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19–pg NON-DIAGONAL METRIC ON A PRODUCT RIEMANNIAN MANIFOLD 5 1 0 Rafik Nasri and Mustapha Djaa 2 n LaboratoryofGeometry, Analysis,ControlandApplications u UniversitédeSa¨ıda J BP138,En-Nasr,20000Sa¨ıda,Algeria 6 2 (Communicated by the associate editor name) ] G Abstract. Inthis paper, weconstruct the symmetrictensor fieldGf1f2 and D hf1f2 on a product manifold and we give conditions under which Gf1f2 be- comesametrictensor,thesestensorsfieldswillbecalledthegeneralizedwarped . h product,andthenwedevelopanexpressionofcurvaturefortheconnectionof t the generalized warped product in relation to those corresponding analogues a of its base and fiber and warping functions. By constructing a frame field in m M1×f1f2 M2 with respect to the Riemannian metric Gf1f2 and hf1f2, then [ we calculate the Laplacian−Beltrami operator of a function on a generalized warped product which may be expressed in terms of the local restrictions of 3 thefunctionstothebaseandfiber. Finally,weconcludesomeinterestingrela- v tionshipsbetweenthegeometryofthecouples(M1,g1)and(M2,g2)andthat 8 of(M1×M2,hf1f2). 0 3 0 1. Introduction. The warped product provides a way to construct new pseudo- 0 rieman nian manifolds from the given ones, see [6],[3] and [2]. This construction . 1 hasusefulapplicationsingeneralrelativity,inthestudyofcosmologicalmodelsand 0 black holes. It generalizes the direct product in the class of pseudo-Riemannian 5 manifolds and it is defined as follows. Let (M ,g ) and (M ,g ) be two pseudo- 1 1 2 2 1 Riemannian manifolds and let f : M −→ R∗ be a positive smooth function on : 1 1 v M , the warpedproductof(M ,g ) and(M ,g )is the productmanifold M ×M 1 1 1 2 2 1 2 Xi equipped with the metric tensor gf1 := π1∗g1+(f ◦π1)2π2∗g2, where π1 and π2 are theprojectionsofM ×M ontoM andM respectively. ThemanifoldM iscalled r 1 2 1 2 1 a the base of (M ×M ,g ) and M is called the fiber. The function f is called the 1 2 f1 2 1 warping function. Thedoublywarpedproductisconstructionintheclassofpseudo-Riemannianman- ifoldsgeneralizingthe warpedproductandthe directproduct,itis obtainedbyho- motheticallydistortingthegeometryofeachbaseM ×{q}andeachfiber{p}×M 1 2 to get a new ”doubly warped” metric tensor on the product manifold and defined as follows. For i ∈ {1,2}, let M be a pseudo-Riemannian manifold equipped with i metric g , and f : M → R∗ be a positive smooth function on M . The well-know i i i i notion of doubly warped product manifold M × M is defined as the product 1 f1f2 2 2010 Mathematics Subject Classification. 53C21,53C50. Key words and phrases. warped products; generalized warped products; Ricci curvature; scalarcurvatureLaplacian−Beltramioperator. 1 2 RAFIK NASRI manifoldM =M ×M equippedwithpseudo-Riemannianmetricwhichisdenoted 1 2 by g , given by f1f2 g =(f ◦π )2π∗g +(f ◦π )2π∗g . f1f2 2 2 1 1 1 1 2 2 Whenthe warpingfunctions f =1orf =1 weobtaina warpedproductordirect 1 2 product. The paper is organized as follows. In section 2, we collect the basic material about Levi-Civita connection, horizontal and vertical lifts. In section 3, we con- sider the metric tensors g and g on manifolds M and M respectively and, for 1 2 1 2 a smooth function f on M , i =1,2, we define the symmetric tensors fields G i i f1f2 and h on M ×M relative to g , g and the warping functions f , f , then f1f2 1 1 1 2 1 1 we give the condition under which G becomes a metric tensor, this tensor field f1f2 will be referred to as the generalized warped product metric, next, we define also its cometric and we compute the gradients of the lifts of f , f . Morever, by con- 1 2 structingaframefieldinM × M withrespecttotheRiemannianmetricG , 1 f1f2 2 f1f2 then we calculate the Laplacian−Beltrami operator of a function on a generalized warped product which may be expressed in terms of the local restrictions of the functions to the base and fiber. To end this section, we conclude with some impor- tant relationships related to the harmonicity of function. In the final section, we compute the curvatures of generalizedwarped product h and we conclude with f1f2 someimportantrelationshipsbetweenthegeometryofthetriples(M ,g ),(M ,g ) 1 1 2 2 and that of (M ×M ,h ). 1 2 f1f2 2. Preliminaries. 2.1. Horizontal and vertical lifts. Throughout this paper M and M will be 1 2 respectivelym andm dimensionalmanifolds,M ×M theproductmanifoldwith 1 2 1 2 thenaturalproductcoordinatesystemandπ :M ×M →M andπ :M ×M → 1 1 2 1 2 1 2 M the usual projection maps. 2 WerecallbrieflyhowthecalculusontheproductmanifoldM ×M derivesfrom 1 2 that of M and M separately. For details see [6]. 1 2 Let ϕ in C∞(M ). The horizontal lift of ϕ to M ×M is ϕh =ϕ ◦π . One 1 1 1 1 2 1 1 1 can define the horizontal lifts of tangent vectors as follows. Let p ∈ M and let 1 1 X ∈ T M . For any p ∈ M the horizontal lift of X to (p ,p ) is the unique p1 p1 1 2 2 p1 1 2 tangent vector Xh in T (M ×M ) such that d π (Xh ) = X (p1,p2) (p1,p2) 1 2 (p1,p2) 1 (p1,p2) p1 and d π (Xh )=0. (p1,p2) 2 (p1,p2) We canalsodefine the horizontallifts ofvectorfields asfollows. LetX ∈Γ(TM ). 1 1 ThehorizontalliftofX toM ×M isthevectorfieldXh ∈Γ(T(M ×M ))whose 1 1 2 1 1 2 value at each (p ,p ) is the horizontal lift of the tangent vector (X )p to (p ,p ). 1 2 1 1 1 2 For (p ,p ) ∈ M ×M , we will denote the set of the horizontal lifts to (p ,p ) of 1 2 1 2 1 2 all the tangent vectors of M at p by L(p ,p )(M ). We will denote the set of the 1 1 1 2 1 horizontal lifts of all vector fields on M by L(M ). 1 1 The verticallift ϕv of a function ϕ ∈C∞(M ) to M ×M and the vertical lift 2 2 2 1 2 Xv of a vector field X ∈ Γ(TM ) to M ×M are defined in the same way using 2 2 2 1 2 the projection π . Note that the spaces L(M ) of the horizontal lifts and L(M ) 2 1 2 of the vertical lifts are vector subspaces of Γ(T(M ×M )) but neither is invariant 1 2 under multiplication by arbitrary functions ϕ∈C∞(M ×M ). 1 2 NON-DIAGONAL METRIC ON A PRODUCT RIEMANNIAN MANIFOLD 3 Observe that if { ∂ ,..., ∂ } is the local basis of the vector fields (resp. ∂x1 ∂xm1 {dx ,...,dx } is the local basis of 1-forms ) relative to a chart (U,Φ) of M and 1 m1 1 { ∂ ,..., ∂ }isthelocalbasisofthevectorfields(resp. {dy ,...,dy }thelocal ∂y1 ∂ym2 1 m2 basisofthe1-forms)relativetoachart(V,Ψ)ofM ,then{( ∂ )h,...,( ∂ )h,( ∂ )v, 2 ∂x1 ∂xm1 ∂y1 ...,( ∂ )v}isthelocalbasisofthevectorfields(resp. {(dx )h,...,(dx )h,(dy )v, ∂ym2 1 m1 1 ...,(dy )v} is the local basis of the 1-forms) relative to the chart (U ×V,Φ×Ψ) m2 of M ×M . 1 2 The following lemma will be useful later for our computations. Lemma 2.1. 1. Let ϕ ∈ C∞(M ), X ,Y ∈ Γ(TM ) and α ∈ Γ(T∗M ), i = 1,2. Let ϕ = i i i i i i i ϕh+ϕv, X =Xh+Xv and α,β ∈Γ(T∗(M ×M )). Then 1 2 1 2 1 2 i/ For all (i,I)∈{(1,h),(2,v)}, we have XI(ϕ)=X (ϕ )I, [X,YI]=[X ,Y ]I and αI(X)=α (X )I. i i i i i i i i i ii/ If for all (i,I)∈{(1,h),(2,v)} we have α(XI)=β(XI), then α=β. i i 2. Let ω and η be r-forms on M , i= 1,2. Let ω =ωh+ωv and η =ηh+ηv. i i i 1 2 1 2 We have dω =(dω )h+(dω )v and ω∧η =(ω ∧η )h+(ω ∧η )v. 1 2 1 1 2 2 Proof. See [4]. Remark 1. Let X be a vector field on M ×M , such that dπ (X)= ϕ(X ◦π ) 1 2 1 1 1 and dπ (X)=φ(X ◦π ), then X =ϕXh+φXv. 2 2 2 1 2 3. About generalized warped products. 3.1. The generalized warped product. let ψ : M → N be a smooth map betweensmoothmanifoldsandletg beametriconk-vectorbundle(F,P )overN. F The metric gψ : Γ(ψ−1F)×Γ(ψ−1F) → C∞(M) on the pull-back (ψ−1F,P ) ψ−1F over M is defined by gψ(U,V)(p)=g (U ,V ), ∀ U,V ∈Γ(ψ−1F), p∈M. ψ(p) p p Given a linear connection ∇N on k-vector bundle (F,P ) over N, the pull-back F ψ connection ∇is the unique linear connection on the pull-back (ψ−1F,P ) over ψ−1F M such that ψ ∇ W ◦ψ =∇NW, ∀W ∈Γ(F), ∀X ∈Γ(TM). (1) X dψ(X) Further, let U ∈ψ−(cid:0)1F and(cid:1) let p∈M, X ∈Γ(TM). Then ψ (∇ U)(p)=(∇N U)(ψ(p)), (2) X dpψ(Xp) where U ∈Γ(F) with U ◦ψ =U. e Now, let π , i=1,2, be the usual projection of M ×M onto M , given a linear i 1 2 i connecteion∇i onvectorebundleTM ,thepull-backconnectionπ∇i istheuniquelinear i connectiononthepull-backM ×M →π−1(TM )suchthatforeachY ∈Γ(TM ), 1 2 i i i i X ∈Γ(TM ×M ) 1 2 πi i ∇ Y ◦π =∇ Y . (3) X i i i dπi(X) (cid:0) (cid:1) 4 RAFIK NASRI Further, let (p ,p )∈M ×M , U ∈π−1(TM) and X ∈Γ(TM ×M ). Then 1 2 1 2 i 1 2 πi i (∇ U)(p ,p )= ∇ U (p ), (4) X 1 2 (dp1π,ip(2X)(p1,p2)) i Now, we construct a symmetric tens(cid:0)or fiels on prod(cid:1)uct manifold and give the e condition under which it becomes a tensor metric. Let c be an arbitrary real number and let g , (i = 1,2) be a Riemannian metric i tensors on M . Given a smooth positive function f on M , we define a symmetric i i i tensor field on M ×M by 1 2 G =(fv)2π∗g +(fh)2π∗g +cfhfvdfh⊙dfv. (5) f1,f2 2 1 1 1 2 2 1 2 1 2 Where π , (i=1,2) is the projection of M ×M onto M and i 1 2 i dfh⊙dfv =dfh⊗dfv+dfv⊗dfh. 1 2 1 2 2 1 For all X,Y ∈Γ(TM ×M ) 1 2 Gf1,f2(X,Y) =(f2v)2g1π1(dπ1(X),dπ1(Y))+(f1h)2g2π2(dπ2(X),dπ2(Y)) +cfhfv X(fh)Y(fv)+X(fv)Y(fh)) . 1 2 1 2 2 1 It is the unique tensor fields(cid:0)such that for any Xi,Yi ∈Γ(T(cid:1)Mi), (i=1,2) (fJ )2g (X ,Y )I, if (i,I)=(k,K) 3−i i i i G (XI,YK)= (6) f1f2 i k  cfIfKX (f )IY (f )K, otherwise  i k i i k k where (i,I),(k,K),(3−i,J)∈{(1,h),(2,v)}.  We call G the generalized warped product relative to g ,g and the warping f1,f2 1 2 functions f ,f . 1 2 If either f ≡1 or f ≡1 but not both, then we obtaina singly warpedproduct. If 1 2 both f ≡ 1 and f ≡ 1, then we have a product manifold. If neither f nor f is 1 2 1 2 constantandc=0,thenwehaveanontrivialdoubly warpedproduct. Ifneither f 1 norf isconstantandc6=0,thenwehaveanontrivialgeneralizedwarpedproduct. 2 Now, Let us assume that (M ,g ), (i = 1,2) is a smooth connected Riemannian i i manifold. The following proposition provides a necessary and sufficient condition for a symmetric tensor fieldG oftype (0,2)oftwo Riemannian metrics to be a f1,f2 Riemannian metric. Proposition 1. Let (M ,g ), (i = 1,2) be a Riemannian manifold and let f be i i i a positive smooth function on M and c be an arbitrary real number. Then the i symmetric tensor field G is Riemannian metric on M ×M if and only if f1f2 1 2 0≤c2g (gradf ,gradf )hg (gradf ,gradf )v <1. (7) 1 1 1 2 2 2 Proof. Let {e ,...,e } and {e ,...,e } be a local, orthonormal basis of 1 m1 m1+1 m1+m2 the vector fields with respect to g and g on an open O ⊂ M and O ⊂ M 1 2 1 1 2 2 respectively. The matrix of G relative to f1f2 1 1 1 1 {v = eh,...,v = eh ,v = ev ,...,v = ev } 1 fv 1 m1 fv m1 m1+1 fh m1+1 m1+m2 fh m1+m2 2 2 1 1 has the form D cfhfvE 1 1 2 . (8) cfhfv tE D (cid:18) 1 2 2 (cid:19) NON-DIAGONAL METRIC ON A PRODUCT RIEMANNIAN MANIFOLD 5 Where D =(fv)2I , D =(fh)2I and 1 2 m1 2 1 m2 e (f )he (f )v ··· e (f )he (f )v 1 1 m1+1 2 1 1 m1+m2 2 E = ... ... ...  e (f )he (f )v ··· e (f )he (f )v  m1 1 m1+1 2 m1 1 m1+m2 2    We can write the matrix (8) as I O D cfhfvE m1 1 1 2 . (9) cfhfv tED−1 −(cfhfv)2 tED−1E+D O I (cid:18) 1 2 1 1 2 1 2 (cid:19)(cid:18) m2 (cid:19) So D cfhfvE det 1 1 2 =(fh)2m2(fv)2m1det I−c2 tEE . cfhfv tE D 1 2 (cid:18) 1 2 2 (cid:19) (cid:0) (cid:1) and we compute λd2−1 λd d λd d ··· λd d 1 1 2 1 3 1 m2 λd d λd2−1 λd d ··· λd d I−c2 tEE =− 1... 2 ·2·· .2..3 ... .2..m2 .    ... ··· ··· ... ...     λd d λd d λd d ... λd2 −1   1 m2 2 m2 3 m2 m2    Where λ=c2m1(e (f )h)2 and d =e (f )v. i 1 j m1+j 2 i=1 By a straightforwardlong computation using a limited recurrence gives P d−1 d d ··· d 12 11 13 1m d d−1 d ··· d   21 22 23 2m  (Pm) det dd2i...11dm......1d2·2d−i·2·1d··m··2··d.d2i.33. ·d.m····.···3·.··· ...···...···...··· dm..−m....1dd2i.ii.−−.11 =(−1dd2i).iim.++.11(cid:18)1−···λ···j···mP=1d2j···(cid:19)······, dd·2im·m·   ... ··· ··· ··· ... ... ... ··· ··· ···  Where didje=tλddiidd−m+i......111d11j.ddiid··m−+··112··22 ddii··dm−+··113··33 ··············· ddii−+···11···ii···−−22 λdid−idm+1..1ii..−i−−..11−1 1 di+did1−m..i1+i..i+1..+−111 di+···.1···.i···.+2 ····.····.····.λdddmii·−·+−m··11mm··1 =(−1)mλd1di. So, D cfhfvE det(M )= det 1 1 2 f1f2 (cid:18) cf1hf2v tE D2 (cid:19) (10) =(fh2)m2(fv2)m1{1−c2g (gradf ,gradf )hg (gradf ,gradf )v}, 1 2 1 1 1 2 2 2 6 RAFIK NASRI where m (i = 1,2) is the dimension of M . Since, f and f are non-constant i i 1 2 smooth functions, then the proposition follows. Corollary 1. If the symmetric tensor field G of type (0,2) on M ×M is f1,f2 1 2 degenerate, then for any i∈{1,2}, g (gradf ,gradf ) is positive constant k with i i i i 1 k = . i c2k (3−i) Proof. Note that if G is degenerate then c is non-zero real number, f ,f is f1,f2 1 2 nonconstant smooth functions on M , M respectively and we have 1 2 c2g (gradf ,gradf )hg (gradf ,gradf )v =1. 1 1 1 2 2 2 Sinceg (gradf ,gradf )dependonlyonM ,(i=1,2)weconcludethatg (gradf ,gradf ) i i i i i i i is constant. Remark 2. Under the same assumptions as in Proposition 1, if f ,f are non- 1 2 constantsmoothfunctionsonM ,M respectivelyandϕissmoothfunctiononM × 1 2 1 M that satisfies −1 < ϕ < 1 , then the symmetric 2 kgradf1khkgradf2kv kgradf1khkgradf2kv tensor fields G =(fv)2π∗g +(fh)2π∗g +ϕfhfvdfh⊙dfv. f1,f2 2 1 1 1 2 2 1 2 1 2 is Riemannian metric on M ×M . 1 2 In all what follows, we suppose that f and f satisfies the inequality (7). 1 2 Lemma 3.1. Let X be an arbitrary vector field of M ×M , if there exist ϕ ,ψ ∈ 1 2 i i C∞(M ) and X ,Y ∈Γ(TM ), (i=1,2) such that i i i i G (X,Zh)=G (ϕvXh+ϕhXv,Zh), f1f2 1 f1f2 2 1 1 2 1 ∀ Z ∈Γ(TM ), i i  G (X,Zv)=hhG (ψvYh+ψhYv,Zv).  f1f2 2 f1f2 2 1 1 2 2 Then we have, X = ϕvXh+ψhYv+cfhfv ψvY (f )h−ϕvX (f )h grad(fv) 2 1 1 2 1 2 2 1 1 2 1 1 2 (11) (cid:8) (cid:9) − cfhfv ψhY (f )v−ϕhX (f )v grad(fh) 1 2 1 2 2 1 2 2 1 Proof. At first, we pu(cid:8)t (cid:9) B =X −ϕvXh−ψhYv and Z =Zh+Zv. 2 1 1 2 1 2 It suffices to observe that −1 1 G (B,Z)= G (ψhYv−ϕhXv,Zh)+G (ϕvXh−ψvYh,Zv) cfhfv f1f2 cfhfv f1f2 1 2 1 2 1 f1f2 2 1 2 1 2 1 2 1 2 (cid:8) (cid:9) = (ψhYv(fv)−ϕhXv(fv))Zh(fh)+(ϕvXh(fh)−ψvYh(fh))Zv(fv) 1 2 2 1 2 2 1 1 2 1 1 2 1 1 2 2 (cid:8)2 (cid:9) i = (−1) G ( ψJ Y (f )I −ϕJ X (f )I grad(fJ ),Z). f1f2 3−i i i 3−i i i 3−i i=1 X (cid:8) (cid:9) With (i,I),(3−i,J)∈{(1,h),(2,v)}. The result follows. NON-DIAGONAL METRIC ON A PRODUCT RIEMANNIAN MANIFOLD 7 3.2. The Levi-Civita Connection. Lemma 3.2. Let (M ,g ), (i = 1,2) be a Riemannian manifold. The gradient of i i the lifts fh of f and fv of f to M × M w.r.t. G are 1 1 2 2 1 f1,f2 2 f1,f2 1 1 cbh grad(fh)= (gradf )h− 1 (gradf )v , (12) 1 1−c2bhbv (fv)2 1 fhfv 2 1 2 2 1 2 1 (cid:8) 1 cbv (cid:9) grad(fv)= (gradf )v− 2 (gradf )h , (13) 2 1−c2bhbv (fh)2 2 fhfv 1 1 2 1 1 2 where b =kgradf k2 (i=1,2). (cid:8) (cid:9) i i Proof. LetZ ∈Γ(TM ),i=1,2,thenfor(i,I),(3−i,J)∈{(1,h),(2,v)},wehave, i i 1 G (grad(fI),ZI)= G ((gradf )I,ZI), f1f2 i i (fJ )2 f1f2 i i 3−i and G (grad(fI),ZJ )=0. f1f2 i 3−i Therefor, the result follows by Equation (6) and Lemma 3.1. Lemma 3.3. Let (M ,g ), (i = 1,2) be a Riemannian manifold and let ϕ be a i i i smooth function on M . The gradient of the lifts ϕ h of ϕ and ϕ v of ϕ to i 1 1 2 2 M × M w.r.t. G are 1 f1,f2 2 f1f2 1 2 (f gradϕ (f ))h grad(ϕh)= (gradϕ )h−c 1 1 1 grad(fv), (14) 1 fv 1 fv 2 (cid:18) 2(cid:19) 2 1 2 (f gradϕ (f ))v grad(ϕv)= (gradϕ )v−c 2 2 2 grad(fh), (15) 2 fh 2 fh 1 (cid:18) 1 (cid:19) 1 Proof. Let Z ∈ Γ(TM ), (i = 1,2) then for (i,I),(3−i,J) ∈ {(1,h),(2,v)}, we i i have, 1 G (grad(ϕI),ZI)= G ((gradϕ )h,ZI), f1f2 i i (fJ )2 f1f2 i i 3−i and G (grad(ϕI),ZJ )=0. f1f2 i 3−i Therefor, the result follows by Equation (6)and Lemma 3.1. Proposition 2. Let (M ,g ), (i = 1,2) be a pseudo-Riemannian manifold and let 1 i f :M →R∗, be a positive smooth function. The cometric G of G is given by i i + f1f2 f1f2 Gf1f2 = f12v 2g1h+ f11h 2g2v+ 1−c21bh1bv2 (cf22vb)v22(gradf1e)h⊙(gradf1)h (16) e +((cid:16)cf21hb)h12(cid:17)(greadf2)(cid:16)v ⊙(cid:17)(greadf2)v − f1hcf2(cid:8)v(gradf1)h⊙(gradf2)v . It is the unique tensor fields such that (cid:9) 1 g (α ,β )I+ c2bJj g (α ,df )Ig (β ,df )I ,if i=k (fJ)2 i i i 1−c2bhbv i i i i i i G (αI,βK)= j (cid:26) 1 2 (cid:27) i k f1f2  e−c g (α ,df )Ige(β ,df )Ke. if i6=k e f1hf2v(1−c2bh1bv2) i i i k k k for any αi,βi ∈Γ(T∗Mi) (i =e1,2 and je= 3 −i). Where gi (i = 1,2) is(1th7e) cometrics of g and (i,I),(k,K),(j,J)∈{(1,h),(2,v)}. i e 8 RAFIK NASRI Proof. AdirectcomputationusingEquation6,thedefinitionofthemusicalisomor- phismes and 1 2 I cfI ♯ (αI)= ♯ (α ) − i g (α ,df )hgrad(fJ ), Gf1f2i (cid:18)f3J−i(cid:19) gi i f3J−i i i i 3−i (cid:16) (cid:17) for (i,I),(3−i,J)∈{(1,h),(2,v)}, leads to gives (17). e let us compute the Levi-Civita connection of M × M associated with the 1 f1f2 2 1 2 metric G in terms of the Levi-Civita connections ∇and ∇associated with the f1f2 metrics g and g respectively. 1 2 Proposition 3. Let (M ,g ), (i=1,2) be a Riemannian manifold. Then we have i i 1 ∇ Yh =(∇ Y )h+fvB(X ,Y )hgrad(fv) (18) X1h 1 X1 1 2 f1 1 1 2 2 ∇X2vY2v =(∇X2Y2)v+f1hBf2(X2,Y2)hgrad(f1h) (19) ∇X1hY2v =∇Y2vX1h =−cX1(f1)hY2(f2)v f2vgrad(f1h)+f1hgrad(f2v)} (20) + Y (lnf ) vXh+ X (lnf ) hYv, 2 2 1 (cid:8)1 1 2 Where B , (i=1,2) the symmetric (0,2) tensor field of f given by fi (cid:0) (cid:1) (cid:0) (cid:1) i Bfi(Xi,Yi)=cfiHfi(Xi,Yi)+cXi(fi)Yi(fi)−gi(Xi,Yi), Hfi is the Hessian of fi. Proof. Let X ,Y ,Z ∈Γ(TM ), i = 1,2. For any (i,I),(k,K)∈{(1,h),(2,v)} we i i i i have 2G (∇ YI,ZK)=XI(G (YI,ZK))+YI(G (XI,ZK))−ZK(G (XI,YI)) f XiI i k i f1f2 i k i f1f2 i k k f1f2 i i +G ([XI,YI],ZK)+G ([ZK,XI],YI)+G ([ZK,YI],XI). f1f2 i i k f1f2 k i i f1f2 k i i (21) 1. Taking (i,I)=(k,K) in this formula, using Formula (6) and Lemma 2.1, we get i 2G (∇ YI,ZI)=2(fJ )2(g (∇ Y ,Z ))I, f1f2 Xih i i 3−i i Xi i i and using (6) again, we get i G (∇ YI,ZI)=G ((∇ Y )I,ZI). f1f2 XiI i i f1f2 Xi i i Similarly, taking (i,I)6=(k,K), we get cX (f Y (f )−g (X ,Y )) I G (∇ YI,ZK)= i i i i i i i G (f gradf )K,ZK f1f2 XiI i k (f2) f1f2 k k k (cid:18) i (cid:19) The result then follows by Lemma 3.1. (cid:0) (cid:1) 2. Taking i6=k. Atfirst,since∇is torsion-freewehave∇ XI =∇ YK+[XI,YK]. By Lemma YK i XI k i k k i 2.1, we have [XI,YK]=0. This implies that ∇ YK =∇ XI. i k XI k YK i i k Using Formula (6) and Lemma 2.1, we get Y (f ) G (∇ YK,ZI)=G ( k k )KXI,ZI , f1f2 XiI j i f1f2 fk i i and (cid:0) (cid:1) X (f ) G (∇ YK,ZK)=G ( i i )IYK,ZK . f1f2 XiI k k f1f2 fi k k (cid:0) (cid:1) NON-DIAGONAL METRIC ON A PRODUCT RIEMANNIAN MANIFOLD 9 Thus the result follows by Lemma 3.1. 3.3. The Laplacian of the lifts to M and M . 1 2 Theorem 3.4. On a generalized warped product (M × M ,G ) with m = dimM andm =dimM ,Letf :M →Randf :M1 →f1fR2 be2asmf1ofo2thfunctio1ns. 1 2 2 1 1 2 1 Then the Laplacian of the horizontal lift f ◦π of f (resp. vertical lift f ◦π of 1 1 1 2 2 f ) to M × M is given by 2 1 f1f2 2 ∆(fh)= 1 1 ∆ (f ) h− cbh1 ∆ (f ) v+ bh1(c(1−m1)bv2+m2) 1 fv(1−c2bhbv) fv 1 1 fh 2 2 fhfv 2 1 2 (cid:26) 2 1 1 2 (cid:27) (cid:0) (cid:1) (cid:0) (cid:1) (22) c2 bv c(b2)h + 2 (gradf (b ))h− 1 (gradf (b ))v . 2fv(1−c2bhbv)2 fv 1 1 fh 2 2 2 1 2 (cid:26) 2 1 (cid:27) ∆(fv)= 1 1 ∆ (f ) v− cbv2 ∆ (f ) h+ bv2 c(1−m2)bh1 +m1 2 f1h(1−c2bh1bv2)(f1h 2 2 f2v 1 1 (cid:0) f1hf2v (cid:1)) (cid:0) (cid:1) (cid:0) (cid:1) (23) c2 bh c(b2)v + 1 (gradf (b ))v− 2 (gradf (b ))h . 2fh(1−c2bhbv)2 fh 2 2 fv 1 1 1 1 2 (cid:26) 1 2 (cid:27) Where b =kgradf k2 (i=1,2). i i Lemma 3.5. On (M × M ,G ), if {e ,...,e } is the local frame field with 1 f1f2 2 f1f2 1 m1 respect to the metric g and {e ,...,e } is the local frame field with respect 1 m1+1 m1+m2 to the metric g , then {u ,...,u ,u ,...,u } is the local frame field with 2 1 m1 m1+1 m1+m2 repect to the metric G , where f f 1 2 1 eh, j∈{1,...,m }; u′j=f2v jcavj − 1 (gradf )h+cbh1Tv + 1 ev, j∈{m +1,.,1m +m }. (24) (1−c2bh1Avj) f2v 1 f1h j f1h j 1 1 2 n o And for j ∈{m +1,...,m +m },  1 1 2 1 1−c2bhAv j−1 j−1 u = u′, ku′k2= 1 j+1, A = a2, T = a e , a =e (f ). j ku′k j j 1−c2bhAv j i j i i i i 2 j 1 j i=m +1 i=m +1 X1 X1 Proof. We knowthat G is Riemannianmetric if andonly if 0<1−bhbv. Then f f 1 2 1 2 if we choose {e ,...,e } to be a local, orthonormal basis of the vector fields with 1 m1 respecttog onanopenO ⊂M and{e ,...,e }tobealocalorthonormal 1 1 1 m1+1 m1+m2 basis of the vector fields with respect to the metric g on an open O ⊂ M , then 2 2 2 the family 1 1 1 1 {v = eh,...,v = eh ,v = ev ,...,v = ev } 1 fv 1 m1 fv m1 m1+1 fh m1+1 m1+m2 fh m1+m2 2 2 1 1 is a local basis of the vector fields with respect to G on an open O ×O ⊂ f1f2 1 2 M ×M . 1 2 Thegradientoff (resp. f )anditsnormkgradf k(resp. kgradf k)canbewritten 1 2 1 2 as m1 m1 gradf = e (f )e , kgradf k2 = (e (f ))2 (25) 1 k 1 k 1 k 1 k=1 k=1 X X 10 RAFIK NASRI m1+m2 m1+m2 rep. gradf = a e , kgradf k2 = a2 . (26) 2 i i 2 i (cid:0) i=Xm1+1 i=Xm1+1 (cid:1) G is positive definite, which implies that f f 1 2 l 1−c2bh (ah)2 >0, ∀l∈{1,...,m }, (27) 2 k 1 k=1 X and j 1−c2bh (av)2 >0, ∀j ∈{m +1,...,m +m }. (28) 1 i 1 1 2 i=m +1 X1 Forthe proofofthe lemma itis actuallyalmostthe mostinteresting resultbecause it provides an algorithm for constructing {u ,...,u ,u ,...,u } from the 1 m1 m1+1 m1+m2 family {e ,...,e } et {e ,...,e }. 1 m1 m1+1 m1+m2 To do so, we use a limited recurrence (The Gram schmidt process). At first, we put u′ =v and u = v1 . For j ∈{2,...,m ,m +1,...,m +m }, 1 1 1 kv k 1 1 1 2 1 j−1 u′ u′ =v − G (v ,u )u and u = j . (29) j j f1f2 j i i j ku′k i=1 j X By virtue of (29), a straightforwardcalculation using (25) and (26) gives 1 u = eh, ku k=1 ∀k ∈{1,...,m }, k fv k k 1 2 for all j ∈{m ,...,m +m }, we have 1 1 2 u′ = −cavj (gradf )h+ 1 ev j j−1 1 fh j fv 1−c2bh (av)2 1 2 c2b1hiP=avm1+i1 ! j−1 v + 1 j a e , j−1 i i fh 1−c2bh (av)2 (cid:18)i=m1+1 (cid:19) 1 1i=m1+i1 ! P P and j 1−c2bh (av)2 1 i ku′jk=vuu(cid:18) Pij=−m11+1 (cid:19). u 1−c2bh (av)2 u 1 i u(cid:18) i=m1+1 (cid:19) t P Remark 3. With the notations above, we have 1) T is the zero vector field on M , A is the zero function on M and m1+1 2 m1+1 2 A is the care of the gradient of f . m1+m2 2 2)For any j ∈{m +1,...,m +m } 1 1 2 T (f )=A =g (T ,T ), j 2 j 2 j j  uu′jj((ff11hh))==−−ccfbf2fvh11hvb(cid:0)h11u−jc(2afbvjh12vA)vj,(cid:1)=−cff1h2vbh1u′j(f2v), (30) 2 