaton of varables and the fact that each τ F : F R s a functon defned on F, for any {1,,m}. Proposton 3.5. Let M = I b1 F 1 bm F m be a multply generalzed Robertson- Walker space-tme wth the metrc g = dt b 1 g F 1 b m g F m. If the space-tme (M,g) has constant scalar curvature τ, then each fber (F,g F ) has constant scalar curvature τ F, for any {1,,m}. As one can notce from the above formula, t s extremely hard to determne general solutons for warpng functons whch produce an Ensten, or wth constant scalar curvature multply generalzed Robertson-Walker space-tme. Note that non-lnear second order dfferental equatons need to be solved accordng Theorem 3.3. Further note that there s only one dfferental equaton and m dfferent warpng functons n Corollary.7. Therefore nstead of gvng a general answer to the exstence of warpng functons to get an Ensten, or wth constant scalar curvature, space-tme, we smplfy ths problem and consder some specfc cases n mentoned Sectons 4 and 5. ψ.

11 CURVATURE OF MULTIPLY WARPED PRODUCTS Generalzed Kasner Space-tme In ths secton we gve an extenson of Kasner space-tmes and consder ther scalar and Rcc curvatures. Defnton 4.1. A generalzed Kasner space-tme (M, g) s a Lorentzan multply warped product of the form M = I p 1F 1 pmf m wth the metrc g = dt p 1 g F1 pm g Fm, where : I (0, ) s smooth and p R, for any {1,,m} and also I = (t 1,t ) wth t 1 < t. Notce that a Kasner space-tme can be obtaned out of a form defned above by takng = Id (0, ) wth m = 3 and I = (0, ), where Id (0, ) denotes for the dentty functon on (0, ) (see [43]). From now on, for an arbtrary generalzed Kasner space-tme of the form n Defnton 4.1, we ntroduce the followng parameters (ζ;η) ζ := s l p l and η := s l p l. l=1 Remark 4.. Note that ζ 0 mples η 0 and n ths case, defnng S = l=1 s l, results η ζ 1. The latter s for example consequence of the Hölder nequalty (compare wth S page 186 of [38]). By applyng Theorem 3.3, we can easly state the followng result and later we wll examne the solvablty of the dfferental equatons theren. Proposton 4.3. Let M = I 1 F 1 m F m be a generalzed Kasner space-tme wth the metrc g = dt p 1 g F1 pm g Fm. Then the space-tme (M,g) s Ensten wth Rcc curvature λ f and only f (1) each fber (F,g F ) s Ensten wth Rcc curvature λ F for any {1,,m}, () λ = ( p l) s l = (η ζ) ( ) l=1 (3) λ F p p p l [(ζ 1) ( ) ] = λ. ζ and Remark 4.4. Moreover, f n Proposton 4.3 we assume that ζ 0 also, then by Remark 4. s η 0. Hence, (3) s equvalent to (E (3) gk ) and () s equvalent to (E () gk ) λ F p p ( ζ ) ζ ζ = λ, λ = ζ η ( η ζ ). Proof. (of Proposton 4.3 and Remark 4.4) In order to prove (3), note that Equaton (E grw ) says λ F p 1 ( ps ) s p (p ) ( p k) s p s k p = λ. k Hence, by Equaton (.), η ζ k=1,k λ F p p (p s 1) ( ) p p k=1,k s k p k = λ, l=1

12 1 FERNANDO DOBARRO AND BÜLENT ÜNAL and from here So, λ F p p (p s 1) λ F p p and by the defnton of ζ [( λ F p p 1 k=1,k s k p k ) s k p k k=1 [(ζ 1) ( ) ( ) p = λ. ( ) ] = λ, ] = λ. If furthermore ζ 0, applyng agan Equaton., results (E (3) gk ). On the other hand, from () of Theorem 3.3 and by Lemma.8 (a), λ = l=1 λ = (η ζ) ( ) ( p l) s l p, l ζ. Hence, f ζ 0 and as consequence η 0, applyng Lemma.8 (b), results (E () gk ). Note that, from now on and also ncludng the prevous result, when we apply Lemma.8, we denote the usual dervatve n equatons by means of the prme notaton. Remark 4.5. Note that the condtons ζ 0 and η 0 agree wth the condtons usually mposed n the classcal Kasner space-tmes, namely p 1 p p 3 = 1 and p 1 p p 3 = 1 (see [49]). It s easy to show that the unque possblty to construct an Ensten classcal Kasner manfold or a constant scalar curvature classcal Kasner manfold wth p 1 p p 3 = 0 s p 1 = p = p 3 = 0, so that we have just a usual product. Indeed, consderng (t) = t, t s possble to apply Proposton 4.3 and later Proposton 4.11, respectvely. Corollary 4.6. Under the hypothess of Proposton 4.3, along wth ζ 0 and η 0. Assume also that for all, ζ p 0 and η p ζ 0. Then, M s Ensten f and only f for any {1,,m}, (F,g F ) s Ensten Rcc curvature λ F and (4.1) where 0 < ψ := η p ζ η p. (ζ p ) η p ζ ψ ψ = λ F ψ ζ p η p ζ p Proof. Indeed, from equatons (E (3) gk ) and (E() gk ), λ F p = ζ ( η ζ ) p ( ζ ) η η ζ ζ ζ. Thus, snce for all, ζ p 0 and η p ζ 0, then applyng Lemma.8, the result just follows. Example 4.7. Under the condtons of the classcal Kasner metrcs, m = 3, p 1 p p 3 = 1 and p 1 p p 3 = 1, we have λ F = 0, ζ = 1 and η = 1. Hence the hypothess ζ p 0 and η p ζ 0, for all, mples that p 1 for all. In ths case, Equaton (4.1) s equvalent to 0 < ψ = and ψ = 0,.e., 0 < (t) = at b wth a,b 0 and,

13 CURVATURE OF MULTIPLY WARPED PRODUCTS 13 a b 0. Hence, from Equaton (E () gk ), (0, ) p 1 R p R p 3 R s Rcc flat space-tme. Corollary 4.8. Let us assume the hypothess of Corollary 4.6 and that for all, (F,g F ) s Rcc flat. Then, M s Ensten f and only f ψ = 0 wth 0 < ψ := η p ζ η p, for all. Proof. It s an mmedate consequence of Corollary 4.6 Corollary 4.9. Assume that (F,g F ) s Rcc flat for all. Let also ζ,η R \ {0} such that ζ = η and ψ(t) = at b wth a,b 0 and a b > 0. If ζ = ζ, η = η, ζ p 0 and η p ζ 0 for all, then M = (0, ) F 1 p 1 pm F m s a Rcc flat space-tme, where = ψ 1 ζ. Proof. It s suffcent to apply Corollary 4.8 and Proposton 4.3. Remark Note that Corollary 4.9 contans the classcal Kasner metrcs except the case n whch at least one p = 1 (really at most one could be 1 because η = p 1 p p 3 = 1). The followng just follows from Corollary.7 and agan we dscuss the exstence of a soluton for the dfferental equaton below. Proposton Let M = I 1 F 1 m F m be a generalzed Kasner space-tme wth the metrc g = dt p 1 g F1 pm g Fm. Then the space-tme (M,g) has constant scalar curvature τ f and only f (1) each fber (F,g F ) has constant scalar curvature τ F for any {1,,m}, and () τ = ζ [(ζ )ζ ) τ η]( F p, Remark 4.1. If ζ 0, then () n Proposton 4.11 s equvalent to τ = 4ζ ( ζ η ζ ) ζ η ζ η ζ τ F p. Proof. (of Proposton 4.11 and Remark 4.1) For each {1,,m}, let γ = p s 1 and ψ = γ, then by (sc grw ) and Equaton. there results τ = 4s s 1 γ [(γ 1) ( ) ] k=1,k γ γ k s k s s 1 s k 1 ( ) τ F 4 s 1 γ

14 14 FERNANDO DOBARRO AND BÜLENT ÜNAL Then we have τ = [( ) s 1 ( ) ] s p p 1 ( ) s k s p p k k=1,k = ζ m s p τ F p ( ) s 1 p 1 = ζ m s p [(ζ ) p ] ( ) = ζ [(ζ )ζ η]( ) k=1,k τ F p. τ F p τ F p s k p k ( ) Snce (ζ )ζ η 1 = (ζ 1) η = 0 f and only f p = 0 for all {1,,m}, f at least one p 0 there results by Equaton. τ = (ζ 1) 1 ( (ζ 1) η ) (ζ 1) η (ζ 1) η Hence, f ζ 0, applyng Lemma.8, τ F p. τ = 4ζ ( ζ η ζ ) ζ η ζ η ζ τ F p. Corollary Under the hypothess of Proposton 4.11 and ζ 0. Then, by changng varables as u = ζ η ζ, we conclude that the space-tme M has constant scalar curvature τ f and only f or equvalently τ = 4ζ u m ζ η u τ F u 4ζ ζ p η 4 1 η ζ u = τu τ F u η ζ p ζ. Remark If ζ 0 and there s only one fber,.e., n a standard warped product, the equaton n the prevous corollary corresponds to those obtaned n [7, 9]. Example Let us assume that ζ 0 and each F s scalar flat, namely τ F = 0. Hence, equaton n the prevous corollary s wrtten as 4ζ ζ η u = τu.

15 CURVATURE OF MULTIPLY WARPED PRODUCTS 15 Thus all the solutons have the form Ae τ ζ η 4ζ t Be τ ζ η 4ζ t f τ < 0, u(t) = At B f τ = 0, Ae τ ζ η 4ζ t Be τ ζ η 4ζ t f τ > 0, wth constants A and B such that u > 0. If ζ = 0, by Proposton 4.11, we look for postve solutons of the equaton τ = η ( ), > 0. Snce η > 0, the latter s equvalent to ( ) τ τ )( = 0, η η > 0. Solutons of the equaton above are gven as, where C s a postve constant. (t) = Ce ± τ η t, Note that ths example nclude the stuaton of the classcal Kasner space-tmes n the framework of scalar curvature. Compare wth the results about Ensten classcal Kasner metrcs n Remark 4.5 and Example Dmensonal Space-tme Models We frst gve a classfcaton of 4-dmensonal warped product space-tme models and then consder Rcc tensors and scalar curvatures of them. Defnton 5.1. Let M = I b1 F 1 bm F m be a multply generalzed Robertson- Walker space-tme wth metrc g = dt b 1 g F 1 b m g F m. (M,g) s sad to be of Type (I) f m = 1 and dm(f) = 3. (M,g) s sad to be of Type (II) f m = and dm(f 1 ) = 1 and dm(f ) =. (M,g) s sad to be of Type (III) f m = 3 and dm(f 1 ) = 1, dm(f ) = 1 and dm(f 3 ) = 1 Note that Type (I) contans the Robertson-Walker space-tme. The Schwarzschld black hole soluton can be consdered as an example of Type (II). Type (III) ncludes the Kasner space-tme Type (I). Let M = I b F be a Type (I) warped product space-tme wth metrc g = dt b g F. Then the scalar curvature τ of (M,g) s gven as τ = τ ( F b b 6 b (b ) ) b. The problem of constant scalar curvatures of ths type of warped products, known as generalzed Robertson-Walker space-tmes s studed n [3], ndeed, explct solutons to warpng functon are obtaned to have a constant scalar curvature. If v s a vector feld on F and x = v, then t Rc(x,x) = Rc F (v,v) ( bb (b ) ) g F (v,v) 3 b b.

16 16 FERNANDO DOBARRO AND BÜLENT ÜNAL In [], explct solutons are also obtaned for the warpng functon to make the space-tme as Ensten when the fber s also Ensten. 5.. Type (II). Let M = I b1 F 1 b F be a Type (II) warped product space-tme wth metrc g = dt b 1 g F 1 b g F. Then the scalar curvature τ of (M,g) s gven as τ = τ F b b 1 4 b b 1 b Note that τ F1 = 0, snce dm(f 1 ) = 1. ( b b ) 4 b 1 b b 1 b. If v s a vector feld on F, for any {1,} and x = t v 1 v, then Rc(x,x) = Rc F (v,v ) b 1 b b 1 b ( b 1 b 1 b 1b ) 1 b g F1 (v 1,v 1 ) b ( b b (b ) b b ) b 1 g F (v,v ) b 1 Note that Rc F1 0, snce dm(f 1 ) = 1. Classfcaton of Ensten Type (II) generalzed Kasner space-tmes: Let M = I p 1 F 1 p F be an Ensten Type (II) generalzed Kasner space-tme. Then the parameters ntroduced before Proposton 4.3 are gven by ζ = p 1 p, η = p 1 p. Hence the latter arses (E K II) (η ζ) ( ) ζ [ = λ p 1 (ζ 1) ( ) ] = λ λ F p p [(ζ 1) ( ) ] = λ. The last equaton mples n partcular that λ F s constant. Let the system { ( σ ( σ ) = ν σ ;ν) 0 <. where ν and σ are real parameters. All ts solutons σ have the form Ae ν t Be ν t f ν < 0, σ (t) = At B f ν = 0, Ae ν t Be ν t f ν > 0, wth constants A and B such that > 0. Furthermore, let the ( σ ;ν) modfed system ( σ ) = ν σ ( σ ;ν; ) [( σ ) ] = ν( σ ) > 0. Note that ν must be > 0. It s easy to verfy that all ts solutons are gven by where A s a postve constant. Consder now two cases, namely σ (t) = Ae ± ν t,

17 CURVATURE OF MULTIPLY WARPED PRODUCTS 17 ζ = 0: Frst of all, note that p = 1 p 1 and η = 3 p 1. η = 0: Thus, p = 0, for all and 0 = λ = λ F. Thus the correspondng metrc s dt g F1 g F. η 0: Then p 1 0, p 0 and η ( ) = λ [ (E K II) p 1 ( ) ] = λ λ F p 1 1 p 1 [ ( ) ] = λ. If (E K II) λ F = 0: then λ = 0 and s constant 0. Thus the correspondng metrc s dt p 1 0 g F1 p 0 g F. λ F 0: then λ F p = 3 λ, as consequence s constant and consderng the system ths gves a contradcton. 1 ζ 0: Hence η 0 and by Remark 4.4 the system reduces to ζ ( ζ η ζ ) = λ η ζ η ζ p 1 ( ζ ) ζ ζ = λ λ F p p ζ ( ζ ) ζ = λ, η = ζ : So p 1 0 and ether p = 0 or p = p 1. If λ = 0: then λ F = 0. Thus, the correspondng metrc s dt p 1 g F1 p g F, where satsfes ( ζ ;0). λ 0: then ζ = p 1 and p = 0. Hence, by the thrd equaton λ F = λ. Thus, the correspondng metrc s where satsfes ( ζ ;λ). dt ζ g F1 g F, η ζ : Then p 0 and p 1 λ F p p = So, f ( ) p1 1 λ. p λ = 0: then the frst equaton mples, ζ = (At B) ζ η and ( ζ ) = ζ ( ) ζ η η 1 (At B) ζ η A. λ F = 0: then applyng the thrd equaton results A = 0, so ζ s constant and s a postve constant 0. Thus the correspondng metrc s dt p 1 0 g F1 p 0 g F.

18 18 FERNANDO DOBARRO AND BÜLENT ÜNAL λ F 0: then p 1 = 0, hence p = ζ, η ζ = 1. So, by the thrd equaton λ F = A < 0. Thus the correspondng metrc s dt g F1 p g F, wth as above. λ 0: then p 1 0, hence λ ( F p = 1 p ) 1 λ. p λ F = 0: then p 1 = p and the system can be reduced to 3 (ζ 1 3) ζ 1 3 = 1 ( ζ ) 3 ζ = λ whch s equvalent to the solvable system ( ζ ;3λ; ). Note that λ must be > 0. λ F 0: then s constant and ths gves a contradcton. The table that follows specfes the only possble Ensten generalzed Kasner spacetmes of Type (II) wth the correspondng parameters. The last column ndcates the functon or the system whch t satsfes. η ζ η ζ λ λ F p 1 p metrc dt g F1 g F p p 1 dt p1 0 g F1 p1 0 g F 0 = cte > p p 1 no metrc - 0 ζ , p 1 dt p1 g F1 p g F ( ζ ; 0) 0 ζ 1 0 λ 0 0 dt p1 g F1 g F ( ζ ; λ) p 1 0 dt p1 0 g F1 p 0 g F 0 = cte > < dt g F1 p g F ( η ζ ; 0) > 0 0 p 0 dt p1 g F1 p1 g F ( ζ ; 3λ; ) p 1 0 no metrc - Table 1 Note that Corollary 4.9 cannot be appled n the stuatons above. Classfcaton of the Type (II) generalzed Kasner space-tmes wth constant scalar curvature Let M = I p 1 F 1 p F be a Type (II) generalzed Kasner space-tme wth constant scalar curvature. Then the parameters ntroduced before Proposton 4.11 satsfy ζ = p 1 p, η = p 1 p and (csc K II.a) τ = ζ [(ζ )ζ η]( ) τ F p. Note that τ F must be constant f there exst a postve soluton of (csc K II.a) (see also Proposton 3.5). We consder two prncpal cases wth dfferent subcases. ζ = 0: If η = 0: then p 1 = p = 0, τ = τ F and the correspondng metrc s dt g F1 g F. η 0: then p = 1 p 1 and η = 3 p 1 = 6p. The equaton (csc K II.a) reduces to (csc K II.b) τ = η ( ) τ F p.

19 CURVATURE OF MULTIPLY WARPED PRODUCTS 19 1 η ζ 0: mples η 0 and consderng 0 < u = ( ζ ζ ), Corollary 4.13 arses the relaton (csc K II.c) 4 1 η ζ u = τu τ F u η p ζ ζ. η = ζ : Then p 1 0, ether p = 0 or p = p 1, and u = ζ. τ F = 0: So the equaton reduces to u = τu. τ F 0: If p = 0: the equaton reduces to p = p 1 : u = (τ τ F )u. u = τu τ F u 1 3. η ζ : Then p 0 and η ζ 1 3. τ F = 0: (5.1) 4 1 η ζ u = τu τ F 0: (csc K II.c) 4 1 η ζ u = τu τ F u η p ζ ζ. Note that a partcular subcase s η ζ = 1 3. In fact, n ths case, p 1 = p = ζ (see Remark 4.) and the latter equaton reduces to the 3 non-homogeneous lnear ordnary dfferental equaton 3u = τu τ F. Synthetcally, rememberng that n each case the correspondng metrc may be wrtten as dt p 1 g F1 p g F, we fnd that the only possbltes to have constant scalar curvature n a generalzed Kasner space-tme of type (II) are generated by η ζ η τ ζ F p 1 p eq τ F 0 0 τ = τ F 3 0 p p 1 τ = η ( ) 3 0 p p 1 τ = η ( ) τ F p ζ 0 ζ u = τu; u = ζ ζ 0 ζ p 1 u = τu; u = ζ ζ 0 ζ u = (τ τ F ); u = ζ ζ 0 ζ p 1 u = τu τ F u 1 3; u = ζ ζ 0 η p 1 0 (5.1); u = ( ζ ) 1 η ζ 1 η ζ ζ 0 η 0 1, p 1 0 (csc K II.c);u = ( ζ ) ζ ζ 0 1 ζ ζ u = τu τ F ; u = 3 ζ Table

20 0 FERNANDO DOBARRO AND BÜLENT ÜNAL where the condtons for τ must be mposed by the exstence of postve solutons of the ordnary dfferental equatons of the last column, on the correspondng nterval I Type (III). Let M = I b1 F 1 b F b3 F 3 be a type (III) warped product space-tme wth metrc g = dt b 1 g F 1 b g F b 3 g F 3. Then the scalar curvature τ of (M,g) s gven as ( b τ = 1 b b 3 b 1 b b b 3 b 1 b b 3 b 1 b Note that τ F = 0, snce dm(f ) = 1, for any {1,,3}. b 1 b 3 b b 3 b 1 b 3 If v s a vector feld on F, for any {1,,3} and x = t v 1 v v 3, then ( Rc(x, x) = b 1 b 1 b 1b 1 b b 1b ) 1 b 3 g F1 (v 1,v 1 ) b b 3 ( b b b b b 1 b b ) b 3 g F (v,v ) b 1 b 3 ( b 3 b 3 b 3b 3 b 1 b 3b ) 3 b g F3 (v 3,v 3 ) b 1 b b 1 b b 3 b 1 b b 3 Note that Rc F 0, snce dm(f ) = 1, for any {1,,3}. Classfcaton of Ensten Type (III) generalzed Kasner space-tmes Let M = I p 1F 1 p F p 3F 3 be an Ensten Type (III) generalzed Kasner spacetme. Then the parameters ntroduced before Proposton 4.3 satsfy ζ = p 1 p p 3, η = p 1 p p 3. Hence the latter arses (η ζ) ( ) ζ [ = λ p 1 (ζ 1) ( ) ] (E K III) = λ [ p (ζ 1) ( ) ] = λ [ p 3 (ζ 1) ( ) ] = λ. Note that addng the last three equatons, there results (5.) ζ [(ζ 1) ( ) ] = 3λ Consder now two cases, namely ζ = 0: Then applyng (5.), we obtan λ = 0. η = 0: Thus p = 0 for all. Hence the correspondng metrc s dt g F1 g F g F3. η 0: the system reduces to η ( ) (E K III) = 0 [ p ( ) ] = 0 for all = 1,,3. ). then s constant 0. Thus the correspondng metrc s dt p 1 0 g F1 p 0 g F p 3 0 g F3

21 (E K III) (5.3) CURVATURE OF MULTIPLY WARPED PRODUCTS 1 ζ 0: Thus η 0 and by Remark 4.4 the system reduces to ζ ( ζ η ζ ) = λ η p ζ ζ η ζ ( ζ ) ζ = λ for all = 1,,3. Addng the last three equatons n (E K III), we obtan that ( ζ ) ζ = 3λ η = ζ : Then (5.) and (5.3), gve λ = 0. Thus, the correspondng metrc s dt p 1 g F1 p g F p 3 g F3, where satsfes ( ζ ;0). η ζ : Then at least two p s are 0. So f λ = 0: then the frst equaton mples ζ = (At B) ζ η and ( ζ ) = ζ ( ) ζ η η 1 (At B) ζ η A. Then by (5.3) results A = 0, so ζ s constant and s a postve constant 0. Thus the correspondng metrc s dt p 1 0 g F1 p 0 g F p 3 0 g F3. λ 0: then all p s are 0 and all of them are equals, so that p 1 = p = p 3 = ζ ζ. So η = 3 3 and η ζ = 1. Thus the system reduces to 3 3 (ζ 1 3) ζ 1 3 = 1 ( ζ ) 3 ζ = λ, whch s equvalent to the solvable system ( ζ ;3λ; ). Note that λ must be > 0. The table that follows specfes the only possble Ensten generalzed Kasner space-tmes of type (III) wth the correspondng parameters. Lke for the table of Type (II), the last column ndcates the functon or the system whch t satsfes. Ths example may be easly generalzed to the stuaton all the F s are Rcc flat, consderng S = m s > 1 nstead of 3. η ζ η ζ λ p 1 p p 3 metrc dt g F1 g F g F p 1 p p 3 dt p1 0 g F1 p 0 g F p3 0 g F3 0 = cte > 0 0 ζ 1 0 p 1 p p 3 dt p1 g F1 p g F p3 g F3 ( ζ ; 0) p 1 p p 3 dt p1 0 g F1 p 0 g F p3 0 g F3 0 = cte > > 0 p 1 p 1 p 1 dt p1 g F1 p1 g F p1 g F3 ( ζ ; 3λ; ) Table 3 Classfcaton of Type (III) generalzed Kasner space-tmes wth constant scalar curvature Let M = I p 1 F 1 p F p 3 F 3 be a Type (III) generalzed Kasner manfold wth constant scalar curvature. Then the parameters ntroduced before Proposton 4.11 satsfy ζ = p 1 p p 3, η = p 1 p p 3. Thus, ths case s already ncluded n the analyss of Example 4.15.

22 FERNANDO DOBARRO AND BÜLENT ÜNAL We wll close ths secton by an example and the followng comment whch gves some prelmnary deas about our future plans on ths topc (see also the last secton for detals). Example 5.. Let M = I p 1 S 3 p S be a generalzed Kasner manfold wth constant scalar curvature. Then the parameters ntroduced before Proposton 4.11 are gven by ζ = 3p 1 p, η = 3p 1 p. Consder now p 1 = 1 and p = 1, then ζ = 1 and η = 5. Hence, applyng Corollary 4.13 the latter condtons arse for u = 3 the problem (5.4) 3 u τu = τ S3 u 1 3 τ S u 1 3, u > 0, where τ S3,τ S > 0 are the constant scalar curvatures of the correspondng spheres. Note that the equaton n (5.4) has always the constant soluton zero and there exsts τ 1 > 0 such that for τ = τ 1 there s only one constant soluton of (5.4) and for any τ > τ 1 there are two constant solutons of (5.4), so that there exsts a range of τ s, (τ 1, ), where the problem (5.4) has multplcty of solutons; whle there s no constant solutons when τ < τ 1. On the other hand, as n Example 5., consderng S 3 nstead of S wth the same values of p 1 and p,.e., M = I p 1 S 3 p S 3, results ζ = 3p 1 3p = 0, η = 3p 1 3p = 6. Hence, applyng Proposton 4.11 the latter condtons arse the problem (5.5) { 6( ) = τ τ S3 τ S3 4 > 0. The equaton n (5.5) does not have the constant soluton zero. Furthermore there s no constant soluton of (5.5) f τ < τ S3, there s only one constant soluton of (5.5) f τ = τ S3 and two constant solutons of (5.5) f τ > τ S3. The cases consdered above are just some examples for the dfferent type of dfferental equatons nvolved n the problem of constant scalar curvature when the dmensons, curvatures and parameters have dfferent values. In a future artcle, we deal wth the problem of constant scalar curvature of a pseudo-remannan generalzed Kasner manfolds wth a base of dmenson greater than or equal to 1. Ths problem carres to nonlnear partal dfferental equatons wth concave-convex nonlneartes lke n (5.4), among others. Nonlnear ellptc problems wth such nonlneartes have been extensvely studed n bounded domans of R n, after the central artcle of Ambrosett, Brezs and Ceram [1], n whch the authors studed the problem of multplcty of solutons under Drchlet condtons. The problem of constant scalar curvature n a generalzed Kasner manfolds wth base of dmenson greater than or equal to 1 s one of the frst examples where those nonlneartes appear naturally. Another related case s the base conformal warped products, studed n [9]. 6. BTZ (1) Black Hole Solutons Now we consder BTZ (1)-Black Hole Solutons and gve another characterzaton of (BTZ) black hole solutons mentoned n Secton (for further detals see [8, 9, 45, 6]) n order to apply the results obtaned n ths paper. All the cases consdered n [45], can be obtaned applyng the formal approach that follows. By consderng the correspondng square lapse functon N, the related 3- dmensonal, (1)-space-tme model can be expressed as a (1) multply generalzed Robertson-Walker space-tme,.e., (6.1) ds = dt b 1 (t)dx b (t)dφ,

23 CURVATURE OF MULTIPLY WARPED PRODUCTS 3 where (6.) { b 1 (t) = N(F 1 (t)) b (t) = F 1 (t), wth (6.3) F(r) = r a 1 N(µ) dµ and F 1 the nverse functon of F (assumng that there exsts) and a s an approprate constant that s most of the tme related to the event horzon. Recallng (6.4) 1 = d dt (F F 1 )(t) = F (F 1 (t))(f 1 ) (t), we obtan the followng propertes by applyng the chan rule. Here, note that all the functons depend on the varable t and the dervatves are taken wth respect to the correspondng arguments. b 1 = N(b ) b = N(F 1 ) = b 1 b = N(b ) b 1 = b = N (b )b = N (b )b 1 b 1 = N (b )b b 1 N (b )b 1 = N (b )b 1 (N (b )) b 1. Thus, (6.5) b 1 b 1 = N (b )b 1 (N (b )) = N (b )N(b ) (N (b )) = (N N) (b ) = 1 (N ) (b ) b = N (b ) N(b ) b b = 1 (N ) (b ) b b 1 b 1 b b = b b On the other hand, by Corollary.7 appled to the metrc (6.1), wth s 1 = s = 1. The scalar curvature of the correspondng space-tme s gven by ( b τ = 1 b b 1 b ) ( b 1 b ) b 1 b b = 1 b (6.6) ( b 1 b = N (b )N(b ) (N (b )) N (b ) N(b ) ) b = (N ) (b ) (N ) (b ) b Note that, the latter s an expresson of the scalar curvature as an operator n the square lapse functon. Remember that b = F 1. About the Rcc tensor, applyng our Proposton 3. and Theorem 3.3 and by consderng agan s 1 = s = 1, Theorem 3.3 says that the metrc (6.1) s Ensten wth λ f

24 4 FERNANDO DOBARRO AND BÜLENT ÜNAL and only f b 1 b = λ b 1 b b (6.7) 1 b 1 b = λ b 1 b 1 b b b b 1 = λ. b b b 1 On the other hand by makng use of (6.5), the system (6.7) s equvalent to (all the functons are evaluated n r = b ) (6.8) or moreover to the followng (6.9) Thus, we have (N ) (N ) = λ r (N ) (N ) = λ r (N ) = λ, r (N ) (N ) = λ r (N ) = λ. r (6.10) (N ) = λ. Hence, (6.11) N (r) = λ r c 1 r c, wth c 1 and c sutable constants. But, snce (N ) (r) = λr c 1, the second equaton of (6.9) s verfed f and only f c 1 = 0. So, we have proved the followng results. Proposton 6.1. Suppose that we have a ( 1)-Lorentzan multply warped product wth the metrc gven by (6.1), where b 1 and b satsfyng both (6.) and (6.3). The space-tme s Ensten wth Rcc curvature λ f and only f the square lapse functon N satsfes (6.11), wth c 1 = 0 and a sutable constant c. Notce that the statc (BTZ) and the statc (ds) black hole solutons consdered n [45] satsfy Proposton 6.1. Thus they are Ensten multply warped product spacetmes. Remark 6.. Remark that f N satsfes (6.11) wth c 1 = 0, then an applcaton of (6.6) gves the constancy of the scalar curvature τ = 3λ, as desred. Note that ths result agrees wth the ones obtaned n [45]. Furthermore, the followng just follows from the soluton of the nvolved second order lnear ordnary dfferental equaton arsen by the expresson (6.6). Proposton 6.3. Suppose that there s a ( 1)-Lorentzan multply warped product wth the metrc gven by (6.1), where b 1 and b verfyng (6.) and (6.3). The spacetme has constant scalar curvature τ = λ f and only f the square lapse functon N has the form (6.1) N (r) = c 1 1 r λ 6 r c, wth sutable constants c 1 and c. Note that Proposton 6.3 agrees wth Remark 6..

25 CURVATURE OF MULTIPLY WARPED PRODUCTS 5 7. Conclusons Now, we would lke to summarze the content of the paper and to make some concludng remarks. In a bref, we studed expressons that relate the Rcc (respectvely scalar) curvature of a multply warped product wth the Rcc (respectvely scalar) curvatures of ts base and fbers as well as warpng functons. By usng expressons obtaned n the paper, we proved necessary and suffcent condtons for a multply generalzed Robertson-Walker space-tme to be Ensten or to have constant scalar curvature. Furthermore, we ntroduced and consdered a knd of generalzaton of Kasner spacetmes, whch s closely related to recent applcatons n cosmology where metrcs of the form k (7.1) ds = dt e α dx, wth α = α (t), are frequently consdered (see [4, 7]; for other recent topcs concerned Kasner type metrcs see for nstance [6, 39, 47, 48, 61, 66, 76, 77]). If each warpng functon e α s expressed as (7.) e α = p, wth = e α p, for sutable p s, then (7.1) takes the form (7.3) ds = dt k p dx. Our generalzaton of Kasner space-tmes corresponds exactly to the case n whch the s are ndependent of. More explctly, α = p α n Equaton (7.), wth α = α(t) for a suffcently regular fxed functon. Note that a classcal Kasner space-tme corresponds to the case of α 1 (see [58] also). By applyng Lemma.8, we obtaned useful expressons for the Rcc tensor and the scalar curvature of generalzed Robertson-Walker and generalzed Kasner space-tmes. These expressons allowed us to classfy possble Ensten (respectvely wth constant scalar curvature) generalzed Kasner space-tmes of dmenson 4. We also obtaned some partal results for grater dmensons. Fnally, n order to study curvature propertes of multply warped product spacetmes assocated to the BTZ ( 1)-dmensonal black hole solutons, we made applcatons of the prevously obtaned curvature formulas. As a consequence, we characterzed the Ensten BTZ (respectvely wth constant scalar curvature), n terms of the square lapse functon. In forthcomng papers we plan to focus on a specfc generalzaton of the structures studed here, whch s partcularly useful n dfferent felds such as relatvty, extra-dmenson theores (Kaluza-Klen, Randall-Sundrum), strng and super-gravty theores, spectrum of Laplace-Beltram operators on p-forms, among others. Roughly speakng, we wll consder a mxed structure between a multply warped product and a conformal change n the base. Naturally, our man nterest s the study of curvature propertes. As we have made progress on ths subject, we realzed that these curvature related propertes are nterestng and worth to study not only for the physcal pont of vew (see for nstance, the several recent works of Gauntlett, Maldacena, Arguro, Schmdt, among many others), but also for exclusve nonlnear partal dfferental equatons nvolved. Indeed, the curvature related questons arse problems of exstence, unqueness, bfurcaton, study of crtcal ponts, etc. (see Example 5. above and the dfferent works of Aubn, Hebey, Yau, Ambrosett, Choquet-Bruat among others).