Invariants and quasi-umbilicity of timelike surfaces in Minkowski space R3,1

Journal of Geometry and Physics 62 (2012) 1697–1713

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Invariants and quasi-umbilicity of timelike surfaces in Minkowski space R3,1 Pierre Bayard a , Federico Sánchez-Bringas b,∗ a

Instituto de Física y Matemáticas, U.M.S.N.H., C.U., CP. 58040 Morelia, Michoacán, Mexico

b

Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico

article

info

Article history: Received 18 October 2011 Received in revised form 15 February 2012 Accepted 12 March 2012 Available online 16 March 2012 MSC: 53C12 53C42 53A55 53C50

abstract We first describe the numerical invariants and the curvature hyperbola attached to the second fundamental form of a timelike surface in four-dimensional Minkowski space: beside the four natural invariants, a new invariant appears at some special points of the surface, which are said to be quasi-umbilic; at such quasi-umbilic points, the curvature hyperbola degenerates to a line with one point removed. We then study the asymptotic lines on a timelike surface, and characterize the quasi-umbilic points of the surface as the points where the asymptotic directions degenerate to a double lightlike line. We also give an interpretation of the new invariant at a quasi-umbilic point, using the Gauss map of the surface. We finally describe the timelike surfaces which are quasi-umbilic at every point. © 2012 Elsevier B.V. All rights reserved.

Keywords: Timelike surfaces Second order invariants Quasi-umbilic surfaces

0. Introduction The second order invariants of a surface immersed in R4 are the Gaussian curvature, the normal curvature, the norm of the mean curvature vector and the function ∆ which, according to [1], measures at each point the local convexity of the surface [2]. These classical invariants, together with the corresponding curvature ellipses, allow the description of the main properties of the extrinsic geometry of the surface [3,1,4,5]. If we consider the surface immersed in the Minkowski space R3,1 , that is, the affine space R4 endowed with the metric g = dx21 + dx22 + dx23 − dx24 , the geometric structure of this ambient space imposes on the second order invariants new relations and even more, in some exceptional cases, the existence of new invariants. We say that a surface M in R3,1 is timelike (or spacelike) if g induces on M a Lorentzian (or Riemannian) metric (see [6] for basic facts and terminologies concerning Minkowski space). Thus, at each point p of a timelike (or spacelike) surface M, the Minkowski space splits in R3,1 = Tp M ⊕ Np M , where the tangent plane Tp M and the normal plane Np M at p are respectively equipped with a metric of signature (1, 1) (or (2, 0)) and (2, 0) (or (1, 1)). In [7], we provide a classification of the points of a spacelike surface immersed in R3,1 by means of other four natural invariants almost equivalent to the classical ones, and we show that in some exceptional cases a new invariant appears (in fact, when the osculating paraboloid of the surface belongs to some degenerate hyperplane of R3,1 ). Moreover, the curvature ellipse at each point of the surface, even if it is degenerate, describes relations between the invariants analogous to those corresponding to the case of a surface immersed in R4 . In the present article, we apply the same approach to classify points on a timelike surface: we introduce

∗

Corresponding author. Tel.: +52 5556224867; fax: +52 5556224859. E-mail addresses: [email protected] (P. Bayard), [email protected] (F. Sánchez-Bringas).

0393-0440/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2012.03.008

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invariants, denoted by a2 , b2 , α and β , which are determined by a quadratic form associated to the second fundamental form; more precisely, the first two invariants are the eigenvalues of the self-adjoint operator of the discriminant quadratic form Φ (see Section 1), meanwhile the other two are the components of the mean curvature vector on the basis of eigenvectors of this operator. At some special points of the surface, where the surface is said to be quasi-umbilic (keeping the terminology introduced by Clelland in [8] for timelike surfaces in three-dimensional Minkowski space), there is a new invariant which does not appear in the case of spacelike surfaces in R3,1 ; see Lemma 1.11, Remark 1.12 and Theorem 1.13, 2(a). On the other hand, since the tangent plane of a timelike surface is equipped with a Lorentzian metric, the relations between the second order invariants and therefore, the main properties of the extrinsic geometry of the surface, are conveniently described by a hyperbola instead of an ellipse; this hyperbola will be naturally called the curvature hyperbola of the surface at the point. The article is organized as follows. In Section 1 our treatment is algebraic, we describe the numerical invariants a2 , b2 , α and β of a quadratic map R1,1 → R2 , modulo the natural action of the groups of isometries, and classify the equivalence classes in terms of them. Since we also determine the relations of these invariants with the classical ones, we obtain, in fact, a classification of these classes in terms of the classical invariants (Theorem 1.13). In the quasi-umbilic case, only one of the classical invariants subsists, and there is a new invariant. Then, we define and describe the curvature hyperbola associated to a quadratic map in Section 2. This curve may degenerate in different ways determining special points on the surface. There are more types of possible degenerations for the curvature hyperbola than those for the curvature ellipse of spacelike surfaces (Proposition 2.3). In Section 3 we give a geometric interpretation of the new invariant using the Gauss map of a timelike surface with values in the Grassmannian of the oriented spacelike 2-planes in R3,1 : the invariant appears to measure a rate of variation of the normal plane along a tangent lightlike direction. Further on, in Section 4, we describe the fields of asymptotic directions and of mean directionally curved directions in terms of the new invariants and give conditions that determine the causal character of the asymptotic lines; a quasi-umbilic point is then characterized as a point where the asymptotic directions degenerate to a double lightlike line (Theorem 4.4). We also precise the relations between asymptotic and mean directionally curved directions. In the last section (Theorem 5.1), we give a complete description of quasi-umbilic timelike surfaces, namely, surfaces such that the second fundamental form is quasi-umbilic at every point (or equivalently, such that the curvature hyperbola degenerates into a line minus a point at every point). 1. Quadratic maps from R1,1 to R2 In this section, we study the vector space Q (R1,1 , R2 ) of quadratic maps from R1,1 to R2 . We suppose that R1,1 and R2 are canonically oriented and time-oriented: a vector of R1,1 will be called future-directed if its second component in the canonical basis is positive. We consider the reduced (connected) groups of euclidean and Lorentzian direct isometries of R1,1 and R2 , SO1,1 (R) and SO2 (R). They act on Q (R1,1 , R2 ) by composition SO2 (R) × Q (R1,1 , R2 ) × SO1,1 (R) → Q (R1,1 , R2 )

(g1 , q, g2 ) → g1 ◦ q ◦ g2 . We are interested in the description of the quotient set SO2 \ Q (R1,1 , R2 )/SO1,1 . We first introduce the algebraic invariants of a quadratic map and then give a description of the quotient set (Theorem 1.13). 1.1. Forms associated to a quadratic map We fix q ∈ Q (R1,1 , R2 ). If ν belongs to R2 , we denote by Sν the symmetric endomorphism of R1,1 associated to the real quadratic form ⟨q, ν⟩, and we define, for ν, ν1 , ν2 ∈ R2 , Lq (ν) :=

1 2

tr(Sν ),

Qq (ν) := det(Sν ) and

Aq (ν1 , ν2 ) :=

 1 S ν1 , S ν2 . 2

Here Sν1 , Sν2 denotes the morphism Sν1 ◦ Sν2 − Sν2 ◦ Sν1 ; it is skew-symmetric on R1,1 , and thus identifies with the real number ϵ such that its matrix in the canonical basis of R1,1 is





S ν1 , S ν2 =







0

ϵ

ϵ



0

.

In the sequel, we will implicitly make this identification. Thus, Lq is a linear form, Qq is a quadratic form, and Aq is a bilinear skew-symmetric form on R2 . These forms are linked together according to the following lemma: Lemma 1.1. The quadratic form

Φq := L2q − Qq

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satisfies the following identity: for all ν1 , ν2 ∈ R2 ,

˜ q (ν1 , ν2 )2 − Aq (ν1 , ν2 )2 , Φq (ν1 )Φq (ν2 ) = Φ

(1.1)

˜ q denotes the symmetric bilinear form such that Φ ˜ q (ν, ν) = Φq (ν) for all ν ∈ R . In particular the signature of Φq is where Φ (r , s) with 0 ≤ r , s ≤ 1. 2

Proof. This may be easily proved by a straightforward computation, using the representation of Sν1 and Sν2 by their matrices in the canonical basis of R1,1 ; see also Remark 1.2.  The forms Lq , Φq , Aq are invariant by the SO1,1 -action on q: for all g ∈ SO1,1 (R), Lq◦g = Lq ,

Φq◦g = Φq and Aq◦g = Aq .

In the next section we examine in what extent the forms Lq , Φq and Aq determine q modulo the SO1,1 -action. 1.2. Reduction of a quadratic map Let us denote by S the vector space of symmetric and traceless operators on R1,1 . S is naturally equipped with a metric of signature (1, 1): if u : R1,1 → R1,1 belongs to S , define its norm 1 

|u|2 =

2 i,j,k,l

gik g jl uij ukl

where (gij ) is the matrix of the natural metric on R1,1 and (g ij ) is its inverse. Taking the canonical basis of R1,1 and using that u is traceless and symmetric, we easily see that

|u|2 = − det u. Define also

 E1 =

1 0

0 −1



 and

E2 =

0 −1



1 . 0

The basis (E1 , E2 ) is a Lorentzian basis of S : |E1 |2 = 1, |E2 |2 = −1, and ⟨E1 , E2 ⟩ = 0. We consider, associated to q belonging to Q (R1,1 , R2 ), the map fq : R 2 → S

ν → Sνo , where Sνo = Sν − Lq (ν)I stands for the traceless part of Sν . Observe that, for all ν, ν ′ ∈ R2 ,

˜ q (ν, ν ′ ) = ⟨fq (ν), fq (ν ′ )⟩ and Aq (ν, ν ′ ) = [fq (ν), fq (ν ′ )], Φ

(1.2)

where, if s and s belong to S , the bracket [s, s ] stands here for the determinant of the vectors s and s in the basis (E1 , E2 ) of S . ′

′

′

Remark 1.2. Using formulae (1.2), the identity (1.1) appears to be a direct consequence of the Lagrange identity in the Lorentz plane S . Remark 1.3. If u belongs to S , u ̸= 0, its norm |u|2 determines its canonical form as follows: u is diagonalizable if and only if |u|2 > 0; in that case, in some positively oriented and orthonormal basis of R1,1 ,



u = ± |u|2 E1 . Here and below, by a positively oriented basis (e1 , e2 ) of R1,1 we mean a basis which has the orientation of the canonical basis of R1,1 , and which is such that e2 is future-directed. If |u|2 < 0, in some positively oriented and orthonormal basis of R1,1 ,



u = ± −|u|2 E2 , and if |u|2 = 0, defining

N1 =

1 2

(E1 + E2 ) ,

N2 =

1 2

(E1 − E2 ) ,

in a convenient positively oriented and orthonormal basis of R1,1 , u = ±N , where N = N1 or N2 .

(1.3)

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For the reduction of q, we distinguish four cases accordingly to the ranks of fq and Φq . Case 1: rank(fq ) = 2; this is equivalent to Aq ̸= 0. In that case Φq is of signature (1, 1). Lemma 1.4. Let νo ∈ R2 be such that Φq (νo ) = 1. There exists an orthonormal and positively oriented basis (e1 , e2 ) of R1,1 such that, for all ν ∈ R2 , the matrix of Sν in (e1 , e2 ) is given by

˜ q (νo , ν)E1 + Aq (νo , ν)E2 . Sν = Lq (ν)I ± Φ 



(1.4)

Proof. The eigenvalues of Sνo are the roots of the polynomial X 2 − 2Lq (νo )X + Qq (νo ) and are thus Lq (νo ) ± 1. We choose a positively oriented orthonormal basis (e1 , e2 ) of R1,1 such that in this basis Sνo = Lq (νo )I ± E1 . For all ν ∈ R2 , the matrix of Sν may be a priori written in the form Sν = Lq (ν)I + aν E1 + bν E2 , where aν and bν belong to R. Recalling (1.2), we get ˜ q (ν0 , ν) and bν = εAq (νo , ν) where ε = ±1, and thus (1.4).  aν = ε Φ Case 2: rank(fq ) = 1 and Φq ̸= 0; thus Φq has rank one, and fq (R2 ) is a line of S , which is spacelike or timelike according to the sign of Φq (see (1.2)). Lemma 1.5. Let νo ∈ R2 be such that Φq (νo ) = +1 or −1. There exists an orthonormal and positively oriented basis (e1 , e2 ) of R1,1 such that, for all ν ∈ R2 , the matrix of Sν in (e1 , e2 ) is given by

˜ q (νo , ν)E Sν = Lq (ν)I ± Φ

(1.5)

where E = E1 if Φq ≥ 0, and E = E2 if Φq ≤ 0. Proof. Suppose that Φq ≥ 0. Choose a positively oriented orthonormal basis (e1 , e2 ) of R1,1 such that Sνo = Lq (νo )I ± E1 in (e1 , e2 ). For all ν ∈ R2 , the matrix of Sν may be a priori written in the form Sν = Lq (ν)I + aν E1 + bν E2 ; necessarily, ˜ q (ν0 , ν) and bν = 0. Analogously, the case where Φq ≤ 0 relies on the fact that it is possible to choose a positively aν = ±Φ oriented orthonormal basis (e1 , e2 ) of R1,1 such that Sνo = Lq (νo )I ± E2 in (e1 , e2 ).  We define Q1 (R1,1 , R2 ) = {q ∈ Q (R1,1 , R2 ) : Φq ̸= 0}. Setting P1 = {(L, Φ , A) : 0 ̸= Φ has a non-positive discriminant, and (1.1) holds}, where L, Φ and A are respectively linear, bilinear symmetric and bilinear skew-symmetric forms on R2 , the following result holds: Lemma 1.6. The map

Θ1 : Q1 (R1,1 , R2 )/SO1,1 → P1   [q] → L[q] , Φ[q] , A[q] is surjective and two-to-one. Proof. Since the sign in (1.4) or (1.5) is not determined by Lq , Φq and Aq , the map Θ1 is clearly two-to-one. For the surjectivity, if L, Φ and A are given, choose first νo such that Φ (νo ) = ±1, and then define Sν in the canonical basis of R1,1 by (1.4) or (1.5), accordingly to the signature of Φ .  By the natural left-action of SO2 on Q1 (R1,1 , R2 )/SO1,1 , the forms L[q] , Φ[q] transform as Lg .[q] = L[q] ◦ g −1 ,

Φg .[q] = Φ[q] ◦ g −1 ,

whereas the form A[q] is invariant. Thus, if SO2 acts on P1 by g .(L, Φ , A) := (L ◦ g −1 , Φ ◦ g −1 , A),

(1.6)

the map Θ1 is SO2 -equivariant and thus induces a twofold map

Θ 1 : SO2 \ Q1 (R1,1 , R2 )/SO1,1 → SO2 \ P1 .

(1.7)

Since the formula (1.1) permits the recovering of A (up to sign) from Φ , the description of the quotient set SO2 \ Q1 (R1,1 , R2 )/SO1,1 will be achieved with the simultaneous reduction of the forms L[q] and Φ[q] in R2 . Case 3: rank fq = 1 and Φq = 0. In that case fq (R2 ) is a line, which is lightlike. Recall the definition of N1 and N2 in (1.3).

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⃗ ∈ R2 and an orthonormal and positively oriented basis (e1 , e2 ) of R1,1 such that, for all Lemma 1.7. There exist a unit vector u ν ∈ R2 , the matrix of Sν in (e1 , e2 ) is given by Sν = Lq (ν)I + ⟨⃗ u, ν⟩N

(1.8)

⃗ and the basis (e1 , e2 ) are uniquely defined. where N = N1 or N2 . The vector u Proof. In the canonical basis of R1,1 , Sνo reads Sνo = λ(ν)N , where N = N1 or N2 , and where λ is a linear form on R2 . Consider the basis of R1,1 obtained from the canonical basis by a Lorentzian rotation of angle ψ . The matrix of Sνo in this basis is e2ψ λ(ν)N . Thus, there is a unique orthonormal and positively oriented basis of R1,1 such that in that basis Sνo = λ(ν)N ⃗ of R2 .  with |λ| = 1. Moreover, λ(ν) = ⟨⃗ u, ν⟩ for some unit vector u We define Q2 (R1,1 , R2 ) = {q ∈ Q (R1,1 , R2 ) : Φq = 0, fq ̸= 0}, and ∗

⃗) : L ∈ R2 , u⃗ ∈ S 1 ⊂ R2 }. P2 = {(L, u Lemma 1.8. The map

Θ2 : Q2 (R1,1 , R2 )/SO1,1 → P2   [q] → L[q] , u⃗[q] is surjective and two-to-one.

⃗) correspond two classes in Q2 (R1,1 , R2 )/SO1,1 , defining Sν in the canonical basis of R1,1 by (1.8), Proof. To each pair (L, u where N may be chosen to be N1 or N2 .  If SO2 acts on P2 by

⃗) := (L ◦ g −1 , g (⃗u)), g · (L, u

(1.9)

the map Θ is SO2 -equivariant and thus induces a twofold map

Θ 2 : SO2 \ Q2 (R1,1 , R2 )/SO1,1 → SO2 \ P2 .

(1.10) 1 ,1

Thus, the description of the quotient set SO2 \ Q2 (R , R )/SO1,1 will be achieved with the simultaneous reduction of the ⃗[q] in R2 . form L[q] and the vector u Case 4: fq = 0. In that case Sν = Lq (ν)I for all ν ∈ R2 . We define 2

Q3 (R1,1 , R2 ) = {q ∈ Q (R1,1 , R2 ) : Φq = 0, fq = 0}. ∗

Setting P3 = R2 , the map

Θ 3 : SO2 \ Q3 (R1,1 , R2 )/SO1,1 → SO2 \ P3 [q] → [L[q] ]

(1.11)

is bijective. 1.3. Invariants on the quotient set For the discussion, we first define invariants on SO2 \ Q (R1,1 , R2 )/SO1,1 associated to L[q] , Q[q] , A[q] and Φ[q] : Definition 1.9. We consider:

⃗ ∈ R2 such that, for all ν ∈ R2 , ⟨H ⃗ , ν⟩ = L[q] (ν), and its norm 1- the vector H ⃗ , H⃗ ⟩; |H⃗ |2 := ⟨H 2- the two real numbers K := tr Q[q]

and ∆ := det Q[q] ,

where tr Q[q] and det Q[q] are the trace and the determinant of the symmetric operator of R2 associated to Q[q] by the scalar product on R2 ;

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3- the real number KN such that 1

A[q] =

2

KN ω 0 ,

where ω0 is the determinant in the canonical basis of R2 (the canonical volume form on R2 ).

⃗ |2 , K , KN and ∆ are invariant by the SO2 -action on [q] ∈ Q (R1,1 , R2 )/SO1,1 , and thus define invariants on The numbers |H SO2 \ Q (R1,1 , R2 )/SO1,1 . The invariants of the quadratic form Φ[q] have the following simple expressions in terms of these invariants: Lemma 1.10. Let UΦ be the symmetric operator on R2 such that

Φ[q] (ν) = ⟨UΦ (ν), ν⟩ for all ν ∈ R2 . Denoting by tr Φ[q] and det Φ[q] its trace and its determinant, we get:

⃗ |2 − K tr Φ[q] = |H

1 det Φ[q] = − KN2 . 4

and

(1.12)

1.4. The simultaneous reduction of Φ[q] and L[q] We now reduce the operator UΦ . Lemma 1.11. The eigenvalues of UΦ are a2 and −b2 , where 2

a =

b2 =

1



2 1

  ⃗ |2 − K + |H



2

  2 2 ⃗ + |H | − K ,

 KN2

(1.13)

  2 ⃗ |2 − K KN2 + |H .



  ⃗ |2 − K + − |H

(1.14)

⃗ |2 − K ̸= 0. In an orthonormal basis (˜u1 , u˜ 2 ) of eigenvectors of UΦ the vector H ⃗ is α u˜ 1 + β u˜ 2 , Suppose first that KN ̸= 0 or |H with 

1

α2 =  KN2

 2 ⃗ |2 − K + |H



1

β =  2

KN2

⃗ |2 + ∆ + a2 |H

 2 ⃗ |2 − K + |H

1 4

KN2



,

 1 2 2 ⃗ −∆ + b |H | − KN . 2

4

(1.15)

(1.16)

⃗ |2 − K = 0 then ∆ = 0; suppose that fq ̸= 0; writing If now KN = 0 and |H ⃗ = α⃗u + β⃗u⊥ , H

(1.17)

⃗ is given by Lemma 1.7, α and β are new invariants. where u Proof. Formulae (1.13) and (1.14) readily follow from Lemma 1.10. We now prove (1.15) and (1.16). We first assume that u˜ a > 0, and we set ν0 := a1 , where u˜ 1 is assumed to be a unit eigenvector of UΦ associated to the positive eigenvalue a2 . We get Φq (ν0 ) = 1, and, by Lemma 1.4 or 1.5 (changing u˜ 1 to −˜u1 if necessary),

˜ q (ν0 , ν)E1 + Aq (ν0 , ν)E2 Sν = Lq (ν)I + Φ ˜ q (νo , u˜ 1 ) = a and Φ ˜ q (νo , u˜ 2 ) = 0, we have Φ ˜ q (ν0 , ν) = a⟨ν, u˜ 1 ⟩; in some positively oriented basis (e1 , e2 ) of R1,1 . Since Φ moreover, since Aq (νo , u˜ 1 ) = 0 and Aq (νo , u˜ 2 )2 = b2 (using (1.1)), we have Aq (ν0 , ν) = b⟨ν, u˜ 2 ⟩, where b may be negative. Thus, in (e1 , e2 ),  q=

α+a 0

0

−(α − a)

 u˜ 1 +

 β b

b

−β



u˜ 2 .

(1.18)

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By straightforward computations we get

⃗ |2 = α 2 + β 2 , |H K = α 2 + β 2 − a2 + b 2 , ∆ = −a2 β 2 + α 2 b2 − a2 b2 , KN = 2ab.

(1.19)

These formulae also hold if a = 0 and b ̸= 0 (by a similar argument using Lemma 1.5 with ν0 = (1.15) and (1.16) hold by direct computations. 

u˜ 2 ). b

We then verify that

Remark 1.12. (i) Eqs. (1.19) show why we can consider that the classical invariants are almost equivalent to the new invariants. ⃗ |2 − K = 0, we have α 2 + β 2 = |H⃗ |2 = K where α and β are given by (1.17). Thus, (ii) In the case where KN = 0 and |H there is in fact only one new invariant, which is equivalent to the ratio of α and β . 1.5. The classification To state the main theorem we introduce the following notation: let G1 and G2 be the groups of transformations of R2 and R1,1 which are respectively defined by

 G1 =

±1 0



0 ±1

⊂ GL(R2 ),

and

 G2 =

±1 0



0 , ±1



0 ±1

 ±1 0

⊂ GL(R1,1 ),

and let G be the subgroup of G1 × G2 formed by the 16 elements (g1 , g2 ) such that the total number of −1 which appear in the writing of (g1 , g2 ) is even. This group naturally acts by composition on Q (R1,1 , R2 ), and also on the quotient set SO2 \ Q (R1,1 , R2 )/SO1,1 since g1 SO2 g1−1 = SO2 and g2 SO1,1 g2−1 = SO1,1 for all g1 ∈ G1 and all g2 ∈ G2 . It is easy to verify

⃗ |2 are preserved by this action. Since (idR2 , −idR1,1 ) acts trivially on a quadratic map that the invariants K , KN , ∆ and |H q ∈ Q (R1,1 , R2 ), the orbit of q under the group G has 8 elements in general. We also denote by G′ the subgroup of transformations of R1,1 defined by G′ =



±1 0



0 ±1

⊂ GL(R1,1 ).

This group preserves the invariants α and β defined in the quasi-umbilic case, and, in general, the orbit of q ∈ Q (R1,1 , R2 ) under G′ has 2 elements. Theorem 1.13. The class [q] ∈ SO2 \ Q (R1,1 , R2 )/SO1,1 is determined by its invariants in the following way.

⃗ |2 − K ̸= 0, then Φ ̸= 0, and the invariants K , KN , |H ⃗ |2 and ∆ determine [q], up to the action of the group G. (1) If KN ̸= 0 or |H ⃗ |2 − K = 0, then Φ = 0, and the following holds: (2) If KN = 0 and |H (a) if fq ̸= 0, the invariants α and β defined in (1.17) determine [q] up to the action of G′ (q is quasi-umbilic);

⃗ |2 determines [q] (q is umbilic). (b) if fq = 0 then |H Proof. First recall the definition of the maps Θ 1 , Θ 2 and Θ 3 in (1.7), (1.10) and (1.11). If KN ̸= 0, by Lemma 1.11, Θ 1 ([q]) is the class of (L, Φ , A) ∈ P1 where the forms L, Φ and A are defined in the canonical basis (u1 , u2 ) of R2 by L = (α, β),

Φ=



a2 0

0 −b 2

 and

A=

1 2

 KN

0 −1



1 , 0

with a2 , b2 , α, β satisfying (1.13) and (1.14), (1.15) and (1.16) (more precisely, recalling (1.6), if g ∈ SO2 is such that g (u1 ) = u˜ 1 , g (u2 ) = u˜ 2 , where (˜u1 , u˜ 2 ) is the basis given by Lemma 1.11, we have g .(L, Φ , A) = (Lq , Φq , Aq )). Since α and β are determined up to sign by (1.15) and (1.16), eight classes correspond to the given set of invariants (two classes correspond to each choice for α, β since the map Θ 1 in (1.7) is two-to-one); note that classes coincide in the particular case where α or β vanishes. It is not difficult to see that these eight classes are obtained from one of them by the action of G. The proofs in the other cases are analogous and are thus omitted. 

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2. The curvature hyperbola of a quadratic map R1,1 → R2 In this section we describe the geometric properties of the curvature hyperbola associated to a quadratic map in terms of its invariants (Propositions 2.1 and 2.3). The curvature hyperbola H associated to q : R1,1 → R2 is defined as the subset of R2

 H :=

q(v)

|v|2

 : v ∈ R1,1 , |v|2 = ±1 .

We have the following descriptions of the hyperbola: Proposition 2.1. If KN ̸= 0, the curvature hyperbola is not degenerate; its axes are directed by the eigenvectors u˜ 1 and u˜ 2 of UΦ , ⃗ , u˜ 1 , u˜ 2 ), the equation of the hyperbola is given by and, in (H 1 a2

ν12 −

1 b2

ν22 = 1

(2.1)

where a2 , − b2 are the eigenvalues of UΦ given by (1.13) and (1.14). Proof. Using expression (1.18) of q in some orthonormal and positively oriented basis of R1,1 , we readily get that the hyperbola H is parameterized by

  ⃗ ± X (ϕ)˜u1 + Y (ϕ)˜u2 ∈ R2 ϕ ∈ R → H ⃗ = α u˜ 1 + β u˜ 2 and with H X (ϕ) = a cosh 2ϕ,

Y (ϕ) = b sinh 2ϕ.

(2.2)



Remark 2.2. If KN ̸= 0 the function UΦ is invertible, and we define

Φ ∗ (ν) := ⟨ν, UΦ−1 (ν)⟩.

(2.3)

⃗ + ν belongs to The function Φ : R → R then furnishes an intrinsic equation of the curvature hyperbola: for all ν ∈ R , H the hyperbola H if and only if ∗

2

2

Φ ∗ (ν) = 1. We also describe the curvature hyperbola in the degenerate case where KN = 0. Proposition 2.3. If KN = 0 and fq ̸= 0, the curvature hyperbola degenerates to (1) the union of two half-lines



  2 ⃗ ⃗ H ± λ |H | − K .˜u1 , 1 ≤ λ ∈ R

⃗ |2 − K > 0, where u˜ 1 is a unit eigenvector of UΦ associated to the positive eigenvalue; in that case ∆ < 0; if |H (2) the line 

⃗ + λ˜u2 , λ ∈ R H



⃗ |2 − K < 0, where u˜ 2 is an eigenvector of UΦ associated to the negative eigenvalue; in that case ∆ > 0; if |H (3) the line with one point removed 

⃗ + λ⃗u, λ ∈ R \ {0} H



⃗ |2 − K = 0, where u⃗ is the unit vector given by Lemma 1.7; in that case ∆ = 0, and q is quasi-umbilic. if |H ⃗ |2 − K ̸= 0 (i.e. Φ ̸= 0), the result follows from (2.2), observing that b = 0 if |H ⃗ |2 − K > 0, and that a = 0 if Proof. If |H 2 2 ⃗ ⃗ |H | − K < 0. If |H | − K = 0 (i.e. Φ = 0), using expression (1.8) we easily get that H is parameterized by 1

⃗ ± e2ϕ u⃗, ϕ ∈ R → H 2

which gives the result in the last case.



⃗ Remark 2.4. If fq = 0 the hyperbola degenerates to the end point of the vector H.

P. Bayard, F. Sánchez-Bringas / Journal of Geometry and Physics 62 (2012) 1697–1713

1705

3. Interpretation of the new invariant using the Gauss map Let M be a timelike surface in R3,1 . We assume that M is space- and time-oriented. At each point x ∈ M, the second fundamental form II : Tx M → Nx M is a quadratic map between planes equipped with a Lorentzian and a Riemannian metric respectively. We suppose here that the surface is quasi-umbilic at x, i.e. that II : Tx M → Nx M is a quasi-umbilic quadratic map (see Theorem 1.13), and we still denote by α and β the invariants of II introduced in the previous sections. Our purpose here is to give an interpretation of α and β as rates of variation of the normal plane at x. 3.1. The Gauss map of a timelike surface in R3,1 Let us consider Λ2 R3,1 , the vector space of bivectors of R3,1 , endowed with its natural metric (of signature (3, 3)). The Grassmannian of the oriented spacelike 2-planes in R3,1 identifies with the submanifold of unit and simple bivectors

Q = {η ∈ Λ2 R3,1 : ⟨η, η⟩ = 1, η ∧ η = 0}, and the oriented Gauss map with the map G : M → Q,

p → G(p) = u1 ∧ u2 ,

where (u1 , u2 ) is a positively oriented and orthonormal basis of Np M. Consider ∗ : Λ2 R3,1 → Λ2 R3,1 the Hodge operator defined by the relation

⟨∗η, η′ ⟩ = η ∧ η′ for all η, η′ ∈ Λ2 R3,1 , where we identify Λ4 R3,1 to R using the canonical volume form on R3,1 . It satisfies ∗2 = −idΛ2 R3,1 and thus defines a complex structure on Λ2 R3,1 . Also define H (η, η′ ) = ⟨η, η′ ⟩ − i η ∧ η′ for all η, η′ ∈ Λ2 R3,1 . This is a C-bilinear map on Λ2 R3,1 , and we have

Q = {η ∈ Λ2 R3,1 : H (η, η) = 1}. If (eo1 , eo2 , eo3 , eo4 ) stands for the canonical basis of R3,1 , the vectors E1 = eo2 ∧ eo3 ,

E2 = eo3 ∧ eo1 ,

E3 = eo1 ∧ eo2

form a basis of Λ2 R3,1 as a vector space over C; this basis is such that H (Ei , Ej ) = δij for all i, j. Using this basis to identify Λ2 R3,1 with C3 , we readily get

Q = {(Z1 , Z2 , Z3 ) ∈ C3 : Z12 + Z22 + Z32 = 1}. We now use this description to define a complex angle along a (small) curve of spacelike planes; for briefness, we omit the proof. Lemma 3.1. Let e ∈ Q and let Γ be a curve on Q such that Γ (0) = e. Assume that H (Γ ′ (0)) ̸= 0. If t is sufficiently small, Γ (t ) may be written in the form

Γ (t ) = cos θ (t ) e + sin θ (t ) ut

(3.1)

for some smooth functions t → θ (t ) ∈ C and t → ut ∈ Te Q such that θ (0) = 0 and H (ut ) = 1 for all t. The complex function θ(t ) is uniquely defined, up to sign; it will be called a smooth determination of the complex angle between the spacelike plane Γ (t ) and the spacelike plane e. Remark 3.2. The derivative θ ′ (0) ∈ C has the following interpretation: differentiating (3.1) we get

Γ ′ (0) = θ ′ (0)uo

(3.2)

where uo ∈ Te Q is such that H (uo ) = 1. Writing

θ ′ (0) = a + ib and uo = u ∧ ξ where a, b ∈ R, u is a unit vector in e and ξ is a unit and spacelike vector in e⊥ , we get

Γ ′ (0) = au ∧ ξ + bu⊥ ∧ ξ ⊥ . This decomposition has the following interpretation: this is the decomposition of the infinitesimal rotation Γ ′ (0) of the plane e into two orthogonal infinitesimal rotations: an infinitesimal rotation in the spacelike hyperplane e ⊕ Rξ , around the direction u, and an infinitesimal Lorentzian rotation in the orthogonal timelike hyperplane e ⊕ Rξ ⊥ , around the direction u⊥ ; the numbers a and b are the respective angular velocities of these infinitesimal rotations.

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Consider, for all x ∈ M, the complex quadratic form on Tx M G∗ Hx = H (dGx , dGx ). This form is analogous to the third fundamental form of the classical theory of surfaces in R3 . It has the following geometric interpretation: Lemma 3.3. Let u ∈ Txo M, and γ a curve on M such that γ ′ (0) = u. Consider Γ (t ) = G(γ (t )) and assume that H (Γ ′ (0)) ̸= 0. If t → θ (t ) ∈ C is a smooth determination of the complex angle between Γ (t ) and e = Γ (0), then G∗ H (u) = θ ′ (0)2 .

(3.3)

Similar results concerning the angle variation of the tangent plane of a surface in a Riemannian space were obtained by Maillot in [9]. Proof. Using (3.2) we readily get G∗ H (u) = H (Γ ′ (0)) = θ ′ (0)2 .



3.2. Study at a quasi-umbilic point We now compute G∗ H at a point x where the surface is quasi-umbilic, i.e. (recalling Theorem 1.13) where its second fundamental form is quasi-umbilic. Lemma 3.4. If M is quasi-umbilic at x and if α and β are its invariants,

⃗ |2 |ξ |2 + (α ± iβ)⟨T , ξ ⟩2 G∗ Hx (ξ ) = |H

(3.4)

for all ξ ∈ Tx M, where T is some lightlike vector belonging to Tx M. In particular, if N is a lightlike vector in Tx M such that ⟨N , T ⟩ = −1, we get G∗ H (N ) = α ± iβ.

(3.5)

The signs in (3.4) and (3.5) are positive if N = N1 in (1.8), and negative if N = N2 . Proof. Suppose that N = N1 in (1.8). We get, in the basis (e1 , e2 ), Su = α



1 0



0 1

+

1



2

1 −1



1 −1

and

Su⊥ = β



1 0



0 . 1

Since G = u ∧ u⊥ and dG(ξ ) = −Su (ξ ) ∧ u⊥ − u ∧ Su⊥ (ξ ), we readily have



dG(e1 ) = − α +

1



2

e 1 ∧ u⊥ +

1 2

e2 ∧ u⊥ + β e1 ∧ u

(3.6)

e1 ∧ u⊥ + β e2 ∧ u.

(3.7)

and



dG(e2 ) = − α −

1 2



e 2 ∧ u⊥ −

1 2

Using the very definition of H, we get (3.4) by straightforward computations (setting T = e1 − e2 ).



Using (3.3) and (3.5) we get the following interpretation of α and β :

α ± iβ = θ ′ (0)2 where θ ′ (0) is the derivative of the complex angle of the normal plane of the surface along a curve which is tangent to N at t = 0. We will see below that the direction T is a double (lightlike) asymptotic direction of the surface; thus, α ± iβ may be interpreted as the rate of variation of the normal plane in the lightlike direction which is distinct to the asymptotic one. More precisely, the argument of the complex number α ± iβ is the double of the argument of θ ′ (0), which, according to Remark 3.2, may be interpreted as the ratio of the angular velocities of two infinitesimal rotations of the normal plane in orthogonal spacelike and timelike hyperplanes respectively. On the other hand, according to Remark 1.12(ii), the modulus of α ± iβ is the squared norm of the mean curvature vector, which moreover coincides with the Gauss curvature of the surface. See Section 5 for further interpretations of α and β .

P. Bayard, F. Sánchez-Bringas / Journal of Geometry and Physics 62 (2012) 1697–1713

1707

4. Asymptotic and mean directionally curved directions We introduce here the notions of asymptotic directions and of mean directionally curved directions on a timelike surface, and study their basic properties. We will need the following family of height functions: Definition 4.1. Let M be a timelike surface embedded in R3,1 . Let us define the family of functions H : M × S1 → R, where S1 = {x ∈ R

3,1

H (p, ν) = ⟨p, ν⟩,

: ⟨x, x⟩ = 1} is the de Sitter space.

This family of height functions was introduced in [10] to define binormal directions and principal asymptotic directions on spacelike submanifolds of codimension two in Minkowski space. For a given vector ν belonging to S1 , consider hν : M → R defined by hν := H (·, ν). This function is singular at p (i.e. dhν p = 0) if and only if ν belongs to Np M. Consider also Hess hν := ∇ dhν , the Hessian of hν , where ∇ is here the Levi-Civita connection of M acting on the 1-forms. We readily get that Hess hν = IIν .

(4.1)

We say that a unit normal vector ν at p is a binormal vector if Hess hν (p) is singular, and that a non-zero vector v ∈ Tp M is an asymptotic direction if it belongs to the kernel of Hess hν (p) for some unit normal vector ν ∈ Np M. Thus, by definition, v is an asymptotic direction with associated binormal vector ν if and only if the contact at p between the surface and the hyperplane ν ⊥ is of order ≥2 in the direction v . By (4.1), we have Lemma 4.2. A non-zero vector v ∈ Tp M is an asymptotic direction if and only if Sν (v) = 0 for some unit vector ν ∈ Np M. We now prove the following Proposition 4.3. A non-zero vector v ∈ Tp M is an asymptotic direction if and only if δ(v) = 0, where δ(v) = dG(v) ∧ dG(v). Proof. Let U be a neighbourhood of p where we define a frame {ν1 , ν2 } of vector fields normal to M, such that at each q ∈ U, ν1 (q) and ν2 (q) is an orthonormal direct basis of Nq M, and such that ∇ ⊥ νj (p) = 0, j = 1, 2. Then, if v ∈ Tp M, dνjp (v) = (dνjp (v))⊤ + (dνjp (v))⊥ = −Sνjp (v). Since G(q) = ν1 (q) ∧ ν2 (q), for q ∈ U we have dGp (v) = dν1p (v) ∧ ν2 (p) + ν1 (p) ∧ dν2p (v),

= −Sν1p (v) ∧ ν2 (p) − ν1 (p) ∧ Sν2p (v).

(4.2)

Therefore, dGp (v) ∧ dGp (v) = 0 if and only if Sν1p (v) and Sν2p (v) are collinear, that is, if and only if there exist a nontrivial pair of real numbers r , s, such that rSν1p (v) + sSν2p (v) = 0, i.e. such that v ∈ Ker Sνp with ν = r ν1 + sν2 . Thus, dGp (v) ∧ dGp (v) = 0 if and only if v is an asymptotic direction.  We now study the causal character of the asymptotic directions. For this, we first write down the quadratic form δ in an adapted basis. We need to consider separately the quasi-umbilic case, and we first assume that the point is not quasi-umbilic: in some convenient basis (e1 , e2 ) of Tp M the second fundamental form reads



α+a 0

0

−(α − a)



 u˜ 1 +

β

b

b

−β

 u˜ 2 ;

(4.3)

see (1.18). The basis (˜u1 , u˜ 2 ) is here an orthonormal basis of Np M. By the very definition of δ and using (4.2) with ν1 = u˜ 1 and ν2 = u˜ 2 , straightforward computations give then the following expression for δ :

  δ(v) = 2 (α + a)bv12 + (α − a)bv22 − 2aβv1 v2

(4.4)

for all v = v1 e1 + v2 e2 ∈ Tp M. We now assume that the point is quasi-umbilic: we then readily get from (3.4) (since δ is the imaginary part of G∗ H) that

δ(v) = ±β⟨T , v⟩2

(4.5)

where T is some lightlike vector in Tp M. Observe that, as claimed at the end of the previous section, the lightlike direction T is here a double asymptotic direction. We now consider 1

δ o := δ − trg δ g 2

(4.6)

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the traceless part of the quadratic form δ . Using (4.4), (4.5) and the relations (1.19), we easily get trg δ = 2KN

and

disc(δ) := − det g δ = 4∆,

and also disc(δ o ) := − det g δ o = 4∆ + KN2 . Note that, as in the case of surfaces immersed in Euclidean 4-space, the existence of asymptotic lines at a point on the surface is equivalent to the fact that ∆ ≤ 0 at this point. By (4.6),

δ(u) = 0 if and only if δ o (u) = −KN |u|2 .

(4.7)

The causal character of the asymptotic directions appears to depend on the signs of the forms δ and δ . There are two main cases. 1- Assume first that disc(δ) < 0. In that case there are two distinct asymptotic directions. We then divide the discussion in four cases, according to the sign of δ o . First case: disc(δ o ) > 0: if δ o is positive (resp. negative), the solutions u of (4.7) are necessarily timelike (resp. spacelike) if KN > 0, and spacelike (resp. timelike) if KN < 0. Second case: disc(δ o ) < 0: we then have |uδ o |2 = − det(uδ o ) < 0, and, in some positively oriented and orthonormal basis (e1 , e2 ) of Txo M, the matrix of uδ o reads o



M (uδ o , (e1 , e2 )) = ± −|uδ o |2



0 −1



1 ; 0

see Remark 1.3. Writing u = u1 e1 + u2 e2 , (4.7) then reads

δ(u) = 0 if and only if

 ± 2 −|uδo |2 u1 u2 = −KN (u21 − u22 ).

(4.8)

Thus, if u = u1 e1 + u2 e2 is a non-trivial solution of δ(u) = 0, so is u := −u2 e1 + u1 e2 . Observe that these solutions are necessarily spacelike or timelike, and that if one of them is spacelike, the other one is timelike; thus, one asymptotic direction is spacelike and the other one is timelike. Third case: disc(δ o ) = 0, δ o ̸= 0: we then have |uδ o |2 = − det(uδ o ) = 0, and the kernel of uδ o is a lightlike line in Txo M; there is thus a unique lightlike line of solutions for the equations in (4.7). The other independent solution is thus a timelike or a spacelike line. But using (4.7) again, if δ o ≥ 0 (resp. δ o ≤ 0) this solution is necessarily timelike (resp. spacelike) if KN > 0 and spacelike (resp. timelike) if KN < 0. ⃗ = 0 in that case: since KN ̸= 0 Fourth case: δ o = 0: then δ(u) = KN |u|2 , and δ(u) = 0 ⇐⇒ |u|2 = 0. Note that H the point is not quasi-umbilic, and, by (4.4), δ o (v) = 2(α b(v12 + v22 ) − 2aβv1 v2 ). Since δ o = 0 and KN = 2ab ̸= 0, we get

⃗ = 0. α = β = 0, i.e. H

2- Assume now that disc(δ) = 0, δ ̸= 0. There is a double asymptotic direction in that case. First case: disc(δ o ) > 0: as above, if δ o is positive (resp. negative) we get that the asymptotic direction is timelike (resp. spacelike) if KN > 0, and spacelike (resp. timelike) if KN < 0. Second case: disc(δ o ) < 0: this is impossible, since disc(δ o ) = 4∆ + KN2 with ∆ = 0. Third case: disc(δ o ) = 0, δ o ̸= 0: as above, there is a lightlike solution of (4.7), namely the kernel of uδ o ; thus the double asymptotic direction is lightlike. Note that the point is quasi-umbilic, with β ̸= 0: first, since 0 = disc(δ o ) = 4∆ + 4KN2

⃗ |2 − K = 0. Suppose by contradiction that |H ⃗ |2 − K = a2 − b2 ̸= 0; with ∆ = 0, we get KN = 0. We want to show that |H in particular a and b do not vanish simultaneously. We have ∆ = −a2 β 2 + α 2 b2 − a2 b2 = 0 and KN = 2ab = 0.

If a = 0, then ∆ = α 2 b2 = 0. Since b ̸= 0, we thus necessarily have α = 0, and, using (4.4), δ o = 0, a contradiction. A similar argument gives the result if b = 0 and a ̸= 0. Moreover, since δ(u) = ±β⟨T , u⟩2 for some lightlike vector T (see (4.5)) we have β ̸= 0. Fourth case: δ o = 0: this is impossible, since it would imply δ = KN g, which is not degenerate. We summarize the results obtained so far in the following table; in the first column appear the different possible values for the signature of δ o . To simplify the presentation we suppose that KN ≥ 0; if KN ≤ 0, we just have to systematically exchange the words ‘‘spacelike’’ and ‘‘timelike’’ below. Signature of δ o (2, 0) (0, 2) (1, 1) (1, 0) (0, 1) (0, 0)

disc(δ) < 0 two distinct asymptotic directions which are Timelike Spacelike 1 spacelike − 1 timelike 1 lightlike − 1 timelike 1 lightlike − 1 spacelike ⃗ =0 Lightlike with H

disc(δ) = 0, δ ̸= 0 a double asymptotic direction which is Timelike Spacelike Not possible Lightlike (quasi-umbilicity, β ̸= 0) Lightlike (quasi-umbilicity, β ̸= 0) Not possible

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1709

The last missing case, δ = 0, is trivial: if δ = 0, x0 is a point of inflection or of umbilicity; all the directions are then asymptotic directions. This includes in particular the quasi-umbilic case with β = 0. We deduce the following characterization of the quasi-umbilic points of a timelike surface in terms of its asymptotic directions: Theorem 4.4. Assume that x0 ∈ M is such that δ ̸= 0 (x0 is not an inflection nor an umbilic point). Then x0 is a quasi-umbilic point if and only if there is a double lightlike asymptotic direction at x0 . We now give another characterization of the asymptotic directions: asymptotic directions are the pull-back of the directions tangent to the curvature hyperbola from the origin, by the second fundamental form (assuming that the hyperbola is not degenerate, i.e. a and b ̸= 0). For this, consider the basis (˜u1 , u˜ 2 ) of eigenvectors of the operator UΦ and the corresponding expression of the curvature hyperbola in Np M with coordinates (y1 , y2 ),

(y1 − α)2 a2

−

(y2 − β)2

= 1.

b2

In order to determine the lines from the origin which are tangent to the curvature hyperbola we consider the gradient vector field

 2

y1 − α a2

,−

y2 − β



b2

,

and the equation



(y1 , y2 ),



y1 − α a2

,−

y2 − β



b2

= 0.

Then, if (y1 , y2 ) lies on the hyperbola, this equation easily reduces to

α(y1 − α) a2

−

β(y2 − β) b2

= −1.

By substituting II (v1 , v2 ) = (α + a(v12 + v22 ), β + 2bv1 v2 ) in this equation and after some straightforward reduction we get b(α + a)v12 − 2aβv1 v2 + b(α − a)v22 = 0,

(4.9)

which is the equation δ(v) = 0 of the asymptotic directions; see (4.4). Let us describe another field of directions related to the asymptotic field of directions, the Mean directionally curved field of directions: this field was studied for the case of surfaces in R4 in [11] and in [7] page 1161; in our setting it is defined as the pull-back by the second fundamental form of the intersection points of the curvature hyperbola with the line determined by the mean curvature vector. This condition is described by

⃗ , II (v)] = 0 [H

(4.10)

where the brackets stand for the determinant of the vectors in a positively oriented and orthonormal basis of the normal plane. Lemma 4.5. Eq. (4.10) is equivalent to

δ(v, v ∗ ) = 0,

(4.11)

where, if v = x1 e1 + x2 e2 ∈ Tp M in a positively oriented and orthonormal basis (e1 , e2 ) of Tp M , v stands for the vector x2 e1 + x1 e2 . ∗

Proof. We first assume that the point is not quasi-umbilic, and, as in the proof of Lemma 1.11, we consider a positively oriented orthonormal basis (e1 , e2 ) of Tp M such that, for all v = x1 e1 + x2 e2 belonging to Tp M,

⃗ + a(x21 + x22 )˜u1 + 2bx1 x2 u˜ 2 . II (v) = (x21 − x22 )H ⃗ = α u˜ 1 + β u˜ 2 , Eq. (4.10) of the mean directionally curved field of directions has thus the following expression in Since H terms of the invariants: aβ(x21 + x22 ) − 2α bx1 x2 = 0.

(4.12)

Using the expression (4.4) of δ in (e1 , e2 ), we readily see that in (e1 , e2 ) Eq. (4.12) may be written as (4.11). We now assume that the point is quasi-umbilic, and, keeping the notation of Lemma 3.4 and its proof, we consider a positively oriented and orthonormal basis (e1 , e2 ) of Tp M such that, for all v = x1 e1 + x2 e2 in Tp M,

⃗+ II (v) = (x21 − x22 )H

1 2

(x1 + x2 )2 u⃗.

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⃗ = α⃗u + β⃗u⊥ , we get that (4.10) is equivalent to Since H 1 2

β (x1 + x2 )2 = 0.

Moreover, since δ(v) = ±β⟨T , v⟩2 with T = e1 − e2 (see the proof of Lemma 3.4), we easily obtain the result in the quasi-umbilic case too.  As a consequence of Lemma 4.5, writing δ(v, v ∗ ) = ⟨uδ (v), v ∗ ⟩ and noting that v ⊥ v ∗ , we readily get:

v ̸= 0 is a solution of (4.11) if and only if v is a proper direction of uδ . Note that if v is a spacelike (resp. timelike) solution of (4.11), then v ∗ is a timelike (resp. spacelike) solution of (4.11), and also that v = v ∗ if v is lightlike. Moreover the discriminant of the quadratic form in (4.12) is



a β −α b =− ∆+ 2

2

2 2

1 4

KN2



1

= − disc δ o . 4

Thus the mean directionally curved directions are spacelike–timelike if disc δ o is positive, or lightlike if disc δ o = 0, δ o ̸= 0; see Remark 4.7 for more details concerning this case. Suppose now that disc δ o > 0. Then uδ is diagonalizable in some positively oriented orthonormal basis (e1 ′ , e2 ′ ), which is thus formed by mean directionally curved directions. Its matrix in (e1 ′ , e2 ′ ) reads M (uδ , (e1 ′ , e2 ′ )) =

1 2

 KN



1 0

0 1

+

 λ

0



−λ

0

,

where λ belongs to R − {0}. Thus, if v = x1 e1 ′ + x2 e2 ′ ,

δ(v) =



1 2

   1 2 KN + λ x 1 − KN − λ x22 , 2

and, assuming for example that 21 KN + λ ̸= 0 (since δ ̸= 0), we get

 δ(v) = 0 if and only if x1 = ±

1 K 2 N 1 K 2 N

−λ +λ

x2 ;

one of the mean directionally curved lines thus bisects the asymptotic directions. Note that if disc δ = 0, δ ̸= 0, we necessarily have 12 KN −λ = 0 in the expressions above, and the double asymptotic line coincides with the mean directionally curved line x1 = 0. We thus proved the following result: Proposition 4.6. The mean directionally curved directions are the proper directions of the quadratic form δ . If disc δ o > 0 there are two distinct directions: one is spacelike and the other one is timelike; in that case the two asymptotic directions are both spacelike, or both timelike, and one of the mean directionally curved line bisects the angle determined by the asymptotic lines. Remark 4.7. Let us assume that disc δ o = 0 and δ o ̸= 0; in that case, the kernel of uδ o is a double mean directionally curved direction, which is moreover lightlike. If we moreover suppose that disc δ = 0 and δ ̸= 0, we are in the quasi-umbilic case: we also have a double lightlike asymptotic direction, which clearly coincides with the mean directionally curved direction. If we now suppose that disc δ < 0 (this is satisfied if and only if KN ̸= 0), according to the discussion concerning the causal character of the asymptotic lines, we also have a lightlike asymptotic direction in that case, which coincides with the double mean directionally curved direction; the other asymptotic direction is then spacelike or timelike. Note that the mean ⃗ belongs to the asymptotes in that case, with H ⃗ ̸= 0. curvature vector H 5. Quasi-umbilic timelike surfaces in R3,1 We study here timelike surfaces which are such that

⃗ |2 = K and KN = ∆ = 0 |H at every point; such surfaces in the 3-dimensional Minkowski space were described in [8] and were called there quasi⃗ ̸= 0). We keep the same terminology. umbilic surfaces (if they are not umbilic, that is if II o := II − Hg

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5.1. Description of the quasi-umbilic surfaces We will prove the following Theorem 5.1. A timelike surface M in R3,1 is umbilic or quasi-umbilic if and only if it is locally parameterized by

ψ(s, t ) = γ (s) + tT (s)

(5.1) 3 ,1

where γ is a lightlike curve in R value of s.

and T is some lightlike vector field along γ such that γ (s) and T (s) are independent for all ′

This result extends Theorem 4.3 in [8] to the four-dimensional Minkowski space. We will need the following lemma: Lemma 5.2. M is umbilic or quasi-umbilic if and only if at each point of M there exists a lightlike tangent vector T ̸= 0 belonging to the kernel of the traceless part of the second fundamental form, i.e. such that II o (·, T ) = 0. Proof. Recalling the notation in Section 1, M is umbilic or quasi-umbilic at xo if and only if the image of the map f : ν → Sνo is zero (umbilicity), or a lightlike line of the space S of traceless symmetric operator of Txo M (quasi-umbilicity); umbilicity or quasi-umbilicity are also equivalent to det(Sνo ) = 0 for all ν ∈ Nxo M. Suppose first that T is such that II o (·, T ) = 0. This implies that T belongs to the kernel of Sνo , for all ν ∈ Nxo M. Thus det(Sνo ) = 0, and the result follows. We now suppose that det(Sνo ) = 0 for all ν ; if f is zero, the existence of T is trivial; we thus assume that f ̸= 0, and using Lemma 1.7, obtain that, in some orthonormal basis (e1 , e2 ) of Txo M , Sνo = ⟨⃗ u, ν⟩N , with N = N1 or N2 . If N = N1 , we may choose T = e2 − e1 and if N = N2 , T = e1 + e2 .  Proof of Theorem. We first suppose that M is umbilic or quasi-umbilic and consider local coordinates (s, t ) → ψ(s, t ) of M such that the metric of M reads g = −2a(s, t )dsdt , where a is some positive function. We suppose that T := ∂∂t is such that II o (·, T ) = 0. Computing the Christoffel symbols we get

∇T T = thus T ′ :=

1 T a

1 ∂a a ∂t

T;

is such that II o (·, T ′ ) = 0 and ∇T T ′ = 0. Since dT ′ = ∇ T ′ + II (·, T ′ ), we get dT ′ (T ) = 0. Thus

T ′ (ψ(s, t )) = T ′ (ψ(s, 0)) +

∂ ′ (T (ψ(s, u)))du = T ′ (ψ(s, 0)), ∂u

t

 0

that is T (ψ(s, t )) =

a(s, t ) a(s, 0)

T (ψ(s, 0)).

Thus

∂ψ (s, u)du ∂t 0  t  a(s, u) = ψ(s, 0) + du T (ψ(s, 0)), 0 a(s, 0)

ψ(s, t ) = ψ(s, 0) +



t

i.e. ψ is of the form

ψ(s, t ) = γ (s) + f (s, t )T (s) where γ (s) := ψ(s, 0) is a lightlike curve and T (s) := T (ψ(s, 0)) is a lightlike vector field along γ . Since ψ(s, 0) = γ (s), we have f (s, 0) = 0; moreover

∂f a(s, t ) = > 0, ∂t a(s, 0) and formulae s′ = s, t ′ = f (s, t ) thus define local coordinates such that (5.1) holds. We now suppose that (5.1) holds. The second fundamental form of M in the coordinates (s, t ) is given by II = (d2 ψ)N , the normal component of the Hessian of ψ . Since



∂ 2ψ ∂ s2

N

 N = γ ′′ + tT ′′ ,



∂ 2ψ ∂t2

N

 = 0 and

∂ 2ψ ∂ s∂ t

N

= (T ′ )N ,

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P. Bayard, F. Sánchez-Bringas / Journal of Geometry and Physics 62 (2012) 1697–1713

we get

⃗= H

1 2

g ij IIij =

1

⟨γ ′ , T ⟩

(T ′ )N ,

and

⃗ = II o = II − Hg



(γ ′′ + tT ′′ )N 0



0 . 0

Thus II (·, T ) = 0 and M is umbilic or quasi-umbilic. o



Remark 5.3. The description of the timelike umbilic surfaces in R3,1 is well-known (see e.g. [12]): M is umbilic if and only if γ and T in (5.1) may be given by

γ (s) = γ0 + su,

T (s) = v + asw +

1 2

a2 s2 u

where γ0 is a point belonging to R3,1 , u and v are lightlike vectors such that ⟨u, v⟩ = −1, w is a spacelike and unit vector orthogonal to u and v and where a is some non-negative number; if a = 0 the surface is an open set of a timelike plane in R3,1 , and if a > 0 the surface is an open set of a sphere {x : |x − x0 |2 = a12 } in some affine timelike hyperplane in R3,1 . Remark 5.4. Quasi-umbilic surfaces may thus be constructed γ ′ in the  s ′ as follows: consider an arbitrary (smooth) curve 3,1 2 ′ lightcone {x ∈ R : |x| = 0}, define γ (s) := γ (0) + 0 γ (u)du and take a lightlike vector field T along γ such that γ ′ and T are independent; formula (5.1) defines a quasi-umbilic surface. Example 5.5. Taking

γ ′ (s) = (cos s, sin s, 1) and T (s) = (− cos s, − sin s, 1) in R2,1 ⊂ R3,1 , formula (5.1) gives a quasi-umbilic timelike surface such that

α=+

1 2

and β = 0,

and taking

γ ′ (s) = (sinh s, 1, cosh s) and T (s) = (sinh s, −1, cosh s) in R2,1 ⊂ R3,1 we get a quasi-umbilic surface such that

α=−

1 2

and β = 0.

The first example was given in [8]. 5.2. Quasi-umbilic surfaces such that β = 0 The invariant β of a surface with regular Gauss map (i.e. such that dG is injective at every point) determines whether the surface belongs to some hyperplane or not: Proposition 5.6. Suppose that M is a quasi-umbilic surface with regular Gauss map, such that β = 0 on M. Then M belongs to some affine hyperplane; conversely, if M belongs to some hyperplane then β = 0 on M. Proof. Recall that the quadratic form δ on Tx M is defined by

δ(u) := dGx (u) ∧ dGx (u) for all u ∈ Tx M. Using (3.4), we get

δ(u) = ±β⟨T , u⟩2 , where T ̸= 0 is some lightlike vector belonging to Tx M. Thus δ = 0 if and only if β = 0, and the result follows since the Gauss map is moreover assumed to be regular.  Remark 5.7. If M is a quasi-umbilic surface such that β = 0, its Gauss map is regular if and only if α ̸= 0, that is, if and only if its Gauss curvature does not vanish: using (3.6) and (3.7) with β = 0, we readily obtain the matrix of dG in the bases (e1 , e2 ), (e1 ∧ u⊥ , e2 ∧ u⊥ ); its determinant is α 2 .

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We describe quasi-umbilic surfaces with regular Gauss map, β = 0 and (non-zero) constant Gauss curvature: Proposition 5.8. Suppose that M is a quasi-umbilic surface with regular Gauss map, such that β = 0 on M. Then its Gauss curvature is constant if and only if M may be parameterized by (5.1) where γ is a lightlike curve in some affine timelike hyperplane H of R3,1 and T is a lightlike vector field in H such that |T ′ | = 1 and ⟨γ ′ , T ⟩ is constant. Proof. This readily follows from the proposition above and from the formula

⃗ |2 = K = |H

|T ′ |2 .  ⟨γ ′ , T ⟩2

(5.2)

We finally describe flat quasi-umbilic surfaces (α = 0, β = 0). Proposition 5.9. A quasi-umbilic timelike surface M is flat if and only if it may be locally parameterized by (5.1) where γ is a lightlike curve in R3,1 and T = T0 is a constant lightlike vector field along γ such that ⟨γ ′ , T0 ⟩ ̸= 0 everywhere. Proof. Suppose that M is a flat quasi-umbilic timelike surface locally parameterized by (5.1). By formula (5.2), we get |T ′ |2 = 0. Since |T |2 = 0 and ⟨T , T ′ ⟩ = 0, we necessarily have T ′ (s) = f (s)T (s) for some smooth function f . Thus, by integration, T (s) = F (s)T (0) where F is a smooth function such that F (0) = 1. Setting s′ = s, t ′ = tF (s) and T0 = T (0), M is parameterized by γ (s′ ) + t ′ T0 , for small values of s′ and t ′ . The converse statement is straightforward.  Remark 5.10. The Gauss map of a flat quasi-umbilic surface is not regular; the constant lightlike vector T0 belongs to the kernel of dGx at each point. Acknowledgement The second author was partially supported by DGAPA-UNAM grant PAPIIT-IN120011. References [1] D.K.H. Mochida, M.C. Romero Fuster, M.A.S. Ruas, The geometry of surfaces in 4-space from a contact viewpoint, Geom. Dedicata 54 (1995) 323–332. [2] C.T.C. Wall, Geometric properties of generic differentiable manifolds, in: J. Palis, M.P. do Carmo (Eds.), Geometry and Topology: III Latin American School of Mathematics, in: Lecture Notes in Math., vol. 597, Springer-Verlag, 1977, pp. 707–774. [3] J.A. Little, On the singularities of submanifolds of higher dimensional Euclidean spaces, Ann. Mat. Pura Appl. (4A) 83 (1969) 261–336. [4] C.L.E. Moore, E.B. Wilson, Differential geometry of two-dimensional surfaces in hyperspaces, Proc. Amer. Acad. Arts Sci. 52 (1916) 267–368. [5] Y.-C. Wong, Contributions to the theory of surfaces in a 4-space of constant curvature, Trans. Amer. Math. Soc. 59 (3) (1946) 467–507. [6] M. Kriele, Spacetime: Foundations of General Relativity and Differential Geometry, in: Lecture Notes in Physics, Monographs, vol. 59, Springer, 1999. [7] P. Bayard, F. Sánchez-Bringas, Geometric invariants and principal configurations on spacelike surfaces immersed in R3,1 , Proc. Roy. Soc. Edinburgh Sect. A 140 (2010) 1141–1160. [8] J. Clelland, Totally quasi-umbilic timelike surfaces in R1,2 , arXiv:1006.4380v2, 2010. [9] H. Maillot, Courbures et basculements des sous-variétés riemanniennes, Mém. Soc. Math. Fr. (NS) 22 (1986). [10] S. Izumiya, J.J. Nuño Ballesteros, M.C. Romero Fuster, Global properties of codimension two spacelike submanifolds in Minkowski space, Adv. Geom. 10 (2010) 51–75. [11] F. Tari, Self-adjoint operators on surfaces in Rn , Differ. Geom. Appl. 27 (2) (2009) 296–306. [12] S.K. Hong, Totally umbilic Lorentzian surfaces embedded in Ln , Bull. Korean Math. Soc. 34 (1) (1997) 9–17.