Spacelike surfaces which admit a nondegenerate null normal section in a Lorentzian space form

Spacelike surfaces which admit a nondegenerate null normal section in a Lorentzian space form Daniel de la Fuente?, Francisco J. Palomo†and Alfonso Romero?∗ ?Departamento de Geometr´ıa y Topolog´ıa, Universidad de Granada, 18071 Granada, Spain E-mails: [email protected],[email protected] †Departamento de Matem´atica Aplicada, Universidad de M´alaga, 29071 M´alaga, Spain E-mail: [email protected] Abstract In this paper, we develop a formula for spacelike surfaces in a 4dimensional Lorentzian space form which involves its mean curvature vector field and the Gauss curvature of the induced metric and the Gauss curvature of the second fundamental form associated to a nondegenerate null normal section. By means of this formula, we stablish several sufficient conditions for compact spacelike surfaces with constant Gauss curvature to have a null umbilical direction. As another application, we give a new proof of the Liebmann rigidity theorem in Euclidean, hemispherical and hyperbolic spaces, and in the De Sitter spacetime. 2010 MSC: 53C24, 53C50; 53C42. Keywords: Spacelike surface; Gauss curvature; Liebmann theorem; Lorentzian space forms. ∗The first and third authors are partially supported by Spanish MINECO and ERDF project MTM2013-47828-C2-1-P. The second one by Spanish MINECO and ERDF proyect MTM2013-47828-C2-2-P. 1 Introduction Consider a spacelike surface Sin a 4-dimensional spacetime. At least locally, there are two future-directed null vector fields orthogonal to it, one of them ξ, pointing out to a direction will be called inwards, and the other one, η, outwards. At each event p∈S, the shape operator associated to ξ, Aξ(see definition in (1) for details), acting on a tangent vector v∈TpS, measures how does the inwards directed family of light rays (orthogonal to S) converge or diverge in the direction of v. Hence, if the mean curvature function associated to Aξis positive (resp. negative), then the inwards light rays tends to converge (resp. diverge) in average. Other physically relevant function is the mean null curvature function of a spacelike surface S, defined as φS= (trAξ)(trAη), where ξ, η are futuredirected null vectors orthogonal to Swith hξ, ηi=−1. The surface Sis called future converging when φS>0 and trAξ>0, i.e., when both families of light rays ortogonal to it converge. If, in addition, Sis compact, then S is a trapped surface. These surfaces are the precursors of the singularities in gravitational collapse (see, for instance, [6], [7], [14]). In this work, we are interested in spacelike surfaces such that at least one of its null-second fundamental forms IIξ(or IIη), defined in (Eq. 5), is non degenerate. It physically means that, in each event of S, there does not exist any direction v∈TpMsuch that the inwards directed light rays neither approach, nor separate, nor rotate. As a consequence, IIξdefines a new metric on the surface, which may be definite or not. If IIξis positive definite (resp. negative definite), then inwards directed light rays orthogonal to Sconverge (resp. diverge) along any direction at each event of S. In the indefinite case, the new metric is Lorentzian and the inwards directed light rays converge in some directions and diverge in others. An analogous interpretation may be done for the null direction η, but in this case the light rays are now directed outwards. Our first aim precisely consists in to find some sufficient conditions to assure that both null-second fundamental forms are non degenerate (Prop. 2.1). On the other hand, it is natural to ask ourselves about what relation is there between both metrics on such surfaces. With this aim, we find a formula which relate the Gauss curvatures of Swhen is endowed with the first and the second fundamental forms (Eq. 17). This formula widely generalizes the given in [2] and [3] for surfaces in the 3-dimensional De Sitter spacetime, and in [12] for surfaces in 4-dimensional Lorentz-Minkowski spacetime through a light cone. By means of this new formula, we stablish some integral conditions to characterize the null umbilical directions for compact 2 spacelike surfaces a null second fundamental form positive definite (Th. 4.1) and marginally trapped surfaces (Th. 4.4) in a Lorentzian space form. Finally, we study spacelike surfaces immersed in a totally umbilical hypersurface of a Lorentzian space form and, by using the previous characterization of umbilical directions of compact spacelike inmmersions, we give new proofs of the rigidity Liebmann theorem for surfaces on the Euclidean, hemispherical and hyperbolic 3-spaces (Th. 5.3, Th. 5.4 and Th. 5.6) and in the 3-dimensional De Sitter spacetime (Th. 5.7). 2 Preliminaries Let x:M2−→ M4 1(c) be a spacelike immersion of a 2-dimensional (connected) manifold M2into a 4-dimensional Lorentzian space form M4 1(c) of constant sectional curvature c. Denote by h,ifor the Lorentzian metric of M4 1(c) as well as the induced on M2via x. We write ∇and ∇the Levi-Civita connections of M2and M4 1(c), respectively, and let ∇⊥be the connection on the normal bundle of the submanifold. The Gauss and Weingarten formulas of xare ∇XY=∇XY+ II(X, Y ) and ∇Xξ=−AξX+∇⊥ Xξ, (1) for any tangent vector fields X, Y on M2and a normal vector field ξ. The shape (or Weingarten) operator Aξis related to the second fundamental form II by hAξX, Y i=hII(X, Y ), ξi. The mean curvature vector field is given by H=1 2trh,iII,and the Gauss and Codazzi equations of xare respectively, R(X, Y )Z=c{hY, ZiX−hX, ZiY}+AII(Y,Z)X−AII(X,Z)Y(2) (∇XII)(Y, Z) = (∇YII)(X, Z),(3) where Rstands for the curvature tensor of the induced metric and (∇XII)(Y, Z) = ∇⊥ XII(Y, Z)−II(∇XY, Z)−II(Y, ∇XZ), for any tangent vector fields X, Y, Z on M2. For each normal vector field ξ, the Codazzi equation provides us that, (∇XAξ)Y−(∇YAξ)X=A∇⊥ XξY−A∇⊥ YξX. (4) 3 Assume ξis globally defined and denote by IIξthe symmetric tensor field on M2, IIξ(X, Y ) = −hAξX, Y i=−hII(X, Y ), ξi.(5) When IIξis nondegenerate everywhere on M2, we will say that ξis a nondegenerate normal section [4, p. 59]. At any p∈M2,Aξis self-adjoint, hence, there exists an orthonormal basis of tangent vectors e1, e2to M2consisting of eigenvectors of Aξ, that is Aξ(ei) = λiei. The eigenvalues λ1, λ2are the principal curvatures and the eigenvectors e1, e2the principal directions of the normal direction ξ. The Casorati curvature of the normal direction ξis defined by, Cξ=tr(A2 ξ) 2=1 2(λ2 1+λ2 2).(6) Clearly Cξ= 0 if and only if the normal direction ξis geodesic. From now on we assume M4 1(c) is time orientable, i.e., there exists a globally defined timelike vector field Zon M4 1(c) (see [15, pp.345–346] for instance). Now, we take the normal component of Z,Z⊥, and we define the following future (with the same orientation that Z) unit timelike vector field orthogonal to M2, N=1 q−hZ⊥,Z⊥i Z⊥∈X⊥(M2). From now on we suppose M4 1(c) and M2to be orientable. Hence, we may construct a new unit future timelike vector field E∈X⊥(M2), such that, at each point p∈M2, (Np, Ep) be a orthonormal basis and for all positive oriented basis of TpM2, (e1, e2), then (Np, Ep, e1, e2) be a positive oriented orthonormal basis of Tx(p)M. Thus, there exist two independent future null vector fields ξ, η ∈X(M2) defined as follows, ξ=1 √2(N+E), η =1 √2(N−E), which trivialize the normal bundle of M2and satisfy hξ, ηi=−1. Moreover, such a pair of null sections is essentially unique, i.e., for any (ξ0, η0) satisfying the previous conditions it is holds that ξ0=f ξ and η0=1 fηfor some differentiable function fon M2. The following formula holds for II, II(X, Y ) = IIη(X, Y )ξ+ IIξ(X, Y )η, (7) 4 for every X, Y ∈X(M2). In particular, H=−hH, ηiξ−hH, ξiη. (8) Contracting in (2) we obtain, Ric(Y, Z) = chY, Zi+ 2hAHY, Zi+h(AξAη+AηAξ)Y, Zi,(9) and (K−c)Id =−2hH, ηiId +AηAξ+−2hH, ξiId +AξAη, where Kis the Gauss curvature of M2. Taking into account that trAξ= 2hH, ξiand analogously for η, we obtain that, (K−c)Id =AξAη+AηAξ+ 2AH,(10) Therefore, 2(K−c)=4hH,Hi+ 2tr(AξAη) = 4hH,Hi−hII,IIi,(11) where, as usual, hII,IIip= Σ2 i,j=1hII(ei, ej),II(ei, ej)ifor {e1, e2}an orthonormal basis of TpM2. In order to obtain a sufficient condition which asserts that the null normal sections ξand ηare nondegenerate we give the following result. Proposition 2.1. Let x:M2−→ M4 1(c)be a spacelike immersion. If the following inequality is satisfied, ϕ:= (K−c)2−4(K−c)hH,Hi+ 4 det(AH)>0, then the null normal sections ξand ηare nondegenerate. Proof. Assume det(Aξ) vanishes at p∈M2. Let e1, e2be the principal directions of the null normal direction ξwith Aξ(e1) = 0 and Aξ(e2) = λe2. A direct computation shows that det(AξAη+AηAξ) = −λ2hAη(e1), e2i2. But taking account (10), we get, det (K−c)Id −2AH= (K−c)2−4(K−c)hH,Hi+ 4 det(AH)≤0, which contradicts our assumption. 5 Remark 2.2. Note that ϕ(p) = 0 if and only if K(p)−cis an eigenvalue of 2AH. On the other hand, assume that µis a log-harmonic function defined on a simply-connected domain U⊂R2. In [6], it is shown that U, 1 µ(dx2+dy2) is a flat surface which can be isometrically immersed in the 4-dimensional Lorentz-Minkowski space L4. Moreover, its second fundamental form satisfies, II(∂x, ∂x) = 1 µ−1H,II(∂x, ∂y) = 0,II(∂y, ∂y) = 1 µ+ 1H, where H= (1,1,0,0). Taking η=Hit is clear that the null normal section ηis degenerate and det(Aξ)=1/µ2−1. Therefore, for suitable choices of µ > 0, we obtain that ξis nondegenerate. In this case, a direct computation shows that ϕ= 0 and thus Proposition 2.1 can not be weakened to ϕ≥0. On the other hand, the polynomial P(t) = t2−4hH, Hit+ 4 det(Aξ) has, at any point, non-negative discriminant as a consequence of the Schwarz inequality. Thus, the assumption P(K−c)>0 does not follows from a condition on P(t) independent of K−c. Corollary 2.3. Let x:M2−→ M4 1(c)be a spacelike immersion. Assume that M2is extremal (H= 0) and not totally geodesic, then the null normal sections ξand ηare nondegenerate, with IIηand IIξLorentzian metrics on M2. Proof. It is a direct consequence of (11). The Lorentzian signature is deduced from trAξ= 2hH, ξi= 0. Recall that a spacelike immersion x:M2−→ M4 1(c) is called pseudoumbilical when AH=ρId for ρ∈C∞(M). Corollary 2.4. Let x:M2−→ M4 1(c)be a pseudo-umbilical spacelike immersion. If K(p)6=c+ 2ρ(p)at every point p∈M2, then ξand ηare nondegenerate null normal sections Proof. Taking into account that det(AH) = ρ2and hH,Hi=ρthe result is a direct consequence of Proposition 2.1. Proposition 2.5. Let x:M2−→ M4 1(c)be a spacelike immersion with null normal vector field ξand Hthe mean curvature vector field of M2. Then, det(Aξ) = 2hH, ξi2−Cξ.(12) 6 Proof. The characteristic equation for the shape operator Aξ, A2 ξ−(trAξ)Aξ+ (detAξ)Id = 0, implies that, 2(detAξ) = (trAξ)2−(trA2 ξ). Taking into account that trAξ= 2hH, ξithe result easily follows. 3 Gauss curvature of IIξ Assume ξis a nondegenerate null normal section on M2into M4 1. In this case, IIξgiven by (5) provides with a new metric on M2. This section is devoted to obtain an explicit formula for the Gauss curvature of the metric IIξ. Let Ddenote the Levi-Civita connection of the metric tensor IIξ. The difference tensor Lbetween the Levi-Civita connections Dand ∇is given by, L(X, Y ) = DXY−∇XY, (13) for all X, Y ∈X(M2). From the Koszul formula for IIξand (4) we have, L(X, Y ) = 1 2A−1 ξh(∇XAξ)Y+A∇⊥ YξX+B(X, Y )i, where hB(X, Y ), Zi=−hA∇⊥ ZξX, Y ifor all X, Y, Z ∈X(M2). Since hξ, ξi= 0 we obtain that h∇⊥ Xξ, ξi= 0 for every X∈X(M2). Therefore there exists a 1-form ωsuch that, ∇⊥ Xξ=ω(X)ξ. (14) We define Θ ∈X(M2) by hΘ, Xi=ω(X) for every X∈X(M2). That is, Θ is the vector field h,i-metrically equivalent to ω. A direct computation shows that, B(X, Y ) = IIξ(X, Y )Θ, for every X, Y ∈X(M2). Therefore the symmetric difference tensor Lcan be written as follows, L(X, Y ) = 1 2A−1 ξh(∇XAξ)Yi+1 2ω(Y)X+1 2IIξ(X, Y )A−1 ξΘ.(15) 7 Remark 3.1. The curvature of the normal connection satisfies, R⊥(X, Y )ξ=dω(X, Y )ξ, R⊥(X, Y )η=−dω(X, Y )η. Therefore, for every normal vector field α∈X⊥(M2) we obtain that, R⊥(X, Y )α=dω(X, Y )¯α, where ¯α=hα, ξiη−hα, ηiξ. It should be pointed out that if we put Jα = ¯α then hJα, Jαi=−hα, αiand J2=Id. Now consider the Riemannian curvature tensor Rξof IIξwhich can be decomposed as follows Rξ=R+Q1+Q2 where, Q1(X, Y )Z= (DXL)(Y, Z)−(DYL)(X, Z), Q2(X, Y )Z=L(Y, L(X, Z)) −L(X, L(Y, Z)), and X, Y, Z ∈X(M2). Therefore we get the following formula for the Gauss curvature Kξof IIξ, 2Kξ= trIIξ(Ric) + trIIξ(c Q1) + trIIξ(c Q2) (16) where c Qi(X, Y ) = tr{Z7→ Qi(Z, X)Y}i= 1,2, and for a symmetric (0,2) tensor T, trIIξTis the ordinary trace of the (1,1)-tensor Tdefined by IIξ(T(X), Y ) = T(X, Y ). Lemma 3.2. The trace with respect to IIξof the Ricci tensor Ric is given by, trIIξ(Ric) = trIIξ(K·h ,i) = −2KhH, ξi det(Aξ). Proof. Let {E1, E2}be a h,i-orthonormal basis of TpM2, such that Aξ(Ei) = λiEiat a point p∈M2. Consider now Fi=|λi|−1/2Ei,i= 1,2, then {F1, F2} is a IIξ-orthonormal basis of TpM2with εi= IIξ(Fi, Fi) = −λi/|λi|. A direct computation gives now the result. Now observe that the vector field −A−1 ξΘ is metrically equivalent to ω with respect to IIξ. The following result relates this vector field with the second right term of (16). Lemma 3.3. The trace with respect to IIξof the tensor c Q1is given by, trIIξ(c Q1) = divIIξ(A−1 ξΘ). 8 Proof. Consider {E1, E2}and {F1, F2}constructed as previously. Extend F1 and F2near the point pas a local be a h,i-orthonormal IIξ-orthonormal frame {F1, F2}satisfying (DFiFj)p= 0. A direct computation shows that, htrIIξ(c Q1)ip=ε1ε2hIIξF2, Q1(F2, F1)F1)p+ IIξF1, Q1(F1, F2)F2)pi. On the other hand, IIξF2, Q1(F2, F1)F1)p= (F2)pIIξF2, L(F1, F1)−(F1)pIIξF2, L(F2, F1), and taking into account (21) we obtain, htrIIξ(c Q1)ip=−ε1(F1)pω(F1)−ε2(F2)pω(F2) = hdivIIξ(A−1 ξΘ)ip Lemma 3.4. The trace with respect to IIξof the tensor c Q2is given by, trIIξ(c Q2) = IIξ(L, L)−1 4 det(A2 ξ)IIξ∇IIξ(det(Aξ)),∇IIξ(det(Aξ)) +ωA−1 ξΘ + ∇IIξ(det(Aξ)) 2 det(Aξ). Proof. Let {E1, E2}be a local h,i-orthonormal frame at a point p∈M2 such that Aξ(Ei) = λiEifor i= 1,2 at p∈M2. Then, construct {F1, F2}as in Lemma 3.3. A direct computation shows, htrIIξ(c Q2)ip=ε1ε2hIIξL(F1, L(F2, F1)) −L(F2, L(F1, F1)), F2 +IIξL(F2, L(F2, F1)) −L(F1, L(F2, F2)), F1ip. Now, from (21) we obtain, IIξ(L(X, Y ), Z)−IIξ(L(X, Z), Y ) = ω(Y)IIξ(X, Z)−ω(Z)IIξ(X, Y ), and therefore, htrIIξ(c Q2)ip=ε1ε2hIIξL(F1, F2), L(F1, F2)−ε1(ω(F2))2 −IIξL(F2, F2), L(F1, F1)−ε2ω(L(F1, F1)) 9 Lemma 5.2. Let φ:M2→P3 εbe an immersion of a surface (spacelike if ε= 1) in P3 ε. If i:P3 ε→M4 1(c)is a totally umbilical hypersurface, then ∇⊥xξ=∇⊥xη= 0. Proof. Since ∇⊥x vξ=ω(v)ξa straightforward computation from (25) shows, ω(v) = −h∇⊥x vξ, ηi=h∇⊥x vT,Ni= 0. Now we apply our technique to give new proofs of results in [2]. Theorem 5.3. (Liebmann classical rigidity theorem [9]) Let φ:M2−→ E3 be a compact connected surface in the 3-dimensional Euclidean space. If the Gauss curvature of M2is a positive constant K, then M2is a totally umbilical round sphere. Proof. From the Gauss-Bonnet Theorem, the surface M2is topologically an sphere. Let i:E3,→L4be the usual totally geodesic embedding at t= 0. Since the scalar αin (24) is zero, we deduce that det(Ax ξ) = det(Ax η) = K 2, making use of (26). Therefore Ax ξand Ax ηare positive definite (otherwise, we make a change of ξor ηto −ξor −η). Now, Lemma 5.2 and Corollary 4.2 can be called to ensure that the null normal sections ξand ηare umbilical and so, M2is a totally umbilical round sphere in L4and so also in E3. Theorem 5.4. Let φ:M2−→ S3 +be a compact connected surface in the 3dimensional north hemisphere S3 +. If the Gauss curvature of M2is a constant (K > 1), then M2is a totally umbilical round sphere. Proof. Again here we have that M2is a topological sphere. Let i:S3,→S4 1 be the totally geodesic embedding at t= 0, where we have considered S4 1as the warped product R×cosh(t)S3. Since the scalar αin (24) is 0, making use of (26) we deduce that det(Ax ξ) = det(Ax η) = 1 2(K−1) >0. The last inequality is due to on a surface in a hemisphere an elliptic point p0is always reached (then, K(p0)>1). Therefore Ax ξand Ax ηare positive definite (otherwise, we make a change of ξor ηto −ξor −η). Now, Lemma 16 5.2 and Corollary 4.2 can be called again to ensure that the null normal sections ξand ηare then umbilical and so, M2is a totally umbilical round sphere in S3 +. Remark 5.5. Let φ:M2−→ S3be a minimal compact connected surface of (arbitrary) Gauss curvature Kand i:S3→S4 1the above embedding. From Theorem 4.1 and (26), we derive that there is at least a point p∈M2with satisfies K(p) = 0 or at least a point with K(p) = 1. In particular, if Kis constant, we can deduce the classical result K= 0 or K= 1. Consider the 4-dimensional anti De sitter spacetime of sectional curvature −1, H4 1={v∈R5:−v2 0−v2 1+v2 2+v2 3+v2 4=−1}. The 3-dimensional hyperbolic space H3, the complete simply connected Riemannian manifold with sectional curvature −1, may be realized as the following totally geodesic spacelike hypersurface of H4 1, H3={v∈H4 1:v0= 0, v1>0}. Theorem 5.6. Let φ:M2−→ H3be a compact connected surface in the 3dimensional hyperbolic space. If the Gauss curvature Kis a positive constant, then M2is a totally umbilical round sphere. Proof. We have that, one more time, M2is a topological sphere. Let i: H3,→H4 1be the above embedding. Since the scalar αin (24) is 0, making use of (26) we deduce that det(Ax ξ) = det(Ax η) = 1 2(K+ 1). Therefore Ax ξand Ax ηare positive definite (otherwise, we make a change of ξor ηto −ξor −η). Now, Lemma 5.2 and Corollary 4.2 can be called one more time to ensure that the null normal sections ξand ηare umbilical and so, M2is a totally umbilical round sphere in H3. We end this article obtaining a new proof of [3, Th. 12]. Theorem 5.7. Let φ:M2−→ S3 1be a compact connected spacelike surface with constant positive Gauss curvature K < 1. Then M2is a totally umbilical round sphere. 17 Proof. From the assumption on K, we have that M2is a topological sphere. Denote by i:S3 1,→S4 1the usual totally geodesic immersion, and consider as bellow x=i◦φ. Since α= 0, we deduce that det(Ax ξ) = det(Ax η) = 1 2(1 −K). Therefore Ax ξand Ax ηare positive definite. Lemma 5.2 and Corollary 4.2 imply that the null normal sections ξand ηare umbilical. Therefore, M2is a totally umbilical round sphere in S3 1. Remark 5.8. 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