Liouville-type theorems for twisted and warped products manifolds
LLiouville-type theorems for twisted and warped products manifolds S TEPANOV S ERGEY
Abstract.
In the present paper we prove Liouville-type theorems: non-existence theorems for complete twisted and warped products of Riemannian manifolds which generalize and complement similar results for compact manifolds.
Keywords : complete Riemannian manifold, twisted and warped product Riemannian manifolds, non-existence theorems. Mathematical Subject Classification С Introduction
In the present paper we will use a generalized the Bochner technique which is based on extensions of the classical theorem of divergence and the maximal principle to complete, non-compact Riemannian manifolds. We will apply this technique to prove Liouville-type theorems for complete, non-compact Riemannian doubly twisted and warped products manifolds. We recall that warped products provide are natural generalization of direct products of Riemannian manifolds.
The notion of warped products plays very important roles in differential geometry and in general relativity.
For example, we recall that many Riemannian manifolds of nonpositive sectional curvature are obtained by using warped products. On the other hand, we recall also that many basic solutions of the Einstein field equations are warped products. For ex-ample, the well known Schwarzschild’s vacuum model and Robertson-Walker’s ex-panding model of universe in general relativity are warped products. On the other hand, t wisted products provide another natural generalization of direct prod-ucts.
The definition of twisted products extends the notion of warped products in a very natural way. Theorems which we prove in our paper generalize and complement similar well known results for compact doubly twisted and warped products manifolds. In addition, we consider an application of our results to the theory of projective mappings. . Doubly twisted products manifolds
Let and ( ) g,M ( ) g,M be two Riemannian manifolds, →× : ММ i λ ℝ be a strictly positive differentiable function and ii MMM →× : π be a canonical or natu-ral projection for an arbitrary i =
1, 2. Then the double-twisted product ММ λλ × of and is the differentiable manifold M = equipped with the Riemannian metric defined by the following equality ( ) g,M ( g,M ) MM × ggg λλ += ( ) ( ) ggg ∗∗ ⋅+⋅= ππλππλ oo , where * denotes the pull-back operator on ten-sors (see [1]). In this case the functions λ and λ are called twisted functions. In par-ticular, for λ= λ = direct product ( ) gg,MM ⊕× . Leaves M × { y } = and { x } × M ( ) y − π = ( ) x − π are totally umbilical submanifolds of (see [1] and [2]). Therefore, the doubly twisted product ММ λλ × ММ λλ × carries two orthogonal complementary totally umbilical foliations. If we denote by ii TMTMTM →× ∗ : π the natural projection then the vector fields ( ) log λπξ ∇−= ∗ V and ( ) log λπξ ∇−= ∗ H are the mean curvature vectors of these foliations (see [1] and [2]). The converse is also true in the following case. Namely, let ( M, g ) be a simply connected n -dimensional Riemannian manifold and V and H be orthogonal complementary integrable distributions with totally umbilical integral manifolds. If the maximal integral manifolds of V are complete and dim V = m ≥
3, then (
M, g ) is isometric a doubly twisted product of two maximal integral manifolds of V and H (see [3]) . We have proved in [2] that the following relation is satisfied on ММ λλ × ( ) =+ HV div ξξ HV тп тптт s ξξ − −−− −− . (1.1) In this formula s mix is the mixed scalar curvature of ММ λλ × which defined as the scalar function ( ) ∑ ∑ = += = ma nm a EEs ,sec α α here is the mixed sectional curvature in direction of the two-plane ( α EE a ,sec ) }{ α π EEspan a , = for the local orthonormal basis { } m EE ,..., of the vertical distribu-tion V and { is orthonormal basis of horizontal distribution H at an arbi-trary point of (see [4, p. 23]; [5]). It is easy to see that this expression is independent of the chosen bases. } nm EE ,..., + ММ λλ × Further, taking ( ) log λπξ grad V ∗ −= and ( ) log λπξ grad H ∗ −= in (1.1), we find that ( ) ( ) ( ) =+ ∗∗ loglog λπλπ gradgraddiv = ( ) ( ) log,loglog,log λλλλ ∇+∇+− ∗∗ − −−− gradggradgs mn mnmm (1.2) where ( ) ( ) log,loglog,log λπλπλλ gradgradggradgradg ∗∗∗ = , ( ) log,log λλ gradgradg ∗ ( ) log,log λπλπ gradgradg ∗∗ = . Let be a complete, noncompact and simply connected Riemannian manifold such that ММ λλ × ( ) ( ) ( gMLgardgrad ,loglog ∈+ ∗∗ λπλπ ) . We recall here that for a smooth vector field X on a connected complete, noncompact and oriented Riemannian manifold ( M, g ) if 0 = Xdiv ( ) gMLX , ∈ and div X ≥ div X ≤
0) everywhere on ( M, g ) (see [6] and [7]). In addition, we also recall that every simply connected manifold M is orientable. Then for ≤
0 from (1.2) we conclude that mix s C =λ and C =λ for some positive constants C and C . In this case MM × has the metric . A doubly twisted product with twisted functions ∗∗ ⋅+⋅= g С g С g ММ λλ × λ = C > 0 and λ = C > 0 can be considered as a direct product ММ × of ( ) g,M and ( ) g,M for g С g = and g С g = . Summarizing, we formulate the statement which generalizes a theorem on two orthogonal complete totally umbilical foliations on compact Riemannian manifold with non positive mixed scalar curvature that has been proved in [2] and [8]. heorem 1 . Let ( M, g ) be a doubly twisted product ММ λλ × of some Riemannian manifolds and ( . If ( M, g ) is a complete and oriented Riemannian manifold ( M, g ) with nonpositive mixed scalar curvature and ( ) g,M ) g,M mix s ( ) ( ) ( ) gMLgardgrad ,loglog ∈+ ∗∗ λπλπ , then the twisted functions λ and λ are positive constants C and C , respectively, and therefore, ( M, g ) is the direct product of MM × ( ) g,M and ( ) g,M for g С g = and g С g = . If is a well known
Cartan-Hadamard manifold (see, for example, [9, p. 90]), i.e. a complete, noncompact, simply connected Riemannian manifold of non-positive sectional curvature, then the above theorem yields the next ММ λλ × Corollary 1 . If a Cartan-Hadamard manifold ( M, g ) is a doubly twisted product such that ММ λλ × ( ) ( ) ( gMLgardgrad ,loglog ∈+ ∗∗ λπλπ ) , then the twisted functions λ and λ are positive constants and therefore, ( M, g ) is a direct product of MM × ( ) g,M and ( ) g,M for g С g = and g С g = .
2. Twisted products and projective submersions of Riemannian manifolds
A doubly twisted product manifold ММ λλ × is called a twisted product if 1 =λ (see [1]). is called the base and the submanifolds M ( , gM ) × { y } = are called leaves of the twisted product manifold ( ) y − π ММ λ × . On the other hand, ( ) , gM is called fiber and the submanifolds { x } × M = ( ) x − π and called fibres of the twisted product manifold . In addition, all leaves are totally geodesic submanifolds and all fibres are totally umbilical submanifolds in ММ λ × ММ λ × . Converse is also true (see [1]). Namely, let ( M, g ) be a simply connected semi-Riemannian manifold with two orthogonal and complementary integrable distributions V and H . Suppose that V is totally geodesic and with complete integrable manifolds. If H is totally umbilical then ( M, g ) is isometric to a twisted product ММ λ × . In this case, we can formu-late the following heorem 2. Let a twisted product ММ λ × be a complete and simply connected Riemannian manifold. If the mixed sectional curvature of is non-negative then it is isometric to a direct product ММ λ × MM × . Proof.
Our theorem is a corollary of the main theorem of [10] where was considered two orthogonal complementary integrable distributions V and H on a complete Rie-mannian manifold ( M, g ) . If in addition, V is totally geodesic and ∑ for each unite vector field E ( ) = ≥ ma a E,Esec αα which belongs to H at each point of M , then H is totally geodesic (see [10]). In this case, by the de Rham decomposition theorem (see [11, p. 187]) we conclude that if ( M, g ) is a simply connected Riemannian manifold then it is isometric to the direct product ( ) gg,MM ⊕× of some Riemannian manifolds and ( for the Riemannian metric and which induced by g on and . ( ) g,M ) g,M g g М М This completes the proof of the theorem.We recall here the definition of pregeodesic and geodesic curves . Namely, a pre-geodesic curve is a smooth curve γ : t ∈ J ⊂ ℝ → ( ) t γ ∈ M on a Riemannian mani-fold ( M, g ), which becomes a geodesic curve after a change of parameter. Let us change the parameter along γ so that t becomes an affine parameter . Then 0 =∇ Х Х for dtd Х γ= , and γ is called a geodesic curve . By analyzing of the last equation, one can conclude that either γ is an immersion, i.e., 0 ≠ dtd γ for all t ∈ J , or ( ) t γ is a point of M . Let (
M, g ) and ( gM , ) be Riemannian manifolds of dimension n and m such that n > m . A surjective map ( ) ( ) gMgMf ,,: → is a projective submersion if it has maximal rank m at any point x of M and if ( ) γ f is a pregeodesic in ( ) gM , for an arbitrary pre-geodesic γ in ( M, g ) (see [12]). In this case, each pregeodesic line М ⊂γ which is an integral curve of the distribution is mapped into a point ∗ fKer ( ) γ f in M . Note that this fact does not contradict the definition of projective submersion. e call the submanifolds ( ) Myf ⊂ − for an arbitrary Му ∈ as fibers . All fibers of an arbitrary projective submersion are totally geodesic submanifolds (see [12] and [13]). Putting V x ( ) х fKer ∗ = , for any Mx ∈ , we obtain an integrable vertical distri-bution V which corresponds to the totally geodesic foliation of M determined by the fibres of f , since each V x ( ) yfT x − = coincides with tangent space of ( ) у f − at x for . Let H be the complementary distribution of V determined by the Rieman-nian metric g, i.e. H ( ) yxf = x = V x ⊥ at each x ∈ M where H x is called the horizontal space at x We have proved in [12] and [13] that the horizontal distribution H is integrable with totally umbilical integral manifolds. So, ( M, g ) admits two complementary totally geodesic and totally umbilical foliations, whose leaves intersect perpendicularly. If ( M, g ) is a simply connected Riemannian manifold and the integral manifolds of H are geodesically complete then ( M, g ) is isometric to a twisted product ММ λ × such that the maximal integral manifolds of V and H correspond to the canonical fo-liations of the product (see [1]). MM × The converse is also true in local case. Namely, for an arbitrary twisted product ММ λ × the natural projection : M ММ →× λ π is a locally projective sub-mersion. We make here two observations before formulating a conclusion. Firstly, if we sup-pose the geodesic completeness of ( M, g ), then the integral manifolds of are geodesi-cally complete automatically. Secondary, it well known by the Hopf–Rinow theorem that any connected geodesically complete Riemannian manifold is a complete Rie-mannian manifold. Then the following statement is a corollary of Theorem 2.
Corollary 2 . Let ( M, g ) be an n-dimensional complete and simply connected Rieman-nian manifold with non-negative sectional curvature. If ( M, g ) admits a projective submersion ( ) ( gMgMf ,,: → ) onto another m-dimensional ( m < n ) Riemannian manifold ( gM , ) then it is isometric to a direct product ( ) gg,MM ⊕× of some iemannian manifolds ( ) and g,M ( ) g,M such that maximal integral manifolds of and correspond to the canonical foliations of the product ∗ fKer ( ⊥∗ fKer ) MM × . Another corollary follows from Theorem 1.
Corollary 3 . Let ( M, g ) be an n-dimensional complete and simply connected Rieman-nian manifold and ( ) ( ) gMgMf ,,: → be a projective submersion onto another m-dimensional ( m < n ) Riemannian manifold ( ) gM , , then ( M, g ) is a twisted product of some Riemannian manifolds ММ λ × ( ) , gM and ( ) , gM . If ММ λ × has a nonpositive mixed scalar curvature s mix and ( ) ( gMLgard ,log ∈ ∗ λπ ) , then it isometric to a direct product ( ) gg,MM ⊕× such that maximal integral mani-folds of and ( correspond to the canonical foliations of ∗ fKer ) ⊥∗ fKer MM × .
3. Double warped products and warped products manifolds A doubly warped product manifold ( M, g ) is a twisted product manifold ММ λλ × where → М : λ ℝ and → М : λ ℝ are positive smooth functions (see [14]). These functions are called warping functions. A doubly warped product ММ λλ × carries two orthogonal complementary totally umbilical foliations with closed mean curvature vectors V ξ and H ξ because in this case we have the equalities ( ) loglog λλπξ gradgrad V −=−= ∗ and ( ) loglog λλπξ gradgrad H −=−= ∗ , re-spectively (see also [1]). Then the formula (1.2) can be rewrite in the following form ( ) log λλ Δ = loglog λλ gradgrads mn mnmm − −−− ++ − (3.1) where ( iii gradgradggrad λλλ log,loglog = ) for an arbitrary i =
1, 2. . There-fore, if ≤ mix s ( ) log λλ is subharmonic because ( ) log λλ Δ ≥
0. At the same time, we know that on a complete Riemannian manifold (
M, g ) each sub-harmonic function → М f : ℝ whose gradient has integrable norm on ( M, g ) must ac-tually be harmonic (see [15, p. 660]). In our case it means that ( log ) λλ Δ =
0. Then rom (3.1) we obtain C = λ and C = λ for some positive constants C and C . Therefore, the following theorem holds. Theorem 3 . Let ( M, g ) be a complete double-warped product of Rie-mannian manifolds ( ) and ММ λλ × g,M ( ) g,M such that the mixed scalar curvature is nonpositive . If the gradient of mix s ( ) log λλ has integrable norm ( M, g ), then C =λ and C = λ for some positive constants C and C and therefore, ( M, g ) is the direct product of ( ) g,M and ( ) g,M for g С g = and g С g = . Remark . We argue that our Theorem 3 complements the results of the paper [15] where was proved that if the mixed sectional curvature of a complete doubly warped product manifold is non-negative then the warping functions ММ λλ × λ and λ are constants. Twisted products are generalizations of warped products. For a warped product the function ММ λ × λ is a smooth positive function → : М λ ℝ (see [16] and [17]). In this case, leaves are totally geodesic submanifolds in and fibres are extrinsic spheres. In addition, we recall that a submanifold of a Riemannian mani-fold is called an extrinsic sphere if it is a totally umbilical submanifold with parallel mean curvature vector (see [18]). In [1] was proved the following statement: Let ( M, ММ λ × g ) be a simply connected semi-Riemannian manifold with two orthogonal and com-plementary integrable distributions V and H such that V is totally geodesic and with complete leaves and integrable manifolds of H are extrinsic spheres. Then ( M, g ) is isometric to a warped product. Let (
M, g ) be a warped product ММ λ × of two Riemannian manifolds ( ) g,M and . The well known curvature identities (see [19, p. 211]) ( g,M ) ( ) λλπ HessRicRic mn − −= ∗ , ( ) ( ) ggradgradgmnRicRic ∗∗∗ ⎟⎟⎠⎞⎜⎜⎝⎛ −−−Δ−=
22 2222 2122 ,1 πλ λλπλλπ , elating the Ricci curvature Ric of (
M, g ) and the Ricci curvatures
Ric and Ric of ( M , g ) and ( M , g ), respectively. In these identities λΔ is the Laplacian of λ for g and . gg λπ = ∗ From the above identities we obtain two equations ( ) ( )
Rictraces g mn ∗ −=Δ − πλλ (3.2) ( )( ) ( ) λλπλλ gradmnRictraces g mn −−−−=Δ ∗ − (3.3) where ( ) , λλλ gradgradggrad = , ( ) ( ∑ =∗ = ma aag EERicRictrace , π )) and . In first case, if we assume that ( ) ( ∑ +=∗ = nmg EERicRictrace , α αα π Rictraces g ∗ ≥ π then from (3.2) we obtain λΔ ≥ and therefore the warped function → : М λ ℝ is subharmonic. It is well known that Yau showed in [20] that every non-negative L p -subharmonic function on a complete Riemannian manifold must be constant for any p > 1. Therefore, if ( is a complete manifold such that its scalar curvature and for some ) g,M Rictraces g ≥ ∫ ∞< M gp dV λ > p then from (3.2) we conclude that the warped function λ is constant. At the same time, a warped product ММ λ × with a constant warping function λ = C > 0 can be considered as a direct product of and ММ × ( ) g,M ( ) g,M for g С g = . Summarizing the above arguments we can formulate the following Theorem 4 . Let ( M, g ) be a warped product ММ λ × of two Riemannian manifolds and such that the base ( ) g,M ( g,M ) ( ) g,M of ММ λ × is a complete mani-fold and for the scalar curvature sRictraces g ∗ ≥ π of ( ) g,M and for the Ricci tensor Ric of ММ λ × . If for some ∫ ∞< M gp dV λ > p then C =λ for some positive constant C and therefore, ( M, g ) is the direct product of ММ × ( ) g,M and ( ) g,M for g С g = . n the second case, if we assume that Rictraces g ∗ ≥ π then from (3.3) we obtain λΔ ≤
0 and therefore the warped function → : М λ ℝ is superharmonic. In this case, if ( M, g ) is a complete Riemannian manifold and ( gMLgrad , ∈λ ) , then su-perharmonic function λ is harmonic (see the Lemma in Appendix). Thus, we can formulate an analogue of the previous theorem. Theorem 5 . Let ( M, g ) be a warped product ММ λ × of two Riemannian manifolds and such that the base ( ) g,M ( g,M ) ( ) g,M of ММ λ × is a complete mani-fold and for the scalar curvature sRictraces g ∗ ≤ π of ( ) , gM and for the Ricci tensor Ric of ММ λ × . If ( ) gMLgrad , ∈λ then C =λ for some positive constant C and therefore, ( M, g ) is the direct product of and ММ × ( g,M )( ) g,M for g С g = . If we assume that ( ) +− = mn u λ then the following equation holds (see [21]) ( ) ( ) ( ) ( ) +−−− +−=Δ +− − mnmn usussu mn mn (3.4) for n ≥
3. In turn, if and then from the equations (3.4) followed ss ≤ ≥ s ≥Δ u , i.e. u is a subharmonic function on ( ) g,M . Therefore, if we suppose that ( ) g,M is a complete manifold and ( ) +− = mn u λ is L p -function on ( ) g,M for some p > 1 then the warped product function λ is a positive constant. In particular, for n = m + ( ) ussu −=Δ ss ≤ then u is a subharmonic function on ( ) g,M . Thus we have the following result. Theorem 6 . Let be a warped product such that its base ( s a complete manifold. If the warping function ММ λ × ) g,M i ( ) +− mn λ ∈ ( ) , gML p for some p > 1 and the scalar curvatures s, s and s of ММ λ × and of its base and fibre, respec-tively, satisfy one of the two following conditions : and
0 for any n ≥ ss ≥ ≤ s . for = then ss ≤ mn − λ is a positive constant C and therefore, ( M, g ) is iso-metric to the direct product ММ × of ( ) g,M and ( ) g,M for g С g = .
4. Einstein warped product manifolds
Let the warped product be an n -dimensional ( n ≥
3) Einstein manifold, i.e. ММ λ × gsRic n = for the constant scalar curvature s of ММ λ × . In this case, ( ) , gM is an Einstein manifold too (see [16]). It means that for we have 3 ≥− mn gsRic mn − = where Ric and s are the Ricci tensor and the constant scalar curva-ture of ( , respectively. In addition, the following equation holds ) , gM ( ) ⎟⎠⎞⎜⎝⎛ −−+−−=Δ mn snsgradmn λλλλ . (4.1) On the other hand, for gsRic n = the equation (3.3) can be rewritten in the form ( ) λλλλ gradmnss nmn −−−⎟⎠⎞⎜⎝⎛ −=Δ − . (4.2) From (4.1) and (4.2) we obtain ( ) ⎟⎠⎞⎜⎝⎛ −−−⎟⎠⎞⎜⎝⎛ −=Δ λλλλ gradmns n . (4.3) and ( ) ( ) ⎟⎠⎞⎜⎝⎛ −−++= − λλλ gradmnss nmn . (4.4) Now let us analyze the equations (4.1) – (4.4). Firstly, if s > 0 then from (4.4) we ob-tain s > 0. In this case, from (4.4) we conclude that ( ) −−≤ smn sn λ . Secondary, we consider the equation (4.1). For this case, we recall the well known statement from [20]. Namely, if f is a smooth function defined on a complete Riemannian manifold ( M, g ) satisfies the equality 0 ≥Δ ff , then either ∫ = M gp dVf
0 for all 1 ≠ p or f = constant . It means that if is complete manifold such that ММ λ × ( ) smn sn −≥ λ for > 0 and for some ∫ ∞< M gp dV λ ≠ p then from (4.1) we obtain that λ ( ) smn s n −= is constant and therefore ( M, g ) is the direct product ММ × of and ( ) g,M ( ) g,M for gg λ = . In addition, we note that from the equalities ( ) smn sn −≥ λ and ( ) −−≤ smn sn λ follows ( ) ≥−≥ smn sn λ . Thirdly, we consider the equation (4.3). In this case, if we assume that s < 0 and for ∫ ∞< M gp dV λ ≠ p and 1 ≤ λ , then we conclude that 0 =Δ λ and 1 = λ . On the other hand, if we assume that s < 0 and ∫ ∞< M g dVgrad λ for ≥λ , then we conclude that =Δλ and =λ (see the Lemma in Appendix). Summarizing the above arguments we can formulate the following statement. Corollary 4 . Let be an n-dimensional ( n ≥ Einstein warped product of two Riemannian manifolds and ММ λ × ( g,M ) ( ) g,M such that the m-dimensional ( ) base f ≥− mn ( ) g,M o ММ λ × is complete manifold. If ( ) ≥−≥ smn sn λ for the positive constant scalar curvatures s and s of and , respectively and ( ) g,M ММ λ × ( ) , gML p ∈λ for some ≠ p , then λ ( ) smn s n −= and therefore ( M, g ) is the direct product of ММ × ( ) g,M and ( ) g,M for gg λ = . 2) If the scalar curvature s of ММ λ × is negative and for some ( , gML p ∈λ ) ≠ p and ≤λ , then =λ and therefore ( M, g ) is the direct product ММ × of ( ) and ( ) g,M , gM . ) If the scalar curvature s of ММ λ × is negative and ( ) , gMLgrad ∈ λ for ≥ λ , then = λ and therefore ( M, g ) is the direct product of ММ × ( ) g,M and ( ) . , gM If the warped product is an n -dimensional ( n ≥
3) Einstein manifold, then (3.2) can be rewritten in the form ММ λ × ⎟⎠⎞⎜⎝⎛ −=Δ − ss nmmn λλ , (4.5) If is a complete manifold such that ( g,M ) ss nm ≥ and for some ∫ ∞< M gp dV λ ≠ p , then from (4.5) we conclude that the warped function λ is constant and ss nm = = constant . On the other hand, ss nm ≥ and ∫ ∞< M g dVgrad λ , then from (4.5) we conclude that the warped function λ is constant and ss nm = = con-stant (see the Lemma in Appendix). We proved the following Corollary 5 . Let be an n-dimensional ( n ≥ Einstein warped product of two Riemannian manifolds ( ) and ММ λ × g,M ( ) g,M such that the base ( ) g,M of is an m-dimensional complete manifold and ММ λ × ss nm ≥ ( resp. ss nm ≤ ) for the scalar curvature s of and for the constant scalar curvature s of . If for some ( g,M ) ) ММ λ × ( , gML p ∈ λ ≠ p ( resp. ( , gMLgrad ∈ λ ) ) then ss nm = = constant and C = λ for some positive constant C and therefore ( M, g ) is the direct product of ММ × ( ) g,M and ( ) g,M for g С g = . Remark . Corollaries 4 and 5 generalize and complement the main theorem on an Einstein warped product with compact base in [16].
5. Appendix . In conclusion, we consider superharmonic function on complete, non-compact Rie- mannian manifolds and prove the following lemma which is an analogy of the Yau roposition from [15, p. 660]) in which he has argued that on a complete non-compact Riemannian manifold each subharmonic function whose gradient has integrable norm on (
M, g ) must be harmonic.
Lemma . If ( M, g ) is a complete Riemannian manifold without boundary, then any su-perharmonic function with MCf ∈ ( ) gMLfgrad , ∈ is harmonic. Proof . On the one hand, if we assume that ϕ = − f for any superharmonic function then the conditions MCf ∈ Δ ≤ f and ( ) gMLfgrad , ∈ which must be satisfy for the superharmonic function f can be written in the form Δ ϕ ≥
0 and ( gMLgrad , ∈ ϕ ) . In this case, using the Yau statement we conclude that Δ ϕ = f = − ϕ is a harmonic function. Acknowledgements
The work of Sergey Stepanov is supported by RBRF grant 16-01-00053- а (Russia). References [1] Ponge R., Reckziegel H., Twisted products in pseudo-Riemannian geometry, Geom. Dedic., 48:1 (1993), 15-25. [2] Stepanov S.E., A class of Riemannian almost-product structures, Soviet Mathe-matics (Izv. VUZ), 33:7 (1989), 51-59. [3] Kim B.H., Warped products with critical Riemannian metric, Proc. Japan Acad., Ser. A, 71:6 (1995), 117-118. [4] Falcitelli M., Ianus S., Pastore A.M., Riemannian submersions and related topics, Word Scientific Publishing, Singapore, 2004. [5] Rocamora A.H., Some geometric consequences of the Weitzenböck formula on Riemannian almost-product manifolds; weak-harmonic distributions, Illinois Journal of Mathematics, 32:4 (1988), 654-671. [6] Caminha A., Souza P., Camargo F., Complete foliations of space forms by hyper- sufaces, Bull. Braz. Math. Soc., New Series, 41:3 (2010), 339-353.
7] Caminha A., The geometry of closed conformal vector fields on Riemannian spaces, Bull. Braz. Math. Soc., New Series, 42:2 (2011), 277-300. [8] Naveira A.M., Rocamora A.H., A geometrical obstruction to the existence of two totally umbilical complementary foliations in compact manifolds, Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Mathematics 1139 (1985), 263-279. [9] Pigola S., Rigoli M., Setti A.G., Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique, Birkhäuser Verlag AG, Ber-lin (2008). [10] Brito F., Walczak P.G., Totally geodesic foliations with integrable normal bun-dles, Bol. Soc. Bras. Mat., 17:1 (1986), 41-46. [11] Koboyashi S., Nomizu K., Foundations of differential geometry, Volume I, Inter- science Publishers, New York, 1963. [12] Stepanov S.E., On the global theory of projective mappings, Mathematical Notes, 58:1 (1995), 752-756. [13] Stepanov S.E., Geometry of projective submersions of Riemannian manifolds, Russian Mathematics (Iz. VUZ), 43:9 (1999), 44-50. [14] Ünal B., Doubly warped products, Differential Geometry and its Applications, 15 (2001), 253-263. [15] Gutierrez M., Olea B., Semi-Riemannian manifolds with a doubly warped struc-ture, Revista Matematica Iberoamericana, 28:1 (2012), 1-24. [16] Kim D.-S., Kim Y.H., Compact Einstein warped product spaces with nonpositive scalar curvature, Proceedings of the American Mathematical Society, 131:8 (2003), 2573-2576. [17] Defever F., Deszcz R., Glogowska M., Goldberg V.V., Verstraelen L., A class of four-dimensional warped products, Demonstatio Mathematica, 35:4 (2002), 853-864. [18] Kozaki M., Ohkubo T., A characterization of extrinsic spheres in a Riemannian manifold, Tsukuba J. Math., 26:2 (2002), 291-297.
19] O’Neill B., Semi-Riemannian geometry with applications to relativity, Academic Press, San Diego, 1983. [20] Yau S.T., Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659-670. [21] Dobarro F., Dozo E.L., Scalar curvature and warped products of Riemannian manifolds, Transactions of the American Mathematical Society, 303:1 (1987), 161-168.19] O’Neill B., Semi-Riemannian geometry with applications to relativity, Academic Press, San Diego, 1983. [20] Yau S.T., Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659-670. [21] Dobarro F., Dozo E.L., Scalar curvature and warped products of Riemannian manifolds, Transactions of the American Mathematical Society, 303:1 (1987), 161-168.