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Hypersurfaces in non-flat Lorentzian space forms satisfying L ψ = Aψ + b k ∗ Pascual Lucas H. Fabiań Ram´ırez-Ospina † Departamento de Matemáticas, Universidad de Murcia 1 Campus de Espinardo, 30100 Murcia SPAIN 1 0 2 January 18, 2011 n a J 7 1 Abstract ] G We study hypersurfaces either in the De Sitter space Sn+1 Rn+2 or in the anti De .D Sitter space Hn1+1 ⊂Rn2+2 whose position vector ψ satisfies1the⊂cond1ition Lkψ =Aψ+b, h whereLk isthelinearizedoperatorofthe(k+1)-thmeancurvatureofthehypersurface,for t afixedk =0,...,n 1,Aisan(n+2) (n+2)constantmatrixandbisaconstantvector a − × m in the corresponding pseudo-Euclidean space. For every k, we prove that when A is self- adjoint and b=0, the only hypersurfaces satisfying that condition are hypersurfaceswith [ zero(k+1)-thmeancurvatureandconstantk-thmeancurvature,openpiecesofstandard 3 pseudo-RiemannianproductsinSn+1 (Sm(r) Sn−m(√1 r2),Hm( r) Sn−m(√1+r2), 1 1 v Sm(√1 r2) Sn−m(r), Hm( √r2 1) ×Sn−m(r)), o−pen pieces−of ×standard pseudo- 8 R1ieman−nian p×roducts in Hn+1−(Hm(−r)× Sn−m(√r2 1), Hm( √1+r2) Sn−m(r), 77 Sm1 (√r2 1) Hn−m( r), 1Hm( √11−r2)× Hn−m( r)−) and ope−n pieces of×a qu1adratic 2 hypersur−face×x Mnc+−1 Rx,x− = d−, wh×ere R is a−self-adjoint constant matrix whose 12. Hmn1in+i1mal Rpon2l+y2n{.omW∈iahlenisHt2k|+ihsacto+nsitba,nat2a}−nd4bb≤is 0a,naonnd-zMeronc+c1onssttaanndtsvfeocrtoSrn1,+w1e⊂shRown1+t2haotr ⊂ 0 the hypersurface is totally umbilical, and then we also obtain a classification result (see 1 Theorem 2). : v i Mathematics Subject Classifications (2010): 53C50, 53B25, 53B30 X r a Keywords: linearizedoperatorLk; isoparametrichypersurface; k-maximalhypersurface; Takahashi theorem; higher order mean curvatures; Newton transformations. ∗ThisworkhasbeenpartiallysupportedbyMICINNProject No. MTM2009-10418, andFundaciońSéneca, Spain Project No. 04540/GERM/06. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundacioń Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007-2010). †Corresponding author. E-mail addresses: [email protected] and [email protected] 1 1 Introduction It is well known that the Laplacian operator of a hypersurface Mn immersed into Rn+1 is an (intrinsic) second-order linear differential operator, which arises naturally as the linearized operator of the firstvariation of the mean curvature for normal variations of the hypersurface. From this point of view, the Laplacian operator ∆ can be seen as the first one of a sequence of operators L = ∆, L ,...,L , where L stands for the linearized operator of the first 0 1 n 1 k { − } variation of the (k+1)th mean curvature, arising from normal variations of the hypersurface (see, for instance, [18]). These operators are given by L (f) = tr(P 2f), for a smooth k k ◦ ∇ function f on M, where P denotes the kth Newton transformation associated to the second k fundamental form of the hypersurface, and 2f denotes the self-adjoint linear operator met- ∇ rically equivalent to the hessian of f. In particular, when k = 1 the operator L is nothing 1 but the operator (cid:3) introduced by Cheng and Yau in [7] for the study of hypersurfaces with constant scalar curvature. Note that, in this context, the scalar curvature of M is nothing but εn(n 1)H , where H stands for the second mean curvature and ε= 1 depends on the 2 2 − ± causal character of the normal vector (see next section for details). From this point of view, and inspired by Garay’s extension of Takahashi theorem and its subsequent generalizations and extensions ([19], [6], [10], [8], [12], [1], [2], [3]), Al´ıas and Gu¨rbu¨z initiated in [4] the study of hypersurfaces in Euclidean space satisfying the general condition L ψ = Aψ + b, where A R(n+1) (n+1) is a constant matrix and b Rn+1 is a k × ∈ ∈ constant vector. They show that the only hypersurfaces satisfying that condition are open pieces of hypersurfaces with zero (k+1)-th mean curvature, or open pieces of a round sphere Sn(r), oropenpieces ofageneralized sphericalcylinderSm(r) Rn m, withk+1 m n 1. − × ≤ ≤ − Following the ideas contained in [4], we have completely extended to the Lorentz-Minkowski space the previous classification theorem obtained by Al´ıas and Gu¨rbu¨z. In particular, the following classification result was given in [14, Theorem 1]. Theorem A. ([14]) Let ψ : M Ln+1 be an orientable hypersurface immersed into the → Lorentz-Minkowski space Ln+1, and let L be the linearized operator of the (k + 1)th mean k curvature of M, for some fixed k = 0,1,...,n 1. Then the immersion satisfies the condition − L ψ = Aψ+b, for some constant matrix A R(n+1) (n+1) and some constant vector b Ln+1, k × ∈ ∈ if and only if it is one of the following hypersurfaces in Ln+1: 1. a hypersurface with zero (k+1)th mean curvature; 2. an open piece of the totally umbilical hypersurface Sn(r) or Hn( r); 1 − 3. an open piece of a generalized cylinder Sm(r) Rn m, Hm( r) Rn m, with k +1 1 × − − × − ≤ m n 1, or L m Sn m(r), with k+1 n m n 1. − ≤ − × ≤ − ≤ − In [5], and as a natural continuation of the study started in [4], Al´ıas and Kashani consider the study of hypersurfaces Mn immersed either into the sphere Sn+1 Rn+2 or into the ⊂ hyperbolic space Hn+1 Rn+2 whose position vector x satisfies the condition L x = Ax+b, ⊂ 1 k forsomeconstant matrix A R(n+2) (n+2) andsomeconstant vector b Rn+2,q = 0,1. They ∈ × ∈ q show the following two results: Theorem B. ([5]) The immersion x satisfies the condition L x = Ax, for some self-adjoint k constant matrix A R(n+2) (n+2), if and only if it is one of the following hypersurfaces: (1) a × ∈ hypersurface having zero (k+1)-th mean curvature and constant k-th mean curvature; (2) an open piece of a standard Riemannian product Sm(√1 r2) Sn m(r) Sn+1, 0 < r < 1; (3) − − × ⊂ an open piece of a standard Riemannian product Hm( √1+r2) Sn m(r) Hn+1, r > 0. − − × ⊂ 2 Theorem C. ([5]) The immersion x satisfies the condition L x = Ax + b, for some self- k adjoint constant matrix A R(n+2) (n+2) and some non-zero constant vector b Rn+2, if and × ∈ ∈ only if it is one of the following hypersurfaces: (1) an open piece of a totally umbilical round sphere Sn(r) Sn+1; (2) an open piece of a totally umbilical hyperbolic space Hn( r) Hn+1, ⊂ − ⊂ r > 1; (3) an open piece of a totally umbilical round sphere Sn(r) Hn+1, r > 0; (4) an open ⊂ piece of a totally umbilical Euclidean space Rn Hn+1. ⊂ The hypersurfaces studied in Theorems B and C are Riemannian, and thus their shape operators are always diagonalizable. However, when the ambient space is a Lorentzian space form Sn+1 or Hn+1, the shape operator of the hypersurface needs not be diagonalizable, 1 1 condition which plays a chief role in the Riemannian case. In this paper we extend, to the indefinite case, the results obtained in [5] for hypersurfaces immersed either into the sphere or intothehyperbolicspace. Forthesakeofsimplifyingthenotation andunifyingthestatements of our main results, let us denote by Mn+1 either the De Sitter space Sn+1 Rn+2 if c = 1, c 1 ⊂ 1 or the anti De Sitter space Hn+1 Rn+2 if c = 1. In this paper, we are able to give the 1 ⊂ 2 − following classification result. Theorem 1 Let ψ : M Mn+1 Rn+2 be an orientable hypersurface immersed into the → c ⊂ q space form Mn+1, and let L be the linearized operator of the (k+1)-th mean curvature of M, c k for some fixed k = 0,1,...,n 1. Then the immersion satisfies the condition L ψ = Aψ, for k − some self-adjoint constant matrix A R(n+2) (n+2), if and only if it is one of the following × ∈ hypersurfaces: (1) a hypersurface having zero (k+1)-th mean curvature and constant k-th mean curvature; (2) anopenpieceofastandard pseudo-Riemannianproduct inSn+1: Sm(r) Sn m(√1 r2), 1 1 × − − Hm( r) Sn m(√1+r2), Hm( √r2 1) Sn m(r). − − − × − − × (3) anopenpieceofastandard pseudo-Riemannianproduct inHn+1: Hm( r) Sn m(√r2 1), 1 1 − × − − Hm( √1+r2) Sn m(r), Sm(√r2 1) Hn m( r), Hm( √1 r2) Hn m( r). − × 1− 1 − × − − − − × − − (4) an open piece of a quadratic hypersurface x Mn+1 Rn+2 Rx,x = d , where R { ∈ c ⊂ q | h i } is a self-adjoint constant matrix whose minimal polynomial is t2+at+b, a2 4b 0. − ≤ Finally, in the case where A is self-adjoint and b is a non-zero constant vector, we are able to prove the following classification result. Theorem 2 Let ψ : M Mn+1 Rn+2 be an orientable hypersurface immersed into the → c ⊂ q space form Mn+1, and let L be the linearized operator of the (k+1)-th mean curvature of M, c k for some fixed k = 0,1,...,n 1. Assume that H is constant. Then the immersion satisfies k − the condition L ψ = Aψ+b, for some self-adjoint constant matrix A R(n+2) (n+2) and some k × ∈ non-zero constant vector b Rn+2, if and only if: ∈ q (i) c= 1 and it is an open piece of a totally umbilical hypersurface in Sn+1 Rn+2: Sn(r), 1 ⊂ 1 r > 1; Hn( r), r > 0; Sn(r), 0 < r < 1; Rn. (ii) c = 1 an−d it is an op1en piece of a totally umbilical hypersurface in Hn+1 Rn+2: − 1 ⊂ 2 Hn( r), r > 1; Hn( r), 0< r < 1; Sn(r), r > 0; Rn. 1 − − 1 1 2 Preliminaries In this section we recall some formulas and notions about hypersurfaces in Lorentzian space forms that will be used later on. Let Rn+2 be the (n+2)-dimensional pseudo-Euclidean space q 3 of index q 1, whose metric tensor , is given by ≥ h i q n+2 , = dx2+ dx2, h i − i j i=1 j=q+1 X X where x = (x ,...,x ) denotes the usual rectangular coordinates in Rn+2. The pseudo- 1 n+2 Euclidean De Sitter space of index q and radius r is defined by Sn+1(r) = x Rn+2 x,x = r2 , q { ∈ q | h i } and the pseudo-Euclidean anti-De Sitter space of index q and radius r is defined by − Hn+1( r)= x Rn+2 x,x = r2 . q − { ∈ q+1 | h i − } Throughoutthispaper,wewillconsiderboththecaseofhypersurfacesimmersedintoLorentzian De Sitter space Sn+1 Sn+1(1), and the case of hypersurfaces immersed into Lorentzian anti 1 ≡ 1 De Sitter space Hn+1 Hn+1( 1). In order to simplify our notation and computations, we 1 ≡ 1 − will denote by Mn+1 the De Sitter space Sn+1 or the anti De Sitter space Hn+1 according to c 1 1 c= 1or c = 1, respectively. We willuseRn+2 todenote thecorrespondingpseudo-Euclidean − q space where Mn+1 lives, so that q = 1 if c = 1 and q = 2 if c = 1. Then its metric is given c − by , = dx2+cdx2+dx2+ +dx2 , h i − 1 2 3 ··· n+2 and we can write Mn+1 = x Rn+2 x2+cx2+x2+ +x2 = c . c { ∈ q | − 1 2 3 ··· n+1 } It is well known that Sn+1 Rn+2 and Hn+1 Rn+2 are Lorentzian totally umbilical hyper- 1 ⊂ 1 1 ⊂ 2 surfaces with constant sectional curvature +1 and 1, respectively. − Let ψ : M Mn+1 Rn+2 be a connected orientable hypersurface with Gauss map N, −→ c ⊂ q N,N = ε = 1. Let 0, and denote the Levi-Civita connections on Rn+2, Mn+1 and h i ± ∇ ∇ ∇ q c M, respectively. Then the Gauss and Weingarten formulas are given by 0 Y = Y +ε SX,Y N c X,Y ψ, (1) ∇X ∇X h i − h i and SX = N = 0 N, −∇X −∇X for all tangent vector fields X,Y X(M), where S : X(M) X(M) stands for the shape ∈ −→ operator (or Weingarten endomorphism) of M, with respect to the chosen orientation N. Let = E ,E ,...,E be a (local) frame in Mn+1. Without loss of generality, we will B { 1 2 n+1} c say that is an orthornormal frame when B E ,E = 1 and E ,E = 0, j = 2,...,n+1, 1 1 1 j h i − h i E ,E = δ , 2 i,j n+1; i j ij h i ≤ ≤ and we will say that is a pseudo-orthornormal frame, when the following conditions are B satisfied: E ,E = 1 and E ,E = E ,E = 0, 1 2 1 1 2 2 h i − h i h i E ,E = 0, i= 1,2, j = 3,...,n+1, i j h i E ,E = δ , 3 i,j n+1. i j ij h i ≤ ≤ 4 It is well-known (see, for instance, [17, pp. 261–262]) that the shape operator S of the hypersurface M can be expressed, in an appropriate frame, in one of the following types: 0 κ b κ 0 − 1 b κ κ   2 I. S ≈  0 ... ; II. S ≈  κ3 ... , b 6= 0;  κ     n 0 κ     n   0 κ 0 0 0 κ 0 0 κ 1 1 κ     1 0 κ III. S ≈ 0 κ3 ...κn; IV. S ≈ −0 κ4 ...κn. (2)   In cases I and II, S is represented with respect to an orthonormal frame, whereas in cases III and IV, the frame is pseudo-orthonormal. The characteristic polynomial Q (t) of the shape operator S is given by S n Q (t) = det(tI S) = a tn k, with a = 1. S k − 0 − k=0 X Making use of the Leverrier–Faddeev method (see [13, 9]), the coefficients of Q (t) can be S computed, in terms of the traces of Sj, as follows: k 1 a = a tr(Sj), k = 1,...,n, with a = 1. (3) k k j 0 −k − j=1 X Bearing in mind the type of shape operator S, we can see that the coefficients of Q (t) for S S of types I, III and IV, are given by n a = κ , 1 i −   Xi=1 n (4)   ak = (−1)k κi1···κik, k = 2,...,n,  i1<X···<ik    whereas if S is of type II then they are given by n a = κ , 1 i −  i=1  X n n (5)   ak = (−1)k κi1···κik + b2 κi1···κik−2 , k = 2,...,n. " #  i1<X···<ik i1<ij·X·6=·<1i,k2−2      5 If S is of type II or III, then we consider that κ = κ = κ, and if S is of type IV we consider 1 2 that κ = κ = κ = κ. From now on, we will write 1 2 3 n n µ = κ κ and µJ = κ κ , k i1··· ik k i1··· ik i1<X···<ik i1<iXj··∈/·<Jik where k 1,...,n and J 1,...,n . Observe that ∈ { } ⊂ { } µ = µ and µ = κ µm +µm, (6) ∅k k k m k−1 k where µm stands for µ{m}. k k Then the coefficients a of characteristic polynomial Q (t), given in equations (4) and (5), k S can be easily written as follows a = ( 1)kµ , in cases I, III, IV; (7) k − k a = ( 1)k(µ +b2µ1,2 ), in case II. (8) k − k k−2 We use here that µ = 1 and µ = 0 if k < 0. 0 k The k-th mean curvature or mean curvature of order k of M is defined by n H = ( ε)ka , (9) k k k − (cid:18) (cid:19) n n! where = . In particular, when k = 1, k k!(n k)! (cid:18) (cid:19) − nH = εa = εtr(S), 1 1 − andsoH isnothingbuttheusualmeancurvatureH ofM,whichisoneofthemostimportant 1 extrinsiccurvaturesof thehypersurface. ThehypersurfaceM issaidto bek-maximal inMn+1 c if H 0. On the other hand, H defines a geometric quantity which is related to the k+1 2 ≡ (intrinsic) scalar curvature of M. Indeed, it follows from the Gauss equation of M that its Ricci curvature is given by Ric(X,Y) = (n 1)c X,Y +nH SX,Y ε SX,SY , X,Y X(M), (10) 1 − h i h i− h i ∈ and then, from (3), the scalar curvature Scal=tr(Ric) of M is Scal = n(n 1)c+ε a tr(S) tr(S2) = n(n 1)(c+εH ). (11) 1 2 − − − − (cid:16) (cid:17) 3 The Newton transformations The k-th Newton transformation of M is the operator P :X(M) X(M) defined by k −→ k P = a Sj. k k j − j=0 X 6 Equivalently, P can be defined inductively by k P = I and P = a I +S P . (12) 0 k k k 1 ◦ − Note that by Cayley-Hamilton theorem we have P = 0. The Newton transformations were n introduced by Reilly [18] in the Riemannian context; its definition was P = ( 1)kP . We k k − have the following properties of P (the proof is algebraic and straightforward). k Lemma 3 Let ψ : Mn Mn+1 be a hypersurface in the Lorentzian space form Mn+1. The → c c Newton transformations P satisfy: k (a) P is self-adjoint and commutes with S. k (b) tr(P )= (n k)a = c H . k k k k − (c) tr(S P ) = (k+1)a = εc H , 1 k n 1. k k+1 k k+1 ◦ − ≤ ≤ − (d) tr(S2 P )= a a (k+2)a = C nH H (n k 1)H , 1 k n 2. k 1 k+1 k+2 k 1 k+1 k+2 ◦ − − − − ≤ ≤ − Here, the constants ck and Ck are given by (cid:0) (cid:1) n n c c = ( ε)k(n k) = ( ε)k(k+1) and C = k . k k − − k − k+1 k+1 (cid:18) (cid:19) (cid:18) (cid:19) Next we are going to describe the covariant derivative of the shape operator S and the kth Newton transformation P . To do that, we will work with a (local) tangent frame of vector k fields E ,E ...,E in which S adopts its canonical form, and we need to distinguish four 1 2 n { } cases, according to the canonical form of the shape operator, see equation (2). Let (wj) be the connection 1-forms, defined by wj(X) = E ,E , so that wj = wi. i i h∇X i ji i − j The following four propositions are technical results that we will use later on. Their proofs are straightforward. Proposition 4 (S is of type I) Suppose that the shape operator S is of type I, and let E ,E ...,E be an orthonormal 1 2 n { } frame such that SE = κ E , i= 1,...,n. Then we have: i i i j ( S)E = X(κ )E + ε (κ κ )ω (X)E , ∇X i i i j i − j i j i=j X6 P E = ( 1)kµiE , k i i − k for every i = 1,...,n, where ε = E ,E . i i i h i Proposition 5 (S is of type II) Suppose that the shape operator S is of type II, and let E ,E ...,E be an orthonormal 1 2 n { } frame such that SE = κE +bE , SE = bE +κE , and SE = κ E , i 3. Then the 1 1 2 2 1 2 i i i − ≥ 7 covariant derivative S is given by ∇ n ( S)E = X(κ)+2bω2(X) E +X(b)E + (κ κ )ωj(X)+bωj(X) E , ∇X 1 1 1 2 − j 1 2 j (cid:0) (cid:1) Xj=3(cid:16) (cid:17) n ( S)E = X(b)E + X(κ)+2bω1(X) E + (κ κ )ωj(X)+bωj(X) E , ∇X 2 − 1 2 2 − j 2 1 j (cid:0) (cid:1) Xj=3(cid:16) (cid:17) ( S)E = κω1(X)+bω2(X) κ ω1(X) E + bω1(X) κω2(X)+κ ω2(X) E ∇X i i i − i i 1 i − i i i 2 n (cid:0) j(cid:1) (cid:0) (cid:1) +X(κ )E + (κ κ )ω (X)E , i 3. i i i − j i j ≥ j=1,2,i 6X The Newton transformation P satisfies k P E = ( 1)k(µ1E +bµ1,2 E ), k 1 1 2 − k k−1 P E = ( 1)k( bµ1,2 E +µ1E ), k 2 1 2 − − k−1 k P E = ( 1)k(µi +b2µ1,2,i)E , i 3. k i i − k k−2 ≥ Proposition 6 (S is of type III) Assumethattheshape operator S isoftype III,andlet E ,E ,...,E beapseudo-orthonormal 1 2 n { } frame such that SE = κE +E , SE = κE , and SE = κ E , i 3. Then the covariant 1 1 2 2 2 i i i ≥ derivative S satisfies ∇ n ( S)E = X(κ)E +2ω2(X)E + (κ κ )ωj(X)+ωj(X) E , ∇X 1 1 1 2 − j 1 2 j Xj=3(cid:16) (cid:17) n j ( S)E = X(κ)E + (κ κ )ω (X)E , ∇X 2 2 − j 2 j j=3 X ( S)E = κω2(X) κ ω2(X) E + ω2(X)+κω1(X) κ ω1(X) E ∇X i i − i i 1 i i − i i 2 n (cid:0) (cid:1) (cid:0) j (cid:1) +X(κ )E + (κ κ )ω (X)E , i 3. i i i − j i j ≥ j=1,2,i 6X The Newton transformation P is given by k P E = ( 1)k µ1E µ1,2 E , k 1 1 2 − k − k−1 PkE2 = ( 1)k(cid:0)µ1E2, (cid:1) − k P E = ( 1)kµiE , i 3. k i i − k ≥ Proposition 7 (S is of type IV) Supposethattheshape operator S isoftypeIV,andlet E ,E ,...,E beapseudo-orthonormal 1 2 n { } tangent frame such that SE = κE E , SE = κE , SE = E +κE , and SE = κ E , 1 1 3 2 2 3 2 3 i i i − 8 i 4. Then the covariant derivative S satisfies ≥ ∇ n ( S)E = X(κ)+ω2(X) E +2ω1(X)E ω2(X)E + (κ κ )ωj(X) ωj(X) E , ∇X 1 3 1 3 2 − 1 3 − j 1 − 3 j (cid:0) (cid:1) Xj=4(cid:16) (cid:17) n ( S)E = X(κ) ω3(X) E + (κ κ )ωj(X)E , ∇X 2 − 2 2 − j i j j=4 (cid:0) (cid:1) X n ( S)E = ω2(X)E + X(κ)+2ω3(X) E + (κ κ )ωj(X)+ωj(X) E , ∇X 3 − 1 2 2 3 − j 3 2 j (cid:2) (cid:3) Xj=4(cid:16) (cid:17) ( S)E = κω2(X) κ ω2(X) E + κω1(X) ω3(X) κ ω1(X) E ∇X i i − i i 1 i − i − i i 2 n + (cid:0)κ ω3(X) κω3(X)(cid:1) ω2(X(cid:0)) E +X(κ )E + ((cid:1)κ κ )ωj(X)E , i 4. i i − i − i 3 i i i − j i j ≥ j=1,2,3,i (cid:0) (cid:1) 6 X The Newton transformation P is given by k P E = ( 1)k(µ1E µ1,2,3E +µ1,2 E ), k 1 1 2 3 − k − k−2 k−1 P E = ( 1)kµ1E , k 2 2 − k P E = ( 1)k( µ1,2 E +µ1E ), k 3 2 3 − − k−1 k P E = ( 1)kµiE , i 4. k i i − k ≥ In the following lemma we present two new properties of the Newton transformations. For any differentiable function f (M), the gradient of f is the vector field f metrically ∞ ∈ C ∇ equivalent to df, which is characterized by f,X = X(f), for every differentiable vector h∇ i field X X(M). The divergence of a vector field X is the differentiable function defined as ∈ the trace of operator X, where X(Y) := X, that is, Y ∇ ∇ ∇ div(X) = tr( X) = gij X,E , ∇ h∇Ei ji i,j X E being any local frame of tangent vectors fields, where (gij) represents the inverse of the i { } metric (g ) = ( E ,E ). Analogously, the divergence of a operator T : X(M) X(M) is ij i j h i −→ the vector field div(T) X(M) defined as the trace of T, that is, ∈ ∇ div(T)= tr( T) = gij( T)E , ∇ ∇Ei j i,j X where T(E ,E )= ( T)E . ∇ i j ∇Ei j Lemma 8 The Newton transformation P , for k = 0,...,n 1, satisfies: k − (a) tr( S P ) = X(a ) = a ,X = εC H ,X . X k k+1 k+1 k k+1 ∇ ◦ − −h∇ i h∇ i (b) div(P ) = 0. k The proof can be found in [14]. Bearing in mind this lemma we obtain div(P ( f))= tr P 2f , k k ∇ ◦∇ (cid:0) (cid:1) 9 where 2f : X(M) X(M) denotes the self-adjoint linear operator metrically equivalent to ∇ −→ the Hessian of f, given by 2f(X),Y = ( f),Y , X,Y X(M). X ∇ h∇ ∇ i ∈ (cid:10) (cid:11) AssociatedtoeachNewtontransformationP ,wecandefinethesecond-orderlineardifferential k operator L : (M) (M) given by k ∞ ∞ C −→ C L (f) = tr P 2f . (13) k k ◦∇ (cid:0) (cid:1) When k = 0, L = ∆ is nothing but the Laplacian operator; when k = 1, L is the operator 0 1 (cid:3) introduced by Chen and Yau, [7]. An interesting property of L is the following. For every couple of differentiable functions k f,g C (M) we have ∞ ∈ L (fg) = div P (fg) = div P (g f +f g) k k k ◦∇ ◦ ∇ ∇ = gL (f)+fL (g)+2 P ( f), g . (14) k(cid:0) k (cid:1) (cid:0)k (cid:1) h ∇ ∇ i 4 Examples The goal of this section is to show some examples of hypersurfaces in the Lorentzian space form Mn+1 satisfying the condition L ψ = Aψ +b, where A is a constant matrix and b is a c k constantvector. Beforethat,wearegoingtocomputeL actingonthecoordinatecomponents k of the immersion ψ, that is, a function given by a,ψ , where a Rn+2 is an arbitrary fixed h i ∈ q vector. A direct computation shows that a,ψ =a = a ε a,N N c a,ψ ψ, (15) ⊤ ∇h i − h i − h i wherea X(M) denotes the tangential component of a. Taking covariant derivative in (15), ⊤ ∈ and using that 0 a = 0, jointly with the Gauss and Weingarten formulae, we obtain ∇X a,ψ = a = ε a,N SX c a,ψ X, (16) X X ⊤ ∇ ∇h i ∇ h i − h i for every vector field X X(M). Finally, by using (13) and Lemma 3, we find that ∈ L a,ψ = ε a,N tr(P S) c a,ψ tr(P I) k k k h i h i ◦ − h i ◦ = c H a,N cc H a,ψ . (17) k k+1 k k h i− h i Then we can compute L ψ as follows, k L ψ = L (δ ψ,e ),...,L (δ ψ,e ) k k 1 1 k n+2 n+2 h i h i (cid:16) (cid:17) = c H δ e ,N ,...,δ e ,N cc H δ e ,ψ ,...,δ e ,ψ k k+1 1 1 n+2 n+2 k k 1 1 n+2 n+2 h i h i − h i h i (cid:16) (cid:17) (cid:16) (cid:17) = c H N cc H ψ, (18) k k+1 k k − where e ,...,e stands for the standard orthonormal basis in Rn+2 and δ = e ,e . { 1 n+2} q i h i ii 10