Timelike surfaces in Minkowski space with a canonical null direction
aa r X i v : . [ m a t h . DG ] A ug Timelike surfaces in Minkowski spacewith a canonical null direction
Victor H. Patty-Yujra ∗ and Gabriel Ruiz-Hern´andez † September 23, 2018
Abstract
Given a constant vector field Z in Minkowski space, a timelike surfaceis said to have a canonical null direction with respect to Z if the projectionof Z on the tangent space of the surface gives a lightlike vector field. Inthis paper we describe these surfaces in the ruled case. For example whenthe Minkowski space has three dimensions then a surface with a canonicalnull direction is minimal and flat. On the other hand, we describe severalproperties in the non ruled case and we partially describe these surfacesin four-dimensional Minkowski space. We give different ways for buildingthese surfaces in four-dimensional Minkowski space and we finally use theGauss map for describe another properties of these surfaces. Keywords:
Timelike surfaces; canonical null direction; principal direction.
Mathematics Subject Classification 2010:
Introduction
We consider R n, the ( n + 1) − dimensional Minkowski space defined by R n +1 endowed with the metric of signature ( n, h· , ·i = − dx + dx + . . . + dx n +1 . A surface M in R n, is said to be timelike if the metric h· , ·i induces a Lorentzianmetric, i.e. a metric of signature (1 , , on M. ∗ email: [email protected], Instituto de Matem´aticas UNAM, Unidad Juriquilla.Quer´etaro, M´exico. † email: [email protected], Instituto de Matem´aticas UNAM, Unidad Juriquilla.Quer´etaro, M´exico. efinition 1. We say that a timelike surface M in R n, has a canonical nulldirection with respect to a constant vector field Z in R n, if the tangent part Z ⊤ of Z is a lightlike vector field along M, i.e. Z ⊤ is nonzero and h Z ⊤ , Z ⊤ i = 0 . We will say that Z defines a null direction on the surface.In this paper, we are interested in the description of timelike surfaces witha canonical null direction in Minkowski space. We will begin by describing thecompatibility equations which determine a canonical null direction on a surfaceand we will see that there exists two different cases for consider: the ruled andthe non ruled case. We give a complete description of these surfaces in theruled case (Theorem 2.2). On the other hand, we give several properties inthe non ruled case and we partially describe these surfaces in four-dimensionalMinkowski space (Proposition 3.7 and Theorem 3.10). We also give differentways for building these surfaces in four-dimensional Minkowski space and wefinally use the Gauss map for describe another properties of these surfaces.The notion of a canonical null direction only makes sense for timelike sub-manifolds in the n + 1-dimensional Minkowski space and it is inspired in theconcept of surfaces with canonical principal direction with respect to a parallelvector field defined by F. Dillen and his collaborators in [4] and [5]. The sec-ond author together with E. Garnica and O. Palmas in [6] investigated the caseof hypersurfaces with a canonical principal direction with respect to a closedconformal vector field.The paper is organized as follows. In Section 1 we describe the compatibilityequations which determine a canonical null direction on a timelike surface andwe give some properties about their geometry. In Corollary 1.8 we proved thatif a surface in R n, has parallel mean curvature then it is minimal. In Section 2we give a classification of these surfaces in Minkowski space in the ruled case. InSection 3 we study the non ruled case: we give some properties and we partiallydescribe these surfaces in four-dimensional Minkowski space. We consider a timelike surface M in R n, with a canonical null direction Z. We can assume that Z is a unit spacelike vector field; therefore, using thenatural decomposition Z = Z ⊤ + Z ⊥ and since h Z ⊤ , Z ⊤ i = 0 we have that h Z ⊥ , Z ⊥ i = 1 . Here and below we denote by h· , ·i the metric on the Minkowskispace, on T M and on the normal bundle
N M.
We will denote by II : T M × T M → N M the second fundamental form ofthe immersion M ⊂ R n, given by II ( X, Y ) = ∇ X Y − ∇ X Y, where ∇ and ∇ are the Levi Civita connections of R n, and M, respectively.As usual, if ν ∈ N M, A ν : T M → T M stands for the symmetric operator such2hat h A ν ( X ) , Y i = h II ( X, Y ) , ν i , for all X, Y ∈ T M.
Finally, we denote by ∇ ⊥ the Levi Civita connection of thenormal bundle N M.
The following lemma is fundamental.
Lemma 1.1.
We have ∇ X Z ⊤ = A Z ⊥ ( X ) and ∇ ⊥ X Z ⊥ = − II ( Z ⊤ , X ) , (1) for all X ∈ T M.
Proof.
Using the Gauss and Weingarten equations, we obtain that0 = ∇ X Z = ∇ X Z ⊤ + ∇ X Z ⊥ = ∇ X Z ⊤ − A Z ⊥ ( X ) + II ( Z ⊤ , X ) + ∇ ⊥ X Z ⊥ ;the result follows by taking tangent and normal parts. Lemma 1.2.
We have A Z ⊥ ( Z ⊤ ) = 0 and ∇ Z ⊤ Z ⊤ = 0 . In particular, Z ⊤ is a canonical principal direction on the surface.Proof. Using (1) we get h A Z ⊥ ( Z ⊤ ) , X i = h II ( Z ⊤ , X ) , Z ⊥ i = −h∇ ⊥ X Z ⊥ , Z ⊥ i = − X h Z ⊥ , Z ⊥ i = 0 , for all X ∈ T M.
Finally, ∇ Z ⊤ Z ⊤ = A Z ⊥ ( Z ⊤ ) = 0.Let us consider W a lightlike vector field tangent to M ( i.e. W is nonzeroand h W, W i = 0) such that h Z ⊤ , W i = − . Remark 1.3.
If we consider the frame ( Z ⊤ , W ) of lightlike vector fields on T M (with h Z ⊤ , W i = − ), the mean curvature vector of the immersion is given by ~H := 12 tr h , i II = − II ( Z ⊤ , W ) . We define the function a := h II ( W, W ) , Z ⊥ i . Lemma 1.4.
The Levi-Civita connection of M satisfies the following relations: ∇ Z ⊤ Z ⊤ = 0 = ∇ Z ⊤ W, ∇ W Z ⊤ = − aZ ⊤ and ∇ W W = aW. In particular, [ Z ⊤ , W ] = aZ ⊤ . roof. The first equality was given in Lemma 1.2. Now, h W, W i = 0 im-plies that h∇ Z ⊤ W, W i = 0; and h Z ⊤ , W i = − h∇ Z ⊤ Z ⊤ , W i + h Z ⊤ , ∇ Z ⊤ W i = h Z ⊤ , ∇ Z ⊤ W i ; therefore, ∇ Z ⊤ W = −h∇ Z ⊤ W, W i Z ⊤ − h∇ Z ⊤ W, Z ⊤ i W = 0 . In a similar way, using h Z ⊤ , Z ⊤ i = 0 we deduce that h∇ W Z ⊤ , Z ⊤ i = 0; using(1) we get h∇ W Z ⊤ , W i = h A Z ⊥ ( W ) , W i = h II ( W, W ) , Z ⊥ i = a ; thus, ∇ W Z ⊤ = −h∇ W Z ⊤ , W i Z ⊤ − h∇ W Z ⊤ , Z ⊤ i W = − aZ ⊤ . On the other hand, since h∇ W W, W i = 0 , and h∇ W W, Z ⊤ i = −h W, ∇ W Z ⊤ i = h W, aZ ⊤ i = − a, we deduce that, ∇ W W = −h∇ W W, W i Z ⊤ − h∇ W W, Z ⊤ i W = aW. Finally, [ Z ⊤ , W ] = ∇ Z ⊤ W − ∇ W Z ⊤ = aZ ⊤ , because ∇ Z ⊤ W = 0 . We have the following relations for the curvature tensors of M . Proposition 1.5.
The curvature tensor R and the normal curvature tensor R ⊥ of M in R n, are given by R ( Z ⊤ , W ) Z ⊤ = Z ⊤ ( a ) Z ⊤ and R ⊥ ( Z ⊤ , W ) Z ⊥ = aII ( Z ⊤ , Z ⊤ ) . Proof.
Using the equalities of Lemma 1.4, we get R ( Z ⊤ , W ) Z ⊤ = ∇ W ∇ Z ⊤ Z ⊤ − ∇ Z ⊤ ∇ W Z ⊤ + ∇ [ Z ⊤ ,W ] Z ⊤ = −∇ Z ⊤ ( − aZ ⊤ ) + ∇ ( aZ ⊤ ) Z ⊤ = Z ⊤ ( a ) Z ⊤ . On other hand, by (1) we have R ⊥ ( Z ⊤ , W ) Z ⊥ = ∇ ⊥ W ∇ ⊥ Z ⊤ Z ⊥ − ∇ ⊥ Z ⊤ ∇ ⊥ W Z ⊥ + ∇ ⊥ [ Z ⊤ ,W ] Z ⊥ = −∇ ⊥ W ( II ( Z ⊤ , Z ⊤ )) + ∇ ⊥ Z ⊤ ( II ( W, Z ⊤ )) − aII ( Z ⊤ , Z ⊤ );by Codazzi equation and the equalities of Lemma 1.4, we obtain that − ∇ ⊥ W ( II ( Z ⊤ , Z ⊤ )) + ∇ ⊥ Z ⊤ ( II ( W, Z ⊤ ))= − (cid:16) ˜ ∇ W II (cid:17) ( Z ⊤ , Z ⊤ ) − II ( ∇ W Z ⊤ , Z ⊤ ) − II ( Z ⊤ , ∇ W Z ⊤ )+ (cid:16) ˜ ∇ Z ⊤ II (cid:17) ( W, Z ⊤ ) + II ( ∇ Z ⊤ W, Z ⊤ ) + II ( W, ∇ Z ⊤ Z ⊤ )= 2 aII ( Z ⊤ , Z ⊤ ) , this finish the proof. 4 orollary 1.6. The Gaussian curvature of M is given by K = h R ( Z ⊤ , W ) Z ⊤ , W i| Z ⊤ | | W | − h Z ⊤ , W i = Z ⊤ ( a ) . Using the formula above for the Gauss curvature K, we will find a relationbetween the norm of the mean curvature vector and the Gaussian curvature. Proposition 1.7.
The mean curvature vector and its derivative satisfies thefollowing relations: ∇ ⊥ W ~H = −∇ ⊥ Z ⊤ ( II ( W, W )) and | ~H | = −h∇ ⊥ W ~H, Z ⊥ i . (2) Moreover, we have K = | ~H | − h II ( W, W ) , II ( Z ⊤ , Z ⊤ ) i . Proof.
By Codazzi equation and the formulae of Lemma 1.4, we have ∇ ⊥ Z ⊤ ( II ( W, W )) = (cid:16) ˜ ∇ Z ⊤ II (cid:17) ( W, W ) + 2 II ( ∇ Z ⊤ W, W )= (cid:16) ˜ ∇ W II (cid:17) ( Z ⊤ , W )= ∇ ⊥ W ( II ( Z ⊤ , W )) − II ( ∇ W Z ⊤ , W ) − II ( Z ⊤ , ∇ W W )= −∇ ⊥ W ~H + aII ( Z ⊤ , W ) − aII ( Z ⊤ , W )= −∇ ⊥ W ~H. On other hand, since h ~H, Z ⊥ i = −h II ( Z ⊤ , W ) , Z ⊥ i = 0 (see Lemma 1.2), from(1) we get0 = W h ~H, Z ⊥ i = h∇ ⊥ W ~H, Z ⊥ i + h ~H, ∇ ⊥ W Z ⊥ i = h∇ ⊥ W ~H, Z ⊥ i − h ~H, II ( Z ⊤ , W ) i = h∇ ⊥ W ~H, Z ⊥ i + | ~H | . Therefore, by Corollary 1.6 and the equalities in (1)-(2) we obtain K = Z ⊤ ( a ) = Z ⊤ h II ( W, W ) , Z ⊥ i = h∇ ⊥ Z ⊤ ( II ( W, W )) , Z ⊥ i + h II ( W, W ) , ∇ ⊥ Z ⊤ Z ⊥ i = −h∇ ⊥ W ~H, Z ⊥ i − h II ( W, W ) , II ( Z ⊤ , Z ⊤ ) i which proves the assertion. Corollary 1.8.
If the mean curvature vector ~H is parallel then the surface M is minimal, i.e. ~H = 0 . Proof.
This is a consequence of the second equality in Proposition 1.7.The normal curvature tensor R ⊥ is determined by the vector II ( Z ⊤ , Z ⊤ ) , which is orthogonal to Z ⊥ (see the proof of Lemma 1.2: h II ( Z ⊤ , Z ⊤ ) , Z ⊥ i = 0).Therefore, we can consider two cases: when II ( Z ⊤ , Z ⊤ ) = 0 (the ruled case)and when II ( Z ⊤ , Z ⊤ ) = 0 (the non ruled case).5 The ruled case
In this section we study the case of a timelike surface M in R n, with a canonicalnull direction Z such that II ( Z ⊤ , Z ⊤ ) = 0 . By Remark 1.3 and Proposition1.5, the Gauss curvature and the normal curvature tensor satisfy the followingrelations: | ~H | − K = 0 and R ⊥ = 0 . (3)The timelike surfaces in four-dimensional pseudo Euclidean space for which (3)is valid are called umbilic (if II −h· , ·i ~H = 0) or quasi-umbilic (if II −h· , ·i ~H = 0).See e.g. [1, 2].The surfaces in R , such that | ~H | − K = 0 were classified in [3]. Remark 2.1.
The normal vector field Z ⊥ is parallel if and only if M is minimal.Let us verify this fact. By (1) , we have ∇ ⊥ Z ⊤ Z ⊥ = − II ( Z ⊤ , Z ⊤ ) = 0 and ∇ ⊥ W Z ⊥ = − II ( Z ⊤ , W ) = ~H, which proves the assertion. The next result gives a local description of a timelike surface M in R n, witha canonical null direction Z such that II ( Z ⊤ , Z ⊤ ) = 0 . We moreover assumethat Z is not orthogonal to the surface; otherwise, M should be any surface ina hyperplane orthogonal to Z . Theorem 2.2.
A timelike surface M in R n, has a canonical null direction withrespect to Z and satisfies the condition II ( Z ⊤ , Z ⊤ ) = 0 if and only if M can belocally parametrized by ψ ( x, y ) = α ( x ) + y Z ⊤ ( x ) , (4) where α ( x ) is a lightlike curve in R n, , Z ⊤ ( x ) is the restriction of the null vectorfield Z ⊤ along α and where the following conditions holds • Z is not orthogonal to α ′ ( x ) for every x, • the vectors α ′ ( x ) and Z ⊤ ( x ) are linearly independent for every x, • the position vectors Z ⊤ ( x ) gives a curve in a timelike hyperplane.Proof. Let us consider a coordinate system ( x, y ) ψ ( x, y ) of M such themetric of M is given by h· , ·i = − λ ( x, y ) dxdy, where λ is some positive function; we moreover assume that Z ⊤ = ∂ψ∂y satisfies II ( Z ⊤ , Z ⊤ ) = 0 . By calculating the Christoffel symbols of the metric we getthat ∇ Z ⊤ Z ⊤ = 1 λ ∂λ∂y Z ⊤ . T := λ Z ⊤ satisfies that ∇ Z ⊤ T = 0 and II ( T, Z ⊤ ) = 0 . Since ∇ Z ⊤ T = ∇ Z ⊤ T + II ( T, Z ⊤ ) = 0 , we have that T ( ψ ( x, y )) = T ( ψ ( x, Z y ∂∂u ( T ( ψ ( x, u ))) du = T ( ψ ( x, . Then, Z ⊤ ( ψ ( x, y )) = λ ( x,y ) λ ( x, Z ⊤ ( ψ ( x, , and therefore, ψ ( x, y ) = ψ ( x,
0) + Z y ∂ψ∂u ( x, u ) du = ψ ( x,
0) + (cid:18)Z y λ ( x, u ) λ ( x, du (cid:19) Z ⊤ ( ψ ( x, . So, ψ can be written as ψ ( x, y ) = α ( x ) + f ( x, y ) Z ⊤ ( x ) , where α ( x ) := ψ ( x,
0) is a lightlike curve in R n, and Z ⊤ ( x ) := Z ⊤ ( ψ ( x, α. Since ψ ( x,
0) = α ( x ), we have that f ( x,
0) = 0;moreover ∂f∂y = λ ( x, y ) λ ( x, > . So, the formulae x ′ = x and y ′ = f ( x, y ) , define local coordinates such that(4) is valid. Moreover, since Z = Z ⊤ + Z ⊥ , we have that Z is a spacelikeconstant vector with h Z, Z ⊤ i = 0 , in particular h Z, Z ⊤ ( x ) i = 0 for all x ; thusthe positions vectors Z ⊤ ( x ) are orthogonal to Z and so they are contained inthe timelike hyperplane orthogonal to Z. Reciprocally, suppose that M is parametrized as in (4). Since the positionsvectors Z ⊤ ( x ) lives in a timelike hyperplane, we can choose a constant spacelikevector in the spacelike line orthogonal to the hyperplane. So, h Z, Z ⊤ ( x ) i = 0for all x. This implies that the tangent part of Z is ∂ψ∂y = Z ⊤ ( x ) because Z isnot orthogonal to α ′ ( x ). Finally, since M is a ruled surface with rules in thedirection Z ⊤ ( x ) , we deduce that II ( Z ⊤ , Z ⊤ ) = 0. R , In this case, the normal vector Z ⊥ is parallel; by (1) we have that II ( Z ⊤ , X ) = 0 , for all X ∈ T M ; see Remark 2.1. Using moreover (3) we get:
Corollary 2.3.
A timelike surface M in R , with a canonical null direction Z is flat and minimal. Theorem 2.4.
A timelike surface M in R , with a canonical null direction Z can be locally parametrized by ψ ( x, y ) = α ( x ) + y T , (5) where α ( x ) is a lightlike curve in R , , T is some constant lightlike vector along α, and the vectors α ′ ( x ) and T are linearly independent for every x. roof. Let us observe that in this case we have that II ( Z ⊤ , Z ⊤ ) = 0 . We canadapt the proof of Proposition 2.2 to obtain that M can be locally parametrizedas in (4). The second fundamental form in the coordinates ( x, y ) is given by II = ( d ψ ) N . The mean curvature vector ~H = g ij II ij = h α ′ ,Z ⊤ i (cid:8) ( Z ⊤ ) ′ (cid:9) N satisfies the relation K = | ~H | = | ( Z ⊤ ) ′ | h α ′ , Z ⊤ i . therefore, the condition K = | ~H | = 0 , is equivalent to | ( Z ⊤ ) ′ | = 0 . Since | Z ⊤ | = 0 and h ( Z ⊤ ) ′ , Z ⊤ i = 0 , we have the relation ( Z ⊤ ) ′ ( x ) = h ( x ) Z ⊤ ( x ) . Thus, by integration we get Z ⊤ ( x ) = H ( x ) Z ⊤ (0) where H is a smooth functionsuch that H (0) = 1 . Using the change of variable x ′ = x, y ′ = yH ( x ) , andwriting T := Z ⊤ (0) , we find that M is parametrized by α ( x ′ ) + y ′ T , for smallvalues of x ′ and y ′ . Reciprocally, suppose that M is parametrized as in (5). Thus, a spacelikeconstant vector Z in R , such that h Z, T i = 0 , defines a canonical null directionon M. Moreover, Z ⊤ ( x ) = H ( x ) T , for some smooth function H ( x ) . In this section we study the case of a timelike surface M in R n, with a canonicalnull direction Z such that II ( Z ⊤ , Z ⊤ ) = 0 . We note that, as a consequence ofProposition 1.5 and Corollary 1.6, we have the following:
Corollary 3.1.
Let us assume that the surface M has normal curvature tensor R ⊥ identically zero (i.e. the function a is identically zero). Then the Gausscurvature K is also constant zero. We note that, if we assume that ∇ a is a multiple of Z ⊤ we get that the Gausscurvature K = Z ⊤ ( a ) = h∇ a, Z ⊤ i (Corollary 1.6) is zero. We will describe theconverse statement. We need some lemmas. Lemma 3.2.
There is a local smooth function f : M → R such that ∇ f = Z ⊤ . Moreover, f is a harmonic function, i.e. ∆ f = 0 . Proof.
We consider the 1 − form θ ( X ) = h X, Z ⊤ i , for all X ∈ T M.
Using theequalities of Lemma 1.4, we get θ is a closed 1 − form, i.e. dθ = 0; thus, thereexists a function f : M → R such that df = θ, that is ∇ f = Z ⊤ . We compute the laplacian of the function f. In the orthonormal frame (cid:16) Z ⊤ + W √ , Z ⊤ − W √ (cid:17) on T M, we get∆ f = − f ( Z ⊤ , W ) = − h∇ Z ⊤ ∇ f, W i = − h∇ Z ⊤ Z ⊤ , W i = 0 , because ∇ Z ⊤ Z ⊤ = 0 (see Lemma 1.2). 8 emma 3.3. The laplacian of the function a = h II ( W, W ) , Z ⊥ i is given by ∆ a = − Ka − W ( K ) , where K is the Gauss curvature of the surface. In particular, if the Gausscurvature is zero, a is a harmonic function.Proof. In the same frame, as in the proof of Lemma 3.2, we get∆ a = − f ( W, Z ⊤ ) = − h∇ W ∇ a, Z ⊤ i . On the other hand, since K = Z ⊤ ( a ) = h∇ a, Z ⊤ i (Corollary 1.6), using Lemma1.4 we obtain W ( K ) = W h∇ a, Z ⊤ i = h∇ W ∇ a, Z ⊤ i + h∇ a, ∇ W Z ⊤ i = −
12 ∆ a − a h∇ a, Z ⊤ i which is the equality of the lemma. Proposition 3.4.
The Gauss curvature K is zero if and only if there exists aharmonic function a : M → R such that ∇ a = a Z ⊤ . Proof.
We assume that the Gauss curvature K is zero: since K = Z ⊤ ( a ) = h∇ a, Z ⊤ i (Corollary 1.6), there exists a smooth function a : M → R such that ∇ a = a Z ⊤ because Z ⊤ is a null vector field. Using Lemmas 3.2 and 3.3, weobtain0 = ∆ a = div( ∇ a ) = div( a ∇ f ) = h∇ a , ∇ f i + a ∆ f = h∇ a , Z ⊤ i , thus, there exists a smooth function a : M → R such that ∇ a = a Z ⊤ . Thelaplacian of the function a is given by∆ a = − h∇ W ∇ a , Z ⊤ i = − h W ( a ) Z ⊤ + a ∇ W Z ⊤ , Z ⊤ i = 0 . Note that, we can continue with this procedure. R , In this case, we consider the normalized vector field ν := II ( Z ⊤ , Z ⊤ ) | II ( Z ⊤ , Z ⊤ ) | ∈ N M.
Note that ν is orthogonal to Z ⊥ (see Lemma 1.2). We recall that Z ⊥ is a space-like vector field with h Z ⊥ , Z ⊥ i = 1 . So, ( Z ⊥ , ν ) defines an oriented orthonormalframe of the normal bundle N M along M . Corollary 3.5.
The normal curvature of the surface M in R , is given by K N = a | II ( Z ⊤ , Z ⊤ ) | . roof. Using the Ricci equation, in the orthonormal frame (cid:16) Z ⊤ + W √ , Z ⊤ − W √ (cid:17) on T M, we obtain K N = (cid:28) ( A Z ⊥ ◦ A ν − A ν ◦ A Z ⊥ ) (cid:18) Z ⊤ + W √ (cid:19) , Z ⊤ − W √ (cid:29) = −h ( A Z ⊥ ◦ A ν − A ν ◦ A Z ⊥ )( Z ⊤ ) , W i = h R ⊥ ( Z ⊤ , W ) Z ⊥ , ν i ;we get the result by replacing the second equality given in Proposition 1.5.Now, we will give a relation between the Gauss curvature, the normal cur-vature and the mean curvature vector of M in R , . Lemma 3.6.
In the orthonormal frame ( Z ⊥ , ν ) orthogonal to M, we have thefollowing relation II ( W, W ) = K N | II ( Z ⊤ , Z ⊤ ) | Z ⊥ + | ~H | − K | II ( Z ⊤ , Z ⊤ ) | ν. In particular, | II ( W, W ) | | II ( Z ⊤ , Z ⊤ ) | = ( | ~H | − K ) + K N . Proof.
We have II ( W, W ) = h II ( W, W ) , Z ⊥ i Z ⊥ + h II ( W, W ) , ν i ν = aZ ⊥ + h II ( W, W ) , II ( Z ⊤ , Z ⊤ ) i| II ( Z ⊤ , Z ⊤ ) | ν, we get the result by using Corollary 3.5 and Proposition 1.7.Using the lemma above we have the following description in a simple case: Proposition 3.7.
Consider a timelike surface M in R , with a canonical nulldirection Z such that II ( Z ⊤ , Z ⊤ ) = 0 . If M is minimal and has flat normalbundle (i.e. K N = 0 ) then it can be parametrized as ψ ( x, y ) = α ( x ) + y W , where α ′ ( x ) = Z ⊤ ( x ) , α ′′ ( x ) = II ( Z ⊤ , Z ⊤ ) ( α is a geodesic of M ), W issome constant lightlike tangent vector along α and the vectors α ′ ( x ) and W arelinearly independent for every x. Proof.
Since a = 0 ( i.e. K N = 0), by Lemma 1.4 we have that Z ⊤ and W are parallel vector fields and [ Z ⊤ , W ] = 0 . So, there exists a coordinate system( x, y ) ψ ( x, y ) of M such that ∂ψ∂x ( x, y ) = Z ⊤ ( ψ ( x, y )) and ∂ψ∂y ( x, y ) = W ( ψ ( x, y )) .
10e have that II ( W, · ) = 0 : indeed, II ( W, Z ⊤ ) = − ~H = 0 and II ( W, W ) = 0because K = K N = | ~H | = 0 in Lemma 3.6. Since ∇ W = 0 , we get that ∇ W = 0; thus, W ( ψ ( x, y )) = W ( ψ ( x, Z y ∂∂u W ( ψ ( x, u )) du = W ( ψ ( x, , (6)this implies that, ψ ( x, y ) = ψ ( x,
0) + Z y ∂ψ∂u ( x, u ) du = ψ ( x,
0) + Z y W ( ψ ( x, du = ψ ( x,
0) + y W ( ψ ( x, . In the same way, let us observe that W ( ψ ( x, y )) = W ( ψ (0 , y )) + Z x ∂∂r W ( ψ ( r, y )) dr = W ( ψ (0 , y )) . (7)The equalities (6)-(7) imply that W ( ψ ( x, y )) =: W is constant, i.e. ψ ( x, y ) = α ( x ) + y W , where α ( x ) := ψ ( x,
0) is a lightlike curve such that α ′ ( x ) = Z ⊤ ( ψ ( x, . The following example describe a timelike surface in R , with a canonicalnull direction Z such that II ( Z ⊤ , Z ⊤ ) = 0 which is minimal but has normalcurvature not zero. Here and below, we denote by { e , e , e , e } the canonicalbasis of the four-dimensional Minkowski space; of course e is a timelike vector. Example 3.8.
Let us consider the surface M in R , parametrized as ψ ( x, y ) = α ( x ) + β ( y ) , where α and β are two lightlike curves contained in the timelike hyperplanesorthogonal to e and e , respectively, and satisfy the following conditions • h α ′ ( x ) , β ′ ( y ) i 6 = 0 for every ( x, y ) , • h e , α ′ ( x ) i 6 = 0 for every x, • β ′′ ( y ) (resp. α ′′ ( x ) ) is not lightlike: in other case, β ′′ ( y ) would be linearlydependent to β ′ ( y ) and thus β would be a lightlike line in R , . We have that M is a minimal timelike surface in R , with normal curvature notzero and has a canonical null direction with respect to e with II ( e ⊤ , e ⊤ ) = 0 . Indeed, note that M is a timelike surface because its tangent plane is gen-erated by the linearly independent lightlike tangent vectors ψ x = α ′ ( x ) and y = β ′ ( y ) . On the other hand, since the curve β is orthogonal to e , the tangentpart of e is given by e ⊤ = h e , β ′ ( y ) ih α ′ ( x ) , β ′ ( y ) i α ′ ( x ) + h e , α ′ ( x ) ih α ′ ( x ) , β ′ ( y ) i β ′ ( y ) = h e , α ′ ( x ) ih α ′ ( x ) , β ′ ( y ) i β ′ ( y ); this proves that e ⊤ = λ ( x, y ) β ′ ( y ) , where λ ( x, y ) := h e ,α ′ ( x ) ih α ′ ( x ) ,β ′ ( y ) i is not zero, is alightlike direction on the surface M. Since ∇ e ⊤ e ⊤ = 0 (Lemma 1.2), we obtain ∇ β ′ ( y ) β ′ ( y ) = − λ ( x, y ) ∂λ∂y β ′ ( y ) . Therefore, β ′′ ( y ) = ∇ β ′ ( y ) β ′ ( y ) + II ( β ′ ( y ) , β ′ ( y )) = − λ ( x, y ) ∂λ∂y β ′ ( y ) + II ( β ′ ( y ) , β ′ ( y )); since β ′ ( y ) and β ′′ ( y ) are linearly independent, we get that II ( β ′ ( y ) , β ′ ( y )) = 0 , that is II ( e ⊤ , e ⊤ ) = λ ( x, y ) II ( β ′ ( y ) , β ′ ( y )) = 0 . Now, since ψ xy = 0 we get that ∇ ψ x ψ y = 0 and II ( ψ x , ψ y ) = 0 , in particular, M is minimal. We finally prove that M has normal curvature not zero: weconsider the lightlike tangent vector W := − λ ( x, y ) h α ′ ( x ) , β ′ ( y ) i α ′ ( x ) = − h e , α ′ ( x ) i α ′ ( x ) which is such that h e ⊤ , W i = − since ∇ W e ⊤ = − a e ⊤ (Lemma 1.4) we obtain a = 1 λ ( x, y ) h e , α ′ ( x ) i ∂λ∂x ; according to Corollary 3.5, K N = 0 if and only if a = 0 , that is, if and only if ∂λ∂x = 0 , the last equality is equivalent to λ = h e , α ′′ ( x ) ih α ′′ ( x ) , β ′ ( y ) i which is valid when α ′′ ( x ) is linearly dependent to α ′ ( x ) . We finally give anexplicit numerical example of this situation: consider α ( x ) = (cosh x, sinh x, x, and β ( y ) = (cosh y, y, , sinh y ) , defined on a domain for ( x, y ) where h α ′ ( x ) , β ′ ( y ) i 6 = 0 . .2 Timelike surfaces in R , as a graph of a function In this section, we will study the situation when a surface is given as the graphof a smooth function.Let f, g : U ⊂ R → R be two smooth functions and consider the surface M := (cid:8) ( f ( x, y ) , g ( x, y ) , x, y ) ∈ R , | x, y ∈ U (cid:9) ⊂ R , (8)given as a graph of the function ( x, y ) → ( f ( x, y ) , g ( x, y )) . A global parametriza-tion of this surface is given by ψ : U ⊂ R → R , , ψ ( x, y ) = ( f ( x, y ) , g ( x, y ) , x, y ) . The tangent vectors to the surface are ψ x = ( f x , g x , ,
0) and ψ y = ( f y , g y , , , and the components of the induced metric h· , ·i in M are given by E := h ψ x , ψ x i = 1 − f x + g x , F := h ψ x , ψ y i = − f x f y + g x g y and G := h ψ y , ψ y i = 1 − f y + g y . The determinant of this metric isdet h· , ·i = EG − F = 1 − |∇ f | + |∇ g | − h∇ f, ∇ g i , where the right hand side is calculated on R with its standard Riemannian flatmetric; in particular, M is a timelike surface if and only if det h· , ·i < . Proposition 3.9.
Let M be a timelike surface in R , given as in (8) . Then M has a canonical null direction with respect to e (resp. e ) if and only if ψ x (resp. ψ y ) is a lightlike vector field along M. In that situation we have e ⊤ = 1 F ψ x (cid:18) resp. e ⊤ = 1 F ψ y (cid:19) . Proof.
We have to calculate the tangent part of e along M (the case for thevector e is similar therefore it will be omitted): writing e ⊤ = aψ x + bψ y , we get h e ⊤ , ψ x i = aE + bF and h e ⊤ , ψ y i = aF + bG ;therefore (cid:18) (cid:19) = (cid:18) h e , ψ x ih e , ψ y i (cid:19) = (cid:18) h e ⊤ , ψ x ih e ⊤ , ψ y i (cid:19) = (cid:18) E FF G (cid:19) (cid:18) ab (cid:19) and thus a = − FEG − F and b = EEG − F . h e ⊤ , e ⊤ i = a E + b G + 2 abF = EEG − F that is, e ⊤ is a lightlike vector field along M if and only if E = h ψ x , ψ x i = 1 − f x + g x = 0 , i.e. if and only if ψ x is a lightlike vector field. Theorem 3.10.
Let M be a timelike surface in R , given as in (8) . Assumethat M has a canonical null direction with respect to e and e . Then M isminimal if and only if M can be locally parametrized by ψ ( x, y ) = α ( x ) + β ( y ) (9) where α and β are two lightlike curves contained in the timelike hyperplanesorthogonal to e and e , respectively.Proof. A global basis for the normal bundle
N M is given by the vector fields ξ := (1 , , f x , f y ) and ξ := (0 , − , g x , g y ) . The components of the induced metric in
N M are given by L := h ξ , ξ i = |∇ f | − , M := h ξ , ξ i = h∇ f, ∇ g i , N := h ξ , ξ i = |∇ g | +1 , and satisfies LN − M > . We are going to calculate the condition for M tobe minimal. Using Proposition 3.9, we have that the tangent vectors ψ x and ψ y of M are lightlike; therefore, M is minimal if and only if II ( ψ x , ψ y ) = 0 . Ingeneral, for i, j ∈ { x, y } , we have II ( ψ i , ψ j ) = ( ∇ ψ i ψ j ) ⊥ = a ξ + b ξ , where ∇ is the Levi Civita connection of R , and (cid:18) ab (cid:19) = 1 LN − M (cid:18) N − M − M L (cid:19) (cid:18) h∇ ψ i ψ j , ξ ih∇ ψ i ψ j , ξ i (cid:19) . Since ∇ ψ i ψ j = ( f ij , g ij , , , we get h∇ ψ i ψ j , ξ i = − f ij and h∇ ψ i ψ j , ξ i = − g ij . We deduce that, II ( ψ x , ψ x ) = − f xx N + g xx MLN − M ξ + f xx M − g xx LLN − M ξ ,II ( ψ y , ψ y ) = − f yy N + g yy MLN − M ξ + f yy M − g yy LLN − M ξ ,II ( ψ x , ψ y ) = − f xy N + g xy MLN − M ξ + f xy M − g xy LLN − M ξ . M is minimal if and only if (cid:18) − N MM − L (cid:19) (cid:18) f xy g xy (cid:19) = (cid:18) − f xy N + g xy Mf xy M − g xy L (cid:19) = (cid:18) (cid:19) ;since LN − M > , we obtain that f xy = 0 = g xy . Thus, by integration we get f ( x, y ) = α ( x ) + β ( y ) and g ( x, y ) = α ( x ) + β ( y )This implies that ψ can be written as in (9) with α ( x ) = ( α ( x ) , α ( x ) , x, e ) and β ( y ) = ( β ( y ) , β ( y ) , , y ) (orthogonal to e ). Let usobserve that in this case ψ x and ψ y are lightlike vectors if and only if α and β are lightlike curves.We consider the isometric embedding of R , in R , given by R , := ( e ) ⊥ , where e is the fourth vector of the canonical basis of R , . Proposition 3.11.
Let M be a Lorentzian surface in R , , f : M → R be agiven smooth function. Let us consider the surface obtained as the graph of f, i.e. M := { ( p, f ( p )) | p ∈ M } ⊂ R , , with the induced metric. Then M has a canonical null direction with respect to e if and only if ∇ f is a lightlike vector field on M . Proof.
The surface M is parametrized by the immersion ψ : M → R , , ψ ( p ) = ( p, f ( p )) . We consider a local orthonormal frame ( X , X ) of T M with ǫ j = h X j , X j i andsuch that ǫ ǫ = − M is Lorentzian). Moreover, X and X are orthogonalto e . Using the immersion ψ, we can get the induced local frame on T M,Y j := dψ ( X j ) = X j + df ( X j ) e , j = 1 , . So, the induced metric h· , ·i on M is given by the matrix h· , ·i = (cid:18) h Y , Y i h Y , Y ih Y , Y i h Y , Y i (cid:19) = (cid:18) ǫ + h∇ f, X i h∇ f, X ih∇ f, X ih∇ f, X ih∇ f, X i ǫ + h∇ f, X i (cid:19) , and its determinant isdet h· , ·i = − ǫ h∇ f, X i + ǫ h∇ f, X i = − (1 + ǫ h∇ f, X i + ǫ h∇ f, X i )= − (1 + h∇ f, ∇ f i )15since ∇ f = ǫ h∇ f, X i X + ǫ h∇ f, X i X ); therefore, M is a timelike surfaceif and only if h∇ f, ∇ f i > − . On the other hand, we have e ⊤ = aY + bY where (cid:18) ab (cid:19) = 1det h· , ·i (cid:18) ǫ + h∇ f, X i −h∇ f, X ih∇ f, X i−h∇ f, X ih∇ f, X i ǫ + h∇ f, X i (cid:19) (cid:18) h e , Y ih e , Y i (cid:19) ;since, h e , Y i = h∇ f, X i and h e , Y i = h∇ f, X i we obtain that (cid:18) ab (cid:19) = 1det h· , ·i (cid:18) ǫ h∇ f, X i ǫ h∇ f, X i (cid:19) = − h· , ·i (cid:18) ǫ h∇ f, X i ǫ h∇ f, X i (cid:19) , therefore, e ⊤ = − h· , ·i ( ǫ h∇ f, X i Y + ǫ h∇ f, X i Y ) . Thus h e ⊤ , e ⊤ i = 1det h· , ·i ( h∇ f, X i ( ǫ + h∇ f, X i ) + h∇ f, X i ( ǫ + h∇ f, X i )) − h∇ f, X i h∇ f, X i = 1det h· , ·i ( h∇ f, ∇ f i + h∇ f, ∇ f i ) . Now, it is clear that e ⊤ is a lightlike vector field on M if and only if h∇ f, ∇ f i is either 0 or −
1; but the case h∇ f, ∇ f i = − h· , ·i = 0 . The following proposition generalize Lemma 3.2.
Proposition 3.12.
Let M be a Lorentzian surface, f : M → R be a givensmooth function. If the gradient ∇ f is a lightlike vector field then the integralcurves of ∇ f are geodesics and f is a harmonic function, i.e. △ f = 0 .Proof. Note that ∇ f = 0 because it is lightlike vector field, in particular f isnot a constant function. By a direct computation we get0 = X h∇ f, ∇ f i = 2 h∇ X ∇ f, ∇ f i = 2Hess f ( ∇ f, X ) , for all X ∈ T M ; in particular, ∇ ∇ f ∇ f = 0 because h∇ X ∇ f, ∇ f i = h∇ ∇ f ∇ f, X i .On the other hand, let W be another lightlike vector field defined locally on M such that h∇ f, W i = − . We compute the laplacian of the function f : in theorthonormal frame (cid:16) ∇ f + W √ , ∇ f − W √ (cid:17) on T M, we get∆ f = − f ( ∇ f, W ) = − h∇ ∇ f ∇ f, W i = 0 , because ∇ ∇ f ∇ f = 0 . xample 3.13. Let us consider the timelike surface M := { ( x, exp iy ) | x ∈ R , y ∈ (0 , π ) } ⊂ R , , and the function f : M → R given by f ( x, exp iy ) = y − x. The level curve γ ( y ) = ( y − c, exp iy ) is a lightlike geodesic in M for all constant c ∈ R . Indeed,we only have to remark that γ ′ ( y ) = (1 , i exp iy ) = ∂ x + ∂ y is a lightlike vectorfield. On the other hand, we compute the gradient of the function f : since M is a Lorentzian product, we have that ∂ x = e and ∂ y = − sin y e + cos y e arean orthonormal frame along M , therefore, ∇ f = − ( ∂ x f ) ∂ x + ( ∂ y f ) ∂ y = ∂ x + ∂ y = γ ′ ( y ) , which is a lightlike vector field on M ; from Proposition 3.12 we obtain that γ is a geodesic. Finally, by Proposition 3.11, the timelike surface M := { ( x, exp iy, y − x ) | x ∈ R , y ∈ (0 , π ) } ⊂ R , has a canonical null direction with respect to e . We consider Λ R , , the vector space of bivectors of R , endowed with its nat-ural metric h· , ·i of signature (3 , . The Grassmannian of the oriented timelike2 − planes in R , identifies with the submanifold of unit and simple bivectors Q := (cid:8) η ∈ Λ R , | h η, η i = − , η ∧ η = 0 (cid:9) , and the oriented Gauss map of a timelike surface in R , with the map G : M −→ Q , p G ( p ) = u ∧ u , where ( u , u ) is a positively oriented orthonormal basis of T p M. The Hodge ∗ operator Λ R , → Λ R , is defined by the relation h ⋆η, η ′ i = η ∧ η ′ for all η, η ′ ∈ Λ R , , where we identify Λ R , to R using the canonical volumeelement e ∧ e ∧ e ∧ e of R , . It satisfies ∗ = − id Λ R , and thus i := −∗ definesa complex structure on Λ R , . We also define the map H : Λ R , × Λ R , → C by H ( η, η ′ ) = h η, η ′ i + i η ∧ η ′ , (10)for all η, η ′ ∈ Λ R , . This is a C − bilinear map on Λ R , , and we have Q = (cid:8) η ∈ Λ R , | H ( η, η ) = − (cid:9) . The bivectors { e ∧ e , e ∧ e , e ∧ e } (11)form an orthomormal basis (with respect to the norm H ) of Λ R , as a complexspace of signature ( − , + , − ) . Using this basis of Λ R , , the Grassmannian Q isidentifies with a complex hyperboloid of one sheet Q ≃ (cid:8) ( z , z , z ) ∈ C | − z + z − z = − (cid:9) . .3.1 Timelike surfaces with a canonical null direction We consider an oriented timelike surface M in R , with a canonical null di-rection Z (with h Z, Z i = 1 and) such that II ( Z ⊤ , Z ⊤ ) = 0 . We recall that W is a lightlike vector field tangent to M such that h Z ⊤ , W i = − , and that Z ⊥ is a unit vector field normal to M. As before, we consider the unit vector fieldnormal to the surface ν := II ( Z ⊤ , Z ⊤ ) | II ( Z ⊤ , Z ⊤ ) | ∈ N M ;recall that ν is orthogonal to Z ⊥ (see the proof of Lemma 1.2). We moreoversuppose that e := Z ⊤ + W √ , e := Z ⊤ − W √ , e := Z ⊥ and e := ν, (12)is an oriented and orthonormal basis of R , , and define the orthonormal basis(11) of Λ R , , with respect to the form H. Lemma 3.14.
The Gauss map of M is given by G = W ∧ Z ⊤ , and satisfies dG ( Z ⊤ ) = − ~H ∧ Z ⊤ + W ∧ II ( Z ⊤ , Z ⊤ ) and dG ( W ) = ~H ∧ W − Z ⊤ ∧ II ( W, W ) . Proof.
We only need to compute G = e ∧ e = 12 ( Z ⊤ + W ) ∧ ( Z ⊤ − W ) = W ∧ Z ⊤ . The differential of the expression above is given by dG ( u ) = ( ∇ u W + II ( W, u )) ∧ Z ⊤ + W ∧ ( ∇ u Z ⊤ + II ( Z ⊤ , u ))for all u ∈ T p M ; using the identities of Lemma 1.4 we conclude the result.We define the bivectors N := 1 √ e ∧ e + e ∧ e ) and N := 1 √ − e ∧ e + e ∧ e ); N and N are linearly independent, they satisfy H ( N , N ) = H ( N , N ) = 0and H ( N , N ) = − . Explicitly, N and N are given by N = Z ⊥ ∧ W and N = Z ⊥ ∧ Z ⊤ ;moreover, with respect to the complex structure i = − ⋆ defined on Λ R , , of adirect computation we get iN = W ∧ ν and iN = − Z ⊤ ∧ ν, and the volume element is given by − iN ∧ N . emma 3.15. We have the following identities: • dG ( Z ⊤ ) = i | II ( Z ⊤ , Z ⊤ ) | N − i h ~H, ν i N , • dG ( W ) = − i h ~H, ν i N + (cid:16) K N | II ( Z ⊤ ,Z ⊤ ) | + i | ~H | − K | II ( Z ⊤ ,Z ⊤ ) | (cid:17) N . Proof.
Since 0 = h II ( Z ⊤ , W ) , Z ⊥ i = −h ~H, Z ⊥ i , we have ~H = h ~H, Z ⊥ i Z ⊥ + h ~H, ν i ν = h ~H, ν i ν, (13)replacing this, and the relation given in Proposition 3.6 in the identities ofLemma 3.14, we easily get the result. The pull-back of the form H by the Gauss map. The pull-back by theGauss map G : M −→ Q ⊂ Λ R , of the form H (defined in (10)) permits todefine, for all p ∈ M, the complex quadratic form on the tangent space T p MG ∗ H p : T p M −→ C , u H ( dG p ( u ) , dG p ( u )) . This form is analogous to the third fundamental form of the classical theory ofsurfaces in Euclidean space. We will describe some properties of this quadraticform for a timelike surface with a canonical null direction.
Lemma 3.16.
We have the following identities • H ( dG ( Z ⊤ ) , dG ( Z ⊤ )) = − | II ( Z ⊤ , Z ⊤ ) |h ~H, ν i , • H ( dG ( W ) , dG ( W )) = − | II ( Z ⊤ ,Z ⊤ ) | h ~H, ν i (cid:16) | ~H | − K − iK N (cid:17) , • H ( dG ( Z ⊤ ) , dG ( W )) = (cid:16) | ~H | − K (cid:17) − iK N . Proof.
The proof of these equalities is obtained by a direct computation usingthe expressions of dG ( Z ⊤ ) and dG ( W ) given in Lemma 3.15. Proposition 3.17.
The discriminant of the complex quadratic form G ∗ H sat-isfies disc G ∗ H := − det G ∗ H = − ( K + iK N ) , where K and K N are the Gauss and normal curvatures of the surface M. Proof.
Using the identities of Lemma 3.16, by a direct computation we getdet G ∗ H = (cid:16) (2 | ~H | − K ) − iK N (cid:17) − | ~H | (cid:16) | ~H | − K − iK N (cid:17) = K + 2 iKK N − K N , which implies the result. 19 roposition 3.18. The complex quadratic form G ∗ H is zero at every point of M if and only if M is minimal and has flat normal bundle.Proof. We recall that M is minimal if and only if h ~H, ν i = 0 (identity (13)),and that normal curvature zero implies Gauss curvature zero (see Corollary3.1). Using the identities of Lemma 3.16, since II ( Z ⊤ , Z ⊤ ) = 0 , we easily getthe result.The interpretation of the condition G ∗ H ≡ p in M, the space dG p ( T p M ) belongs to G ( p ) + (cid:8) ξ ∈ Λ R , | H ( G ( p ) , ξ ) = 0 = H ( ξ, ξ ) (cid:9) ⊂ T G ( p ) Q ;this set is the union of two complex lines through G ( p ) in the Grassmannian Q of the oriented and timelike planes of R , ; explicitly, these complex lines aregiven by G ( p ) + C N and G ( p ) + C N . In particular, the first normal space in p is 1 − dimensional, i.e. the osculatorspace of the surface is degenerate at every point p of M. Asymptotic directions on the surface.
For all p ∈ M, we consider the realquadratic form δ : T p M −→ R , u dG p ( u ) ∧ dG p ( u ) , where Λ R , is identified with R by means of the volume element − iN ∧ N ≃ . A non-zero vector u ∈ T p M defines an asymptotic direction at p if δ ( u ) = 0 . The opposite of the determinant of δ, with respect to the metric on M, ∆ := − det δ, is a second order invariant of the surface; ∆ ≤ R , . We will compute the invariant ∆ and describe the asymptotic directions ofa timelike surface with a canonical null direction.
Proposition 3.19.
At every point of M we have: ∆ = − K N where K N is the normal curvature of M. In particular, there exists asymptoticdirections at every point of M. roof. Since δ is the imaginary part of the quadratic form G ∗ H, we have δ ( Z ⊤ ) = 0 (first equality of Lemma 3.16). Using the identities of Lemma 3.15,by a direct computation we get∆ = − (cid:2) dG ( Z ⊤ ) ∧ dG ( W ) (cid:3) + δ ( Z ⊤ ) δ ( W ) = − K N since dG ( Z ⊤ ) ∧ dG ( W ) = − K N iN ∧ N ≃ K N . Proposition 3.20.
At every point of
M, Z ⊤ is an asymptotic direction; more-over, W is an asymptotic direction if and only if M is minimal or has flatnormal bundle.Proof. Since δ is the imaginary part of G ∗ H, using the identities of Lemma 3.16we have δ ( Z ⊤ ) = 0 and δ ( W ) = − | II ( Z ⊤ , Z ⊤ ) | h ~H, ν i K N which implies the results.According to Proposition 3.19, if the normal curvature K N is not zero, thereexists two distinct asymptotic directions at every point of the surface. FromProposition 3.20, Z ⊤ is an asymptotic direction; by a direct computation, wedescribe the other asymptotic direction. Proposition 3.21.
If the surface M has not zero normal curvature, there existstwo different asymptotic directions given by Z ⊤ and h ~H, ν i| II ( Z ⊤ , Z ⊤ ) | Z ⊤ + W at every point of the surface. Acknowledgements.
The first author was supported by the project FORDE-CyT: Conacyt 265667. The second author acknowledges support from DGAPA-UNAM-PAPIIT, under Project IN115017.
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