On Lorentzian surfaces in R 2,2
aa r X i v : . [ m a t h . DG ] M a r On Lorentzian surfaces in R , ∗ , Victor Patty † , Federico S´anchez-Bringas ‡ Abstract: We study the second order invariants of a Lorentzian surface in R , , and the curvaturehyperbolas associated to its second fundamental form. Besides the four natural invariants, newinvariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentziansurface and give an extrinsic proof of the vanishing of the total Gauss and normal curvatures ofa compact Lorentzian surface. The Gauss map and the second order invariants are then used tostudy the asymptotic directions of a Lorentzian surface and discuss their causal character. We alsoconsider the relation of the asymptotic lines with the mean directionally curved lines. We finallyintroduce and describe the quasi-umbilic surfaces, and the surfaces whose four classical invariantsvanish identically. Introduction
Let R , be the space R with the metric g = − dx + dx − dx + dx . A surface M ⊂ R , is said to be Lorentzian if the metric g induces a Lorentzian metric, i.e. ametric of signature (1 , , on M : the tangent and the normal bundles T M and
N M of a Lorentziansurface are equipped with Lorentzian fibre metrics. The second fundamental form at a point p of a Lorentzian surface M is a quadratic map T p M → N p M. The numerical invariants of thesecond fundamental form are second order invariants of the surface at p, and locally determine theextrinsic geometry of the surface in R , . The first purpose of the paper is to completely determinethese invariants: additionally to the 4 natural invariants | ~H | , K, K N and ∆ which are the normof the mean curvature vector, the Gauss curvature, the normal curvature and the resultant of thesecond fundamental form traducing the local convexity of the surface, new invariants appear in somedegenerate cases. A systematic study of the numerical invariants of a quadratic map R , → R , isnecessary for this complete description. The second order invariants of surfaces and their geometricmeaning have been extensively studied in different settings. In [12] J. Little studied them in thecase of a surface immersed in 4-dimensional Euclidian space. The second order invariants of aspacelike and a timelike surface in 4-dimensional Minkowski space were systematically studied in[4] and [5]. With the study of the quadratic maps between two Lorentzian planes, the presentpaper thus completes the description of the second order invariants of surfaces in 4-dimensionalpseudo-Euclidian spaces.We then introduce the notion of curvature hyperbola associated to a quadratic map R , → R , , which is analogous to the classical notion of curvature ellipse introduced in the Euclidian setting[12, 15]. Its geometric properties may be naturally given in terms of the invariants of the quadraticmap. When applied to the second fundamental form of a Lorentzian surface in R , , the curvature ∗ [email protected], Facultad de Ciencias, Universidad Nacional Aut´onoma de M´exico, M´exico † [email protected], Instituto de F´ısica y Matem´aticas, U.M.S.N.H., Ciudad Universitaria, CP. 58040 More-lia, Michoac´an, M´exico ‡ [email protected], Facultad de Ciencias, Universidad Nacional Aut´onoma de M´exico, M´exico R , . Wefirst introduce the Gauss map of an oriented Lorentzian surface. We show that the Gauss andthe normal curvatures are obtained taking the pull-back by the Gauss map of the Lie bracket inΛ R , ; as a consequence of this formula we obtain an extrinsic proof of the well-known fact thatthe total Gauss and normal curvatures vanish for a compact Lorentzian surface in R , . We then use the preceding results to introduce the notion of asymptotic directions of a Lorentziansurface in R , and in Anti de Sitter space; we especially discuss the causal character of the asymp-totic lines in terms of the invariants. Moreover, we relate these directions with the contact direc-tions associated to the family of height functions on a Lorentzian surface M in R , [6]. We alsointroduce the mean directionally curved lines on a Lorentzian surface and specify their relationwith the asymptotic lines.We finally study the quasi-umbilic surfaces in R , , which are defined as the Lorentzian sur-faces whose curvature hyperbolas degenerate at every point to a line with one point removed;alternatively, they are non-umbilic surfaces such that | ~H | = K and K N = ∆ = 0at every point. We then describe the Lorentzian surfaces in R , whose classical invariants | ~H | ,K, K N and ∆ vanish identically: they are surfaces in degenerate hyperplanes or flat umbilic orquasi-umbilic surfaces. In [7], J. Clelland introduced and described the quasi-umbilic surfacesin 3-dimensional Minkowski space. The results of this last paper were then extended to the 4-dimensional Minkowski space in [5]; in the present paper, the results concerning the quasi-umbilicsurfaces in R , may also be considered as extending the main results of [7].The outline of the paper is as follows: we first study the quadratic maps from the Lorentz plane R , into itself and their numerical invariants in Section 1, and describe the curvature hyperbolaassociated to such a quadratic map in Section 2; we then study the Gauss map of a Lorentzian sur-face in Section 3, and the asymptotic lines and the mean directionally curved lines of a Lorentziansurface in R , and in Anti-de Sitter space in Section 4. In Section 5, we finally introduce thenotion of quasi-umbilic surfaces and describe the surfaces which are umbilic or quasi-umbilic, andalso the surfaces whose classical invariants vanish identically. R , → R , and their numerical invari-ants Let R , be the vector space R equipped with the Lorentzian metric h· , ·i := − dx + dx . We will say that a non-zero vector X belonging to R , is spacelike (resp. timelike, or lightlike) ifits Lorentzian norm h X, X i is positive (resp. negative, or null).We denote by Q ( R , , R , ) the vector space of quadratics maps from R , to R , . We supposethat R , is canonically oriented in space and in time: the canonical basis of R , defines theorientation and a timelike vector in R , is said to be future-directed if its first component in thecanonical basis is positive. We consider the reduced (connected) group SO (1 ,
1) of Lorentziandirect isometries of R , . This group acts on Q ( R , , R , ) by composition (on the left and on the2ight) SO (1 , × Q ( R , , R , ) × SO (1 , → Q ( R , , R , )( g , q, g ) → g ◦ q ◦ g . In this section, we are interested in the description of the quotient set SO (1 , \ Q ( R , , R , ) /SO (1 , R , → R , up to the actions of SO (1 ,
1) (Theorem 1.14). The notion of quasi-umbilic quadratic map will also emerge naturally.
We fix q ∈ Q ( R , , R , ) . If ν ∈ R , , we denote by S ν : R , → R , the symmetric endomor-phism associated to the real quadratic form h q, ν i , i.e. such that h S ν ( x ) , x i = h q ( x ) , ν i for all x ∈ R , . For ν, ν , ν ∈ R , we define L q ( ν ) := 12 tr( S ν ) , Q q ( ν ) := det( S ν ) and A q ( ν , ν ) := 12 [ S ν , S ν ] , where [ S ν , S ν ] denotes the morphism S ν ◦ S ν − S ν ◦ S ν ; this morphism is skew-symmetric on R , , and thus identifies with the real number ǫ such that its matrix in the canonical basis of R , is (cid:18) ǫǫ (cid:19) . In the sequel, we will implicitly make this identification. We note that L q is a linear form, Q q is a quadratic form and A q is a bilinear skew-symmetric form on R , . These forms are linkedaccording to the following lemma:
Lemma 1.1.
The quadratic form Φ q := L q − Q q satisfies the following identity: for all ν , ν ∈ R , , Φ q ( ν )Φ q ( ν ) = ˜Φ q ( ν , ν ) − A q ( ν , ν ) , (1) where ˜Φ q ( · , · ) denotes the symmetric bilinear form such that ˜Φ q ( ν, ν ) = Φ q ( ν ) for all ν ∈ R , . Inparticular the signature of Φ q is ( r, s ) with ≤ r, s ≤ . This lemma may be proved by a direct computation, using the representation of S ν and S ν bytheir matrices in the canonical basis of R , . An alternative argument will also be given in Remark1.3 below.
Remark . The forms L q , Φ q and A q are invariant by the right-action of SO (1 ,
1) on q : for all g ∈ SO (1 ,
1) we have L q ◦ g = L q , Φ q ◦ g = Φ q and A q ◦ g = A q . They are thus also defined on the quotient set Q ( R , , R , ) /SO (1 , . In the next section we will show the following: if Φ q = 0 , the forms L q , Φ q and A q determine q up to the right-action of SO (1 , q ≡ , q is determined, up to the right-actionof SO (1 , , by the form L q together with some additional vector µ q ∈ R , (Lemmas 1.5 and 1.7below). 3 .2 Reduction of a quadratic map We denote by S the vector space of the traceless symmetric endomorphisms of R , . S isnaturally equipped with a metric tensor of signature (1 ,
1) : if u belongs to S , we define its normas | u | := 12 tr( u ) . Expressing u in the canonical basis of R , , we also easily get | u | = − det u. Setting E := (cid:18) − (cid:19) and E := (cid:18) − (cid:19) , we have that ( E , E ) is a Lorentzian basis for S such that | E | = −| E | = 1 . Now, associatedto a given quadratic map q ∈ Q ( R , , R , ) , we consider the linear map f q : R , → S ν S ν := S ν − L q ( ν ) I ;for ν ∈ R , , f q ( ν ) is thus the traceless part S ν of the symmetric operator S ν . Remark . It is not difficult to prove the following: for all ν , ν ∈ R , we have˜Φ q ( ν , ν ) = h f q ( ν ) , f q ( ν ) i and A q ( ν , ν ) = det ( E ,E ) ( f q ( ν ) , f q ( ν )) , (2)where, if s and s ′ belong to S , h s, s ′ i and det ( E ,E ) ( s, s ′ ) stand respectively for the scalar productand for the determinant in the basis ( E , E ) of s and s ′ (considered as vectors of the Lorentzianplane S ). Formulas (2) and the Lagrange identity in the Lorentzian plane ( S , h· , ·i ) give a directproof of (1).We recall the following convention concerning the orientation of R , : a basis ( e , e ) of R , is positively oriented if it has the orientation of the canonical basis and if the vector e is timelikeand future-directed, i.e. is such that its first component in the canonical basis is positive (see theintroduction of this section). Remark . If u belongs to S , u = 0 , its norm | u | = − det( u ) determines its canonical form asfollows: u is diagonalizable if and only if | u | > , i.e. if and only if u ∈ S is spacelike; in that case, u = ± p | u | E in some positively oriented and orthonormal basis of R , . If | u | < u is timelike in S ), then u = ± p −| u | E (3)in some positively oriented and orthonormal basis of R , . Finally, if | u | = 0 , setting N := 12 ( E + E ) and N := 12 ( E − E ) , then u = ± N i , i = 1 or 2 (4)in some positively oriented and orthonormal basis of R , .
4e now consider the reduction of a quadratic map q ∈ Q ( R , , R , ) , and divide the discussionin three cases, according to the ranks of f q and Φ q . rang ( f q ) = , or rang ( f q ) = with Φ q = . In that case, there is an orthonormal andpositively oriented basis ( e , e ) of R , such that, in ( e , e ) , S ν , for all ν ∈ R , , has thefollowing canonical form:Rank f q Signature of Φ q Canonical form of S ν S ν = L q ( ν ) I ± ( ˜Φ q ( ν , ν ) E + A q ( ν , ν ) E )1 (1,0) S ν = L q ( ν ) I ± ˜Φ q ( ν , ν ) E S ν = L q ( ν ) I ± ˜Φ q ( ν , ν ) E In the table, ν is some vector belonging to R , . We only give brief indications of the proof,since similar results are proved in [5]. In the first case, we consider ν such that Φ( ν ) = 1;from the remark above, S ν = L ( ν ) I ± E in some orthonormal and positively oriented basis( e , e ) of R , . In ( e , e ) and for an arbitrary ν ∈ R , , S ν may be a priori written S ν = L ( ν ) I ± ( a ν E + b ν E )for some a ν , b ν belonging to R . Straightforward computations using (2) then give a ν =˜Φ q ( ν , ν ) and b ν = A q ( ν , ν ) and thus the required expression. The other cases may beproved similarly (taking ν such that Φ( ν ) = − Q ( R , , R , ) := { q ∈ Q ( R , , R , ) : Φ q = 0 } . Setting P = { ( L, Φ , A ) : Φ is not zero, has a non-positive discriminant, and (1) holds } where L, Φ and A are respectively linear, bilinear symmetric and skew-symmetric forms on R , , the following result holds: Lemma 1.5.
The map Θ : Q ( R , , R , ) /SO (1 , −→ P given by [ q ] ( L [ q ] , Φ [ q ] , A [ q ] ) is surjective and two-to-one. We refer to [5] for details, where a similar result is proved.By the natural left-action of SO (1 ,
1) on Q ( R , , R , ) /SO (1 , , the forms L [ q ] , Φ [ q ] trans-form as L g ◦ [ q ] = L [ q ] ◦ g − , Φ g ◦ [ q ] = Φ [ q ] ◦ g − , whereas the form A [ q ] is invariant. Thus, if SO (1 ,
1) acts on P by g. ( L, Φ , A ) = ( L ◦ g − , Φ ◦ g − , A ) , (5)the map Θ is SO (1 , : SO (1 , \ Q ( R , , R , ) /SO (1 , −→ SO (1 , \ P . (6)Since the formula (1) permits the recovering of A (up to sign) from Φ , the description ofthe quotient set SO (1 , \ Q ( R , , R , ) /SO (1 ,
1) will be achieved with the simultaneousreduction of the forms L [ q ] and Φ [ q ] . This is the aim of the first part of Section 1.4.5. rang ( f q ) = and Φ q = . In that case, f q ( R , ) is a line in S , which is lightlike, and wehave: Lemma 1.6.
There is a vector µ q ∈ R , , unit spacelike or timelike, or lightlike distinguished,and an orthonormal and positively oriented basis ( e , e ) of R , such that, for all ν ∈ R , , the matrix of S ν in ( e , e ) is given by S ν = L q ( ν ) I + h µ q , ν i N, (7) where N = N or N (see Remark 1.4). The vector µ q and the basis ( e , e ) are uniquelydefined.Proof. In the canonical basis of R , , we have S ν = λ q ( ν ) N, where N = N or N , and where λ q is a linear form on R , . We define µ q ∈ R , such that λ q ( ν ) = h µ q , ν i for all ν ∈ R , . We now consider the basis of R , obtained from the canonical basis by a Lorentzian rotationof angle ψ. The matrix of S ν in this basis is S ν = h e ψ µ q , ν i N. Thus, there is a uniqueorthonormal and positively oriented basis of R , such that in that basis S ν = h µ q , ν i N with | µ q | = +1 , − µ q = ( ± , ± . We now set Q ( R , , R , ) := { q ∈ Q ( R , , R , ) : Φ q = 0 , f q = 0 } and P := R , ∗ × H where R , ∗ stands for the set of linear forms on R , and H := (cid:26) µ ∈ R , : | µ | = ± µ = 12 ( ± , ± (cid:27) . Lemma 1.7.
The map Θ : Q ( R , , R , ) /SO (1 , −→ P [ q ] ( L [ q ] , µ [ q ] ) is surjective and two-to-one.Proof. To each pair (
L, µ ) ∈ P correspond two classes in Q ( R , , R , ) /SO (1 , , defining S ν in the canonical basis of R , by (7), where N may be chosen to be N or N . If SO (1 ,
1) acts on P by g. ( L, µ ) = ( L ◦ g − , g.µ ) , where g.µ = g ( µ ) if | µ | = ± , and g.µ = µ if | µ | = 0 , the map Θ is SO (1 , : SO (1 , \ Q ( R , , R , ) /SO (1 , −→ SO (1 , \ P . Thus, the description of the quotient set SO (1 , \ Q ( R , , R , ) /SO (1 ,
1) will be achievedwith the simultaneous reduction of the form L [ q ] and the vector µ [ q ] ∈ H . This is the aim ofthe second part of Section 1.4. 6. f q = . In that case, S ν = L q ( ν ) I for all ν ∈ R , . We define Q ( R , , R , ) := { q ∈ Q ( R , , R , ) : Φ q = 0 , f q = 0 } . Setting P := R , ∗ , the mapΘ : SO (1 , \ Q ( R , , R , ) /SO (1 , −→ SO (1 , \ P [ q ] [ L [ q ] ]is bijective, where the action of SO (1 ,
1) on P is given by g.L = L ◦ g − .We finally define the notions of quasi-umbilic and umbilic quadratic maps, which correspond tothe last two cases considered above: Definition 1.8.
A quadratic map q : R , → R , is said to be quasi-umbilic if rank ( f q ) = 1 and Φ q = 0; this equivalently means that f q ( R , ) is a lightlike line in S . A quadratic map q : R , → R , issaid to be umbilic if f q = 0 . In this section, we define invariants on the quotient set SO (1 , \ Q ( R , , R , ) /SO (1 ,
1) asso-ciated to L [ q ] , Q [ q ] , A [ q ] and Φ [ q ] . Definition 1.9.
Let q : R , → R , be a quadratic map, and [ q ] ∈ Q ( R , , R , ) /SO (1 , its classup to the right action of SO (1 , . We consider1. the vector ~H ∈ R , such that, for all ν ∈ R , , L [ q ] ( ν ) = h ~H, ν i , and its norm | ~H | := h ~H, ~H i ;
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 endomorphismof R , associated to Q [ q ] by the metric h· , ·i on R , ;
3. the real number K N such that A [ q ] = 12 K N ω , where ω is the determinant in the canonical basis of R , (the canonical area form on R , ). The numbers | ~H | , K, K N and ∆ are kept invariant by the left-action of SO (1 ,
1) on [ q ] ∈ Q ( R , , R , ) /SO (1 ,
1) and thus define invariants on the quotient set SO (1 , \ Q ( R , , R , ) /SO (1 , . Remark . When the element of the quotient is given by the second fundamental form ofa Lorentzian surface in R , (see Section 3), ~H, K and K N correspond to the mean curvaturevector, the Gauss curvature and the normal curvature of the surface; the invariant ∆ is similarto the invariant ∆ introduced in [12] for surfaces in R . This is naturally the motivation for thesedefinitions. 7 emark . Let U Φ be the symmetric endomorphism on R , associated to the quadratic formΦ [ q ] . Denoting by tr Φ [ q ] and det Φ [ q ] its trace and its determinant, we havetr Φ [ q ] = | ~H | − K and det Φ [ q ] = 14 K N . (8)These formulas may be proved by direct computations using the very definitions of Φ [ q ] and theinvariants; they will be useful below. Accordingly to the previous sections, we have to consider two cases:
1. Case Φ [ q ] = 0 . In this case, [ q ] ∈ Q ( R , , R , ) /SO (1 ,
1) is determined by Φ [ q ] and L [ q ] (Lemma 1.5); we thus reduce the operator U Φ , together with the mean curvature vector ~H : Proposition 1.12. U Φ is diagonalizable if and only if U = 0 or ( | ~H | − K ) − K N > . In that last case, there is a unique orthonormal and positively oriented basis ( u , u ) of R , suchthat the matrix of U Φ in ( u , u ) is (cid:18) a b (cid:19) (9) where a := | ~H | − K ± q ( | ~H | − K ) − K N and b := | ~H | − K ∓ q ( | ~H | − K ) − K N . (11) Moreover, defining α, β ∈ R such that ~H = αu + βu , we have α = 1 b − a (cid:18) a | ~H | + ∆ − K N (cid:19) (12) and β = 1 b − a (cid:18) b | ~H | + ∆ − K N (cid:19) . (13) Proof.
The first part of the proposition follows from the fact that U Φ is diagonalizable if and onlyif U = 0 or 1 / U Φ ) > det U Φ , together with (8) ( U is spacelike in S , see Remark 1.4). Forthe second part of the statement, we consider the quadratic form Q = L − Φ and its associatedsymmetric operator U Q : R , → R , ; its matrix in ( u , u ) is U Q = (cid:18) − α − a αβ − αβ β − b (cid:19) . The formulas tr U Q = K, det U Q = ∆ (Definition 1.9) and (8) easily give (12) and (13).8 roposition 1.13. U Φ is not diagonalizable if and only if U = 0 and ( | ~H | − K ) − K N ≤ , (14) and we have: if ( | ~H | − K ) − K N < , there is a unique orthonormal and positively oriented basis ( u , u ) of R , such that the matrix of U Φ in ( u , u ) is | ~H | − K (cid:18) (cid:19) ± q K N − ( | ~H | − K ) (cid:18) − (cid:19) . (15) Writing ~H = αu + βu , we have α = 12 (cid:18) −| ~H | + q | ~H | + 4 u (cid:19) , β = 12 (cid:18) | ~H | + q | ~H | + 4 u (cid:19) , (16) where u = 1 q K N − ( | ~H | − K ) (cid:18) − ∆ + 14 K N − | ~H | ( | ~H | − K ) (cid:19) . if ( | ~H | − K ) − K N = 0 , there is a unique orthonormal and positively oriented basis ( u , u ) of R , such that the matrix of U Φ in ( u , u ) is | ~H | − K (cid:18) (cid:19) + (cid:18) ε ε − ε − ε (cid:19) (17) where ε = ± , ε = ± . Writing ~H = αu + βu , we have α = ε (cid:16) | ~H | − ε v (cid:17) v , β = ε (cid:16) | ~H | + ε v (cid:17) v , (18) if v := − ∆ + K N − | ~H | ( | ~H | − K ) is not 0. Moreover, v = 0 if and only if | ~H | = 0; in thatcase, ∆ = 14 K N = 14 K , | ~H | = 0 (19) and ~H = αu + βu , with α = ± β (20) defines new invariants α, β. Proof. In that case U is timelike in S , and its reduction is given by (3) in Remark 1.4, whichproves (15). Formulas (16) may then be proved as formulas (12) and (13) in Proposition 1.12above. Here U is lightlike in S , and its reduction is given by (4) in Remark 1.4, which proves (17).We also get formulas (18) as in Proposition 1.12 above. Further, computing ∆ = det U Q as in theproof of Proposition 1.12, with U Φ given here by (17), we may easily get v = ε ( α + β ) + 2 ε αβ. Thus v = 0 if and only if α = ± β, i.e. | ~H | = 0; formulas (19) then easily follow.9 . Case Φ [ q ] = 0 . In this case, and if f q = 0 , [ q ] ∈ Q ( R , , R , ) /SO (1 ,
1) is determined by theform L [ q ] together with the vector µ [ q ] (Lemma 1.6), and we need to simultaneously reduce L [ q ] and µ [ q ] . We recall that µ [ q ] is normalized so that | µ [ q ] | = ± , or µ [ q ] = ( ± , ± , and we definethe vector µ ∗ [ q ] := ( µ ⊥ [ q ] , if | µ [ q ] | = ± µ ′ [ q ] , if µ [ q ] = ( ± , ±
1) (21)where µ ⊥ [ q ] denotes the reflection of µ [ q ] with respect to the principal diagonal of R , in the firstcase, and µ ′ [ q ] the unique lightlike vector such that h µ [ q ] , µ ′ [ q ] i = in the second case. Now ( µ [ q ] , µ ∗ [ q ] )is a basis of R , , and we define α and β such that ~H = αµ [ q ] + βµ ∗ [ q ] . (22)The numbers α and β are new invariants. We will give an interpretation of these invariants inSection 3 below. We gather the results obtained in the previous sections and give the classification of thequadratic maps R , → R , in terms of their numerical invariants. For sake of simplicity, wewill say that a set of invariants essentially determines a class in SO (1 , \ Q ( R , , R , ) /SO (1 , finite number of classes (corresponding to choices of signs in theformulas given in the previous sections). Theorem 1.14.
The class [ q ] ∈ SO (1 , \ Q ( R , , R , ) /SO (1 , is determined by its invariantsin the following way:1. If Φ = 0 then the following holds:(a) if ( | ~H | − K ) − K N = 0 , the invariants K, K N , | ~H | , ∆ essentially determine [ q ]; (b) if ( | ~H | − K ) − K N = 0 , then we have:i. if | ~H | = 0 , the invariants K, | ~H | , ∆ essentially determine [ q ]; ii. if | ~H | = 0 , the invariant K together with the new invariants α, β defined in (20)essentially determine [ q ] .
2. If
Φ = 0 then K N = 0 , | ~H | − K = 0 and ∆ = 0 , and we have the following:(a) if f q = 0 , the invariants α and β defined in (22) essentially determine [ q ] ( q is quasi-umbilic );(b) if f q = 0 , then | ~H | determine [ q ] ( q is umbilic ).Proof. We only consider the case ( | ~H | − K ) − K N > in (6), Section 1.2. By Proposition 1.12, Θ ([ q ]) isthe class of ( L, Φ , A ) ∈ P where the forms L, Φ and A are defined in the canonical basis ( u , u )of R , by L = ( α, β ) , Φ = (cid:18) a b (cid:19) and A = 12 K N (cid:18) − (cid:19) , with a, b, α and β satisfying (10)-(13) (more precisely, recalling (5), if g ∈ SO (1 ,
1) is such that g ( u ) = ˜ u , g ( u ) = ˜ u , where (˜ u , ˜ u ) is the basis given by Proposition 1.12, we have g. ( L, Φ , A ) =( L q , Φ q , A q )). Since we can choose a sign in the definitions (10) and (11) of a and b, and since α and β are determined up to sign by (12) and (13), sixteen classes correspond to the given set ofinvariants (two classes correspond to each one of the eight possible choices for a, b, α and β sincethe map Θ in (6) is two-to-one). 10 The curvature hyperbola of q : R , → R , The curvature hyperbola H associated to q : R , → R , is defined as the subset of R , H := (cid:26) q ( v ) | v | : v ∈ R , , | v | = ± (cid:27) ;this is the natural analog of the curvature ellipse associated to a quadratic map R → R , where R is the Euclidian plane. Denoting by O the origin of R , , the center of H is the point C suchthat −→ OC = ~H ( ~H is the mean curvature vector of q, see Definition 1.9). We will say that a point P of the hyperbola is spacelike (resp. timelike) if the vector −→ CP is a spacelike (resp. timelike) vectorof R , . Generically, the curvature hyperbola is given as follows: ✟✟✟✟✟✟✟✟✟✟❆❆❆❆❆❆❆❆❆❆ q❛ CO q We have the following descriptions of the hyperbola:
Proposition 2.1. If K N = 0 , the curvature hyperbola H is not degenerate, and the following holds:1- if U = 0 , the curvature hyperbola is H = ( ~H + ν : | ν | = | ~H | − K ) ; its asymptotes are two null lines in R , ;
2- if U = 0 and U Φ is diagonalizable with eigenvalues a and b given by Proposition 1.12, theaxes of H are directed by the eigenvectors u and u of U Φ , and, its equation in ( ~H, u , u ) is ν b − ν a = 1; if a > b (resp. a < b ), its asymptotes are timelike (resp. spacelike) lines, and moreover, thehyperbola contains timelike and spacelike points (resp. contains only spacelike points) if a, b > ,and contains only timelike points (resp. contains timelike and spacelike points) if a, b < ;3- if U Φ is not diagonalizable, we have two cases which correspond to the cases in Proposition1.13:a- if U is timelike (in S ), the equation of H is − | ~H | − K ) K N ν ∓ q K N − ( | ~H | − K ) K N ν ν + 2( | ~H | − K ) K N ν = 1;11 ne of the asymptotes is timelike and the other one is spacelike, and the hyperbola contains timelikeand spacelike points;b- if U is lightlike (in S ), the equation of H is − aa ν + 2 εa ν ν + 1 + aa ν = 1 or − aa ν + 2 εa ν ν − − aa ν = 1 where a = | ~H | − K and ε = ± . If H is given by the first equation (resp. the second equation), it hasa lightlike asymptote, which is the line ν = − εν (resp. the line ν = εν ); its other asymptote istimelike (resp. spacelike) if a > , and is spacelike (resp. timelike) if a < moreover, H containstimelike and spacelike points (resp. only spacelike points, or only timelike points). We omit the proof, which is quite long and elementary.
Remark . If K N = 0 , the function U Φ is invertible, and we may define Φ ∗ ( ν ) := h ν, U − ν i . Itturns out that the function Φ ∗ : R , → R then furnishes an intrinsic equation of the curvaturehyperbola: for all ν ∈ R , , ~H + ν ∈ H if and only if Φ ∗ ( ν ) = 1 . This gives an efficient device to write down the equation of the curvature hyperbola in specificcases, since U Φ may be easily written in terms of the second fundamental form.We also describe the curvature hyperbola in the degenerate case ( K N = 0). Here again, forsake of brevity we omit the proofs. Proposition 2.3. If K N = 0 and Φ q = 0 , we have two possibilities:1- the image of f q is a spacelike line; in this case the hyperbola degenerates to the union of twohalf-lines:(a) if | ~H | − K = 0 , the hyperbola is (cid:26) ~H ± λ q | ~H | − K u , ≤ λ < + ∞ (cid:27) or (cid:26) ~H ± λ q K − | ~H | u , ≤ λ < + ∞ (cid:27) , depending on the sign of | ~H | − K ; (b) if | ~H | − K = 0 , the hyperbola is n ~H ± λ ( u + u ) , ≤ λ < + ∞ o or n ~H ± λ ( u − u ) , ≤ λ < + ∞ o ; this occurs when U Φ is given by (17) with ε = −
2- the image of f q is a timelike line; in that case the hyperbola degenerates to a straight line:(a) if | ~H | − K = 0 , the hyperbola is ~H + R u or ~H + R u , where the first case occurs if | ~H | − K > and the second case if | ~H | − K < (b) if | ~H | − K = 0 , the hyperbola is ~H + R ( u − u ) or ~H + R ( u + u );12 his occurs when U Φ is given by (17) with ε = 1 . For both cases 1 and 2, the case ( a ) corresponds to U Φ diagonalizable and the case ( b ) to U Φ nondiagonalizable, and the basis ( u , u ) is given by Proposition 1.12 and Proposition 1.13 respectively.We moreover note that ∆ ≥ in the case 1, and that ∆ ≤ in the case 2. Proposition 2.4. If K N = 0 and Φ q = 0 , we consider two cases:1. f q = 0; in that case the hyperbola degenerates to a straight line with one point removed n ~H + λµ q , λ ∈ R \{ } o , where µ q is the distinguished vector defined in Lemma 1.6; in that case ∆ = 0 , and q isquasi-umbilic;2. f q = 0; the hyperbola then degenerates to the end point of the vector ~H ; q is umbilic. In the figure below, the hyperbolas ( a ) and ( b ) correspond to the first and to the second casein Proposition 2.3 respectively, and the hyperbola ( c ) to the first case in Proposition 2.4. O q (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) ❞ rr C O q (a) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ✉ C O q (b) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) q❡ C (c) R , M be a Lorentzian surface immersed in R , . We will assume that M is oriented in spaceand in time: the tangent and the normal bundles T M and
N M are oriented, and for all p ∈ M, a component of { X ∈ T p M, g ( X, X ) < } and a component of { X ∈ N p M, g ( X, X ) < } aredistinguished; a vector (tangent or normal to M ) belonging to such a component will be called future-directed . We will moreover adopt the following convention: a basis ( u, v ) of T p M or N p M will be said positively oriented (in space and in time) if it has the orientation of T p M or N p M and if g ( u, u ) < g ( v, v ) > u future-directed. The second fundamental form II : T p M → N p M at each point p ∈ M is a quadratic map between two (oriented) Lorentzian planes: such a quadraticmap naturally defines an element of SO (1 , \ Q ( R , , R , ) /SO (1 , , given by its representationin positively oriented and orthonormal frames of T p M and N p M ; the numerical invariants and thecurvature hyperbola introduced in the previous sections are thus naturally attached to the secondfundamental form II.
Let us consider Λ R , , the vector space of bivectors of R , , endowed with its natural metric h ., . i , which has signature (2 , R , identifies with the submanifold of unit and simple bivectors Q = { η ∈ Λ R , : h η, η i = − , η ∧ η = 0 } , G : M → Q , p G ( p ) = u ∧ u , where ( u , u ) is a positively oriented and orthonormal basis of T p M (we recall that u is timelikeand u is spacelike). We also consider the Lie bracket[ ., . ] : Λ R , × Λ R , → Λ R , . Its restriction to the submanifold Q is a 2-form with values in Λ R , . It appears that its pull-backby the Gauss map gives the Gauss and the normal curvatures of the surface:
Proposition 3.1. If ∇ denotes the Levi-Civita connection on T M and ∇ ′ the normal connectionon N M, we have G ∗ [ ., . ] = R ∇⊕∇ ′ , (23) where R ∇⊕∇ ′ is the curvature tensor of the connection ∇ ⊕ ∇ ′ on T M ⊕ N M, considered as a2-form on M with values in Λ T M ⊕ Λ N M ⊂ M × Λ R , . We refer to [1] for a much more general result, in the Riemannian setting, where the bracket [ ., . ]is interpreted as the curvature tensor of the tautological bundles on the Grassmannian. Althoughsuch an interpretation should be also possible here (and explain the result), we give a more directproof.
Proof.
We assume that ( e , e ) is a local frame of T M in a neighborhood U of p ∈ M such that | e | = − , | e | = 1 on U and ∇ e = ∇ e = 0 at p, and, since G = e ∧ e , we readily get dG ( e ) = II ( e , e ) ∧ e + e ∧ II ( e , e )and dG ( e ) = II ( e , e ) ∧ e + e ∧ II ( e , e ) . For the computation, it is convenient to consider Λ R , as a subset of the Clifford algebra Cl (2 ,
2) :the bracket [ ., . ] is then simply given by[ η, η ′ ] = 12 ( η ′ · η − η · η ′ )for all η, η ′ ∈ Λ R , , where the dot ” · ” stands for the Clifford product; see [9] for the basicproperties of the Clifford algebras. We then compute[ dG ( e ) , dG ( e )] = 12 ( dG ( e ) · dG ( e ) − dG ( e ) · dG ( e ))with dG ( e ) = II ( e , e ) · e + e · II ( e , e )and dG ( e ) = II ( e , e ) · e + e · II ( e , e ) , and easily get [ dG ( e ) , dG ( e )] = A + B A = (cid:0) −h II ( e , e ) , II ( e , e ) i + | II ( e , e ) | (cid:1) e · e = K e · e and B = 12 {− II ( e , e ) · II ( e , e ) + II ( e , e ) · II ( e , e )+ II ( e , e ) · II ( e , e ) − II ( e , e ) · II ( e , e ) } = K N e · e where ( e , e ) is a positively oriented and orthonormal frame of N p M ( | e | = −| e | = − e · e = e · e = 1 and e · e = e · e = − , together with the formulas K = xy − z − uv + w and K N = − w ( x + y ) + z ( u + v )if the second fundamental form is given by II = (cid:18) x zz y (cid:19) e + (cid:18) u ww v (cid:19) e in ( e , e ) . See [3] for details, where a similar computation is carried out. Thus G ∗ [ ., . ] = ( K e ∧ e + K N e ∧ e ) ω M , (24)which is equivalent to (23). Corollary 3.2.
Let us consider the 2-forms ω T and ω N defined on Q by ω T p ( η, η ′ ) := −h p, [ η, η ′ ] i and ω N p ( η, η ′ ) := −h∗ p, [ η, η ′ ] i (25) for all p ∈ Q , η, η ′ ∈ T p Q . Then G ∗ ω T = K ω M and G ∗ ω N = K N ω M , (26) where ω M is the area form of M. In the statement of the corollary and below, h ., . i denotes the natural scalar product on Λ R , and ∗ : Λ R , → Λ R , is the Hodge operator, i.e. the symmetric operator of Λ R , such that η ∧ η ′ = h η, ∗ η ′ i e ∧ e ∧ e ∧ e for all η, η ′ , where e ∧ e ∧ e ∧ e is the canonical volume element. Proof.
By definition, we have [ ., . ] p = ω T p + ω N ( ∗ p )for all p ∈ Q , and the result readily follows from (24).We deduce an extrinsic proof of the following well-known results:15 orollary 3.3. Assume that M is a compact Lorentzian surface immersed in R , , such that T M and
N M are oriented (in space and in time). Then Z M K ω M = 0 and Z M K N ω M = 0 , where ω M is the area form of M. Proof.
Since ∗ = id Λ R , and similarly to the Euclidian case, we have the splittingΛ R , = Λ + R , ⊕ Λ − R , , where Λ + R , and Λ − R , are the eigenspaces of ∗ associated to the eigenvalues +1 and − Q = H × H , (27)where H and H are the hyperboloids H = { η ∈ Λ + R , : h η, η i = − / } and H = { η ∈ Λ − R , : h η, η i = − / } . Let us write G = ( g , g ) in the decomposition (27). We have G ∗ ω T = 12 ( g ∗ ω + g ∗ ω ) and G ∗ ω N = 12 ( g ∗ ω − g ∗ ω ) , where ω and ω are the 2-forms on H and H such that[ X, Y ] = ω p ( X , Y ) p + ω p ( X , Y ) p for all X = X + X and Y = Y + Y ∈ T p Q ≃ T p H ⊕ T p H ( ω and ω are in fact thenatural area forms on H and H ). Now, since H and H are not bounded, we necessarily havedeg g = deg g = 0 and Z M g ∗ ω = Z M g ∗ ω = 0;thus Z M G ∗ ω T = Z M G ∗ ω N = 0 , and (26) implies the result.We finish this section with an interpretation using the Gauss map of the vector µ II and of thenew invariants α and β defined at a quasi-umbilic point of a Lorentzian surface M, i.e. at a point p where the second fundamental form is quasi-umbilic (Definition 1.8). First, for all unit vector u belonging to T p M, if u ⊥ is a vector such that u, u ⊥ is a positively oriented Lorentzian basis of T p M, then dG ( u ) = − ~H ∧ u ⊥ + II ( u, u ) ∧ u ⊥ + u ∧ II ( u, u ⊥ ) (28)where the traceless second fundamental form II is given by II ( e , e ) = ± µ II , II ( e , e ) = ± µ II and II ( e , e ) = 12 µ II (29)(Lemma 1.6). We interpret each term in (28) as an infinitesimal rotation of the tangent plane inthe direction u : the first term − ~H ∧ u ⊥ represents a mean infinitesimal rotation of the tangent16lane (the mean is with respect to the tangent directions) in the hyperplane T p M ⊕ ~H, around thetangent direction u ⊥ and with velocity ~H, whereas the term II ( u, u ) ∧ u ⊥ (resp. u ∧ II ( u, u ⊥ ))represents an infinitesimal rotation of the tangent plane in the hyperplane T p M ⊕ R II ( u, u ) (resp. T p M ⊕ R II ( u, u ⊥ )) around the tangent direction u ⊥ (resp. u ), with velocity II ( u, u ) (resp. II ( u, u ⊥ )). Using (29) we may easily get II ( u, u ) = − II ( u, u ⊥ ) = − µ II h u, N i or II ( u, u ) = II ( u, u ⊥ ) = µ II h u, N i (30)depending the sign in (29), where N and N are the null tangent vectors √ ( e + e ) and √ ( e − e ) . In fact the formulas (30) characterize a quasi-umbilic point: the two infinitesimal rotations II ( u, u ) ∧ u ⊥ and u ∧ II ( u, u ⊥ ) take place in the same hyperplane T p M ⊕ R µ II , with the samevelocities, proportional to the squared of the projection of the direction u onto one of the two nulllines of T p M. Finally the invariants α and β determine the mean infinitesimal rotation once thevector µ II is known. R , R , by meansof its Gauss map, give an intrinsic equation for the asymptotic lines on a Lorentzian surface,discuss their causal characters and show that the asymptotic directions correspond to directionsof degeneracy of natural height functions defined on the surface. We then introduce the meandirectionally curved directions on a Lorentzian surface in R , and mention some of their relationswith the asymptotic directions. We finally study the asymptotic directions of Lorentzian surfacesin Anti de Sitter space. We still assume that M is an oriented Lorentzian surface in R , and denote by G : M → Q itsGauss map. Let us consider the quadratic map δ : T p M → Λ R , , δ ( v ) = 12 dG ( v ) ∧ dG ( v ) , where Λ R , is the space of 4-vectors of R , . Since Λ R , naturally identifies to R (using thecanonical volume element e ∧ e ∧ e ∧ e ), δ may also be considered as a quadratic form on T p M. Definition 4.1.
A non-zero vector v ∈ T p M defines an asymptotic direction at p if δ ( v ) = 0 .Remark . If v, v ′ ∈ T p M are such that G ( p ) = v ∧ v ′ , then dG = II ( v, . ) ∧ v ′ + v ∧ II ( v ′ , . )and δ ( v ) = v ∧ v ′ ∧ II ( v, v ′ ) ∧ II ( v, v ) . (31)Thus v is an asymptotic direction if and only if II ( v, v ′ ) and II ( v, v ) are linearly dependent.We analyze in detail the case where rank f II = 2 and the signature of Φ II is (1 , | ~H | − K ) − K N > | ~H | − K > . (32)Let us first describe the second fundamental form in the basis of eigenvectors ( u , u ) of U φ givenby Proposition 1 .
12. The second hypothesis in (32) implies that the eigenvalues a and b are17ositive, and we set b = √ b . The normal vector ν = b u is such that Φ II ( ν ) = 1. Moreover,straightforward computations yield ˜Φ II ( ν , u ) = 0 and ˜Φ II ( ν , u ) = b and thus˜Φ II ( ν , ν ) = b h ν, u i for all ν ∈ N p M. On the other hand, A II ( ν , u ) = 0 and Equation (1) yields A II ( ν , u ) = a ;thus A II ( ν , ν ) = a h ν, u i where a = √ a or −√ a. Therefore, in some positively oriented andorthonormal basis ( e , e ) of T p M,S u = − αI ± A II ( ν , u ) E = (cid:18) − α ∓ a ± a − α (cid:19) (33)and S u = βI ± ˜Φ II ( ν , u ) E = (cid:18) β ± b β ∓ b (cid:19) (34)(recall the normal form of S ν in the table Section 1.2), and we get II = (cid:20)(cid:18) − α α (cid:19) ∓ (cid:18) aa (cid:19)(cid:21) u + (cid:20)(cid:18) − β β (cid:19) ∓ (cid:18) b b (cid:19)(cid:21) u (35)(keeping in mind the relations h II ( X ) , u i i = h S u i ( X ) , X i , i = 1 , , with | u | = −| u | = − a , b , α and β : we have | ~H | = − α + β , K = − α + β − a − b , (36)∆ = − a β + a b + α b and K N = 2 ab . Further, since dG ( e ) = II ( e , e ) ∧ e + e ∧ II ( e , e )and dG ( e ) = II ( e , e ) ∧ e + e ∧ II ( e , e ) , we easily get δ ( e , e ) = ± a ( β ± b ) , δ ( e , e ) = ± a ( β ∓ b ) and δ ( e , e ) = ∓ α b . Thus, if v = xe + ye , δ ( v ) = ± a ( β ± b ) x ± a ( β ∓ b ) y ∓ α b xy, (37)which proves the following: Proposition 4.3.
Assuming that (32) holds, then, in a positively oriented and orthonormal basis ( e , e ) of T p M such that II ( e ) = II ( e ) = ∓ b u and II ( e , e ) = ∓ a u (38) where II is the traceless second fundamental form, the equation of the asymptotic directions is a ( β ± b ) x + a ( β ∓ b ) y − α b xy = 0 , (39) where a , b , α and β are numerical invariants satisfying (36). We will give applications of this intrinsic equation below.18 emark . The conditions in (38) have the following simple interpretation in terms of the cur-vature hyperbola: the vectors e and e appear to be the preimages by the map v II ( v ) / | v | ofthe points of the hyperbola belonging to the spacelike axis.We now discuss the causal character of the asymptotic directions. We consider δ o := δ −
12 tr g δ g, (40)the traceless part of the quadratic form δ. Using (37) and the relations (36), we easily gettr g δ = − K N and disc ( δ ) := − det g δ = − ∆ , (41)and also disc ( δ o ) := − det g δ o = 14 K N − ∆ . Contrasting with the cases of Riemannian and Lorentzian surfaces in 4-dimensional Minkowskispace R , [4, 5], the existence of asymptotic lines at a point on the surface is equivalent here tothe condition ∆ ≥ δ ( u ) = 0 if and only if δ o ( u ) = 12 K N | u | . (42)The causal character of the asymptotic directions appears to depend on the signs of the forms δ , δ o and their discriminants. The results are similar to the case of the Lorentzian surfaces in4-dimensional Minkowski space [5, p. 1708], and we only briefly describe them below. There aretwo main cases, depending on disc ( δ ). Let us analyze only the case when disc ( δ ) < , that is,when two distinct asymptotic directions are defined. We then divide the discussion in four cases,according to the sign of δ o . First case: disc ( δ o ) > : if δ o is positive (resp. negative), the solutions u of (42) are necessarilyspacelike (resp. timelike) if K N > , and timelike (resp. spacelike) if K N < . Second case: disc ( δ o ) < : let us denote by u δ o the traceless symmetric operator of T p M associated to δ o ; we then have | u δ o | = − det( u δ o ) < e , e ) of T p M, the matrix of u δ o reads M ( u δ o , ( e , e )) = ± p −| u δ o | (cid:18) − (cid:19) ;see Remark 1.4. Writing u = xe + ye , (42) then reads δ ( u ) = 0 if and only if ± p −| u δ o | xy = K N ( x − y ) . (43)Thus, if u = xe + ye is a non trivial solution of δ ( u ) = 0, so is u := − ye + xe . Observe thatthese solutions are necessarily spacelike or timelike, and that if one of them is spacelike, the otherone 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 | u δ o | = − det( u δ o ) = 0 , and the kernel of u δ o is a null line in T p M ; there is thus a unique lightlike line of solutions for the equations in (42). Theother independent solution is thus a timelike or a spacelike line. But using (42) again, if δ o ≥ δ o ≤
0) this solution is necessarily spacelike (resp. timelike) if K N > K N < . Fourth case: δ o = 0 : then δ ( u ) = − K N | u | , and δ ( u ) = 0 ⇐⇒ | u | = 0 . Note that ~H = 0 inthat case: since K N = 0 the point is not quasi-umbilic, and, by (37), δ o ( v ) = ± a β ( x + y ) ∓ α b xy. Since δ o = 0 and K N = 2 ab = 0 , we get α = β = 0 , i.e. ~H = 0 .
19e describe the causal character of the asymptotic directions in the following table; in the firstcolumn appear the different possible values for the signature of δ o . To simplify the presentation wesuppose that K N ≥
0; if K N ≤ , we just have to systematically exchange the words “spacelike”and “timelike” in the table.signature of δ o disc ( δ ) < disc ( δ ) = 0 , δ = 0a double asymptoticdirection which is(2,0) spacelike spacelike(0,2) timelike timelike(1,1) 1 spacelike - 1 timelike Not possible(1,0) 1 lightlike - 1 spacelike lightlike(0,1) 1 lightlike - 1 timelike lightlike(0,0) lightlikewith ~H = 0 Not possibleWe finish this section with a characterization of a quasi-umbilic point of a Lorentzian surfacein terms of its asymptotic directions. This characterization is very similar to a result given in [5];since the proof is also very similar, we only state the result, and refer to [5] for details: Theorem 4.5.
Assume that p ∈ M is such that δ = 0 . Then p is a quasi-umbilic point if and onlyif there is a double lightlike asymptotic direction at p . Let us define the family of height functions on a Lorentzian surface M in R , as H : M × R , → R , H ( p, ν ) = h p, ν i + c, where c ∈ R . The function h ν : M → R defined as h ν = H ( · , ν ) is singular at p ∈ M , that is dh ν p = 0, if and only if ν is normal to M at p . Consider also Hess h ν := ∇ dh ν , the Hessian of h ν , where ∇ is here the Levi-Civita connection of M acting on the 1-forms. Wereadily get that Hess h ν = II ν . (44)We say that a non-zero normal vector ν at p is a binormal vector if the quadratic form Hess h ν is degenerate at p , and that a non-zero vector v ∈ T p M defines a contact direction if it belongs tothe kernel of Hess h ν at p. Thus, by definition, v is a contact direction with associated binormalvector ν if and only if the contact at p between the surface and the hyperplane ν ⊥ is of order ≥ v. We now prove that v ∈ T p M is a contact direction if and only if it is an asymptotic direction.By (44), we readily get the following result: Lemma 4.6.
A non-zero vector v ∈ T p M defines a contact direction if and only if S ν ( v ) = 0 forsome non-zero vector ν normal to M at p , where S ν is the symmetric operator associated to theform II ν . Observe that the normal vector ν given by the lemma is a binormal vector with associatedcontact direction v . 20 roposition 4.7. A vector v ∈ T p M defines a contact direction if and only if it defines anasymptotic direction.Proof. Recalling (31), δ ( v ) = 0 if and only if II ( v, v ′ ) and II ( v, v ) are linearly dependent, that is,if and only if the linear map II ( v, · ) : T p M → N p M has a non trivial kernel; this is equivalentto the existence of a non trivial vector ν ∈ N p M normal to the image of this map, i.e. such that h II ( v, · ) , ν i = 0 . Since h II ( v, w ) , ν i = h S ν ( v ) , w i for all w ∈ T p M, we conclude that v defines an asymptotic direction if and only if S ν ( v ) = 0 forsome non-zero normal vector ν. By Lemma 4.6 this also characterizes a contact direction.
Remark . The notion of contact direction has been used before in different settings, see forinstance [8], [5]. It is usually used to define the notion of asymptotic direction. In this paper, werather defined the asymptotic directions by means of the Gauss map, and finally proved that thetwo notions coincide.
Now let us analyze the mean directionally curved field of directions , studied for surfaces im-mersed in R in [11] and [14], and for timelike surfaces in Minkowski space R , in [5]. In R , thesedirections are defined as the pull-back by the second fundamental form of the intersection pointsin the normal plane of the curvature hyperbola with the line generated by the mean curvaturevector. More precisely, the condition is [ ~H, II ( v )] = 0 , (45)where the brackets stand for the determinant of the vectors in a positively oriented and orthonormalbasis of the normal plane. This is also [ ~H, II ( v )] = 0 , where II is the traceless part of the secondfundamental form. For sake of simplicity, here again we assume that( | ~H | − K ) − K N > | ~H | − K > e , e ) of T p M (seeProposition 4.3 and Remark 4.4 above), the second fundamental form is given by (35), and (45)reads [ ~H, a (2 xy ) u + b ( x + y ) u ] = 0 . Thus, we obtain the following intrinsic equation of these directions:
Proposition 4.9. In ( e , e ) , the equation of the mean directionally curved directions in terms ofthe invariants a , b , α and β is α b ( x + y ) − a βxy = 0 . (46)Moreover, using this equation and the expression (37) of δ, we deduce the following: Lemma 4.10.
Equation (45) is equivalent to δ ( v, v ∗ ) = 0 , (47) where v = xe + ye and v ∗ = ye + xe in a positively oriented and orthonormal basis ( e , e ) of T p M . orollary 4.11. Under the hypotheses above, the mean directionally curved directions bisect theasymptotic directions.Proof.
Let v, v ∗ be the mean directionally curved directions. Assuming moreover that | v | = −| v ∗ | = ± , the vectors v and v ∗ form a Lorentzian basis of T p M ; then, a unit direction v ψ := cosh ψ v + sinh ψ v ∗ , ψ ∈ R is an asymptotic direction if and only if δ ( v ψ ) = cosh ψ δ ( v ) + sinh ψ δ ( v ∗ ) = 0(Definition 4.1 and Equation (47)). Thus v ψ is an asymptotic direction if and only if so is v − ψ , which gives the result. Let us apply these results to the analysis of the Lorentzian surfaces immersed in the
Anti deSitter 3-space . This space is defined by H = { x ∈ R , : h x, x i = − } . It is the 3-dimensional Lorentzian space form with negative curvature. The geometry of Lorentziansurfaces in this space has been studied with an approach of Singularities by analyzing the contactsof the surfaces with some models [6]. Following [2], we consider the φ − de Sitter height function defined on a Lorentzian surface M in H by H φ : M × S (sin φ ) → R , H φ ( p, ν ) = h p, ν i + cos φ, φ ∈ [0 , π/ , where S (sin φ ) := { x ∈ R , : h x, x i = sin φ } is the pseudo sphere with index 2 centeredat the origin and with radius sin φ if φ = 0; if φ = 0 this set is the null cone at the origin { x ∈ R , : h x, x i = 0 } .Let ϕ : U → H be an immersion of an open set U ⊂ R with coordinates u = ( u , u ), whoseimage M = ϕ ( U ) is a Lorentzian surface. The vector field N ( u ) = ϕ ( u ) ∧ ϕ u ( u ) ∧ ϕ u ( u ) | ϕ ( u ) ∧ ϕ u ( u ) ∧ ϕ u ( u ) | is by definition unitary, normal to M and tangent to H , i.e. is such that ( ϕ u , ϕ u , ϕ, N ) is a frameon M whose first two vectors generate the tangent bundle and the last two vectors the normalbundle of M in R , . The φ ± - de Sitter duals of M are defined as N φ ± : U → S (sin φ ) , N φ ± ( u ) = cos φ ϕ ( u ) ± N ( u ) . Since N φ + ( u ) = − N φ + φ/ − ( u ) we only consider φ ∈ I = [0 , π/ H φ ) φ ∈ I is a generating family of a natural Legendrian embeddingof the surface into a contact manifold ∆ ± whose structure is similar to that defined in [2]. Moreover,the image of the φ ± - de Sitter dual is the wave front set of this Legendrian map . Furthermore, thefields N φ ± are normal to M and the φ ± -Gauss-Kronecker curvature at each point of M is definedas the determinant of the linear operator − dN φ ± : T p M → T p M . A point p where this curvature22anishes is called a N φ ± -parabolic point; such a point is characterized as a point where the normal N φ ± is a binormal vector.The results proved at the beginning of the section imply a rigidity property of the contactdirections associated to the different binormal de Sitter duals N φ ± parameterized by φ . Indeed, if N φ + is a family of binormal vectors at p parameterized by φ ∈ I, then there is a family of contactdirections v φ associated to the corresponding family of height functions. If the discriminant of theform δ satisfies ∆ p = 0, there is only one asymptotic direction, and Proposition 4 . v φ coincides with it for any φ . If ∆ p >
0, there are two asymptotic lines at p, l and l say. Weassert that l (or l ) is a contact direction of one of the two families of binormals ( N φ + ) φ ∈ [0 , π ] or( N φ − ) φ ∈ [0 , π ] , that is, is a contact direction for the binormals N φ + for all φ ∈ [0 , π ], or is a contactdirection for the binormals N φ − for all φ ∈ [0 , π ]. Indeed, let v be the contact direction of theheight function defined by the binormal N φ + for some φ ∈ I, and let ( φ k ) k ∈ N be a real sequenceconverging to φ ; we can choose, associated to each shape operator of the sequence ( S φ k ) k ∈ N definedby the binormals ( N φ k + ) k ∈ N , an eigenvector v k corresponding to its null eigenvalue and such thatthe sequence v k converges to v . The possibility of such a continuous choice implies the following:
Proposition 4.12.
Let p be a N φ + -parabolic point on M for all φ ∈ I . Then, the contact directionsof the height functions defined by the binormal vectors N φ + , parameterized by φ ∈ I at p coincide. R , Quasi-umbilic (Lorentzian) surfaces in 3 and 4-dimensional Minkowski space were described in[7] and [5] respectively. We are interested here in quasi-umbilic surfaces in R , : similarly to [5],we will say that a Lorentzian surface M in R , is quasi-umbilic if its second fundamental formis quasi-umbilic at every point of M, which means that the curvature hyperbola degenerates to astraight line with one point removed at every point of M , or equivalently that | ~H | = K and K N = ∆ = 0 (48)together with Φ II = 0 and II = ~Hg (49)on M ; see Proposition 2.4 - above. Similarly to [5, Theorem 5.1], the quasi-umbilic surfaces in R , are described as follows: Theorem 5.1.
A Lorentzian surface M in R , is umbilic or quasi-umbilic if and only if it isparameterized by ψ ( s, t ) = γ ( s ) + tT ( s ) (50) where γ is a lightlike curve in R , and T is some lightlike vector field along γ such that γ ′ ( s ) and T ( s ) are independent for all value of s. This result generalizes the main result of [7] to the space R , . We omit the proof since it isidentical to the proof of Theorem 5.1 in [5] (note that a lemma similar to the key lemma [5, Lemma5.2] is also valid here).
Remark . To our knowledge, the natural problem of the description of the Lorentzian surfacesin R , which are umbilic at every point is an open question (note that Lorentzian umbilic surfacesin 4-dimensional Minkowski space are well-known, see e.g. [10]).23 emark . It may occur that (48) holds, but with Φ II = 0 (Proposition 2.3, -( b ) or -( b ) , with∆ = 0). In that case, we have in fact | ~H | = K = K N = ∆ = 0: indeed, we are in the context ofProposition 1.13 2., with M ( U Φ , ( u , u )) = (cid:18) ǫ ǫ − ǫ − ǫ (cid:19) , ǫ = ± , ǫ = ± u , u ) is the positively oriented and orthonormal basis of N M given by the proposition); moreprecisely, writing ~H := αu + βu the second fundamental form is in fact given by1. ǫ = − II = (cid:18) − α ± α ± (cid:19) u + (cid:18) − β ± ǫ β ± ǫ (cid:19) u , which implies that∆ = ( α − ǫ β ) ≥ , ǫ = 1 : II = (cid:18) − α ∓ ∓ α (cid:19) u + (cid:18) − β ± ǫ ± ǫ β (cid:19) u , which implies that∆ = − ( α + ǫ β ) ≤ , we deduce that α = ± β i.e. | ~H | = 0 . If the Gauss mapof the surface is regular, the surface belongs in fact to a degenerate hyperplane (see Theorem 5.4below). | ~H | = K = K N = ∆ = 0 We describe here the Lorentzian surfaces in R , whose classical invariants are zero. We willsay that an hyperplane of R , is degenerate if the metric of R , induces on it a degenerate metric.We state the main result of the section: Theorem 5.4.
Let M be an oriented Lorentzian surface in R , with regular Gauss map and suchthat K = K N = ∆ = 0 . Then1. if Φ II = 0 , M belongs to a degenerate hyperplane;2. if Φ II ≡ , M is a flat umbilic or quasi-umbilic surface.In both cases, we have in fact | ~H | = K = K N = ∆ = 0 . Conversely, if M belongs to a degenerate hyperplane or is a flat umbilic or quasi-umbilic surfacethen | ~H | = K = K N = ∆ = 0 . Proof.
We assume that M satisfies the hypotheses of the theorem, and that Φ II = 0 (if Φ II ≡ ,M is umbilic or quasi-umbilic by the very definition). The quadratic form Φ II is degenerate since K N = 0. We first prove by contradiction that U Φ is not diagonalizable; we assume that it isdiagonalizable, and consider two cases: Φ has signature (1,0): U Φ is then given by (9) with a = | ~H | , b = 0 if | ~H | < , or a = 0 , b = | ~H | if | ~H | > U = 0 since Φ II is degenerate and not zero); β = 0 i.e. ~H = αu in the first case, and α = 0 i.e. ~H = βu in the second case (formulas (12)-(13)). Thecurvature hyperbolas are given by Proposition 2.3 1- (a), and, in each case, the vector 0 ∈ N M appears to be an extremal point of the (degenerate) hyperbola: if u ∈ T M, | u | = ± I ( u, u ) = 0 , we thus also have II ( u, v ) = 0 for all v ∈ T M, that is dG ( u ) = 0 , a contradictionwith the hypothesis that G is regular. Φ has signature (0,1): we then have a = | ~H | , b = 0 if | ~H | > , or a = 0 , b = | ~H | if | ~H | < α = −| ~H | , β = 0 in the first case,and α = 0 , β = | ~H | in the second case; this is not possible since α and β are necessarilynon-negative.Thus U Φ is not diagonalizable, and ~H is zero or lightlike (by conditions (14) in Proposition1.13). Recalling the normal forms in the table Section 1.2, we have S ν = h ~H, ν i (cid:18) (cid:19) ± ˜Φ II ( ν , ν ) E i in some positive oriented basis e , e of T M, where ν is a vector belonging to N M and E i = E or E , that is II = ~H (cid:18) − (cid:19) ± U Φ ( ν ) ˜ E i , (51)where ˜ E = (cid:18) − − (cid:19) and ˜ E = (cid:18) − − (cid:19) . Thus II ( e , e ) = − ~H − εU Φ ( ν ) , II ( e , e ) = ~H − εU Φ ( ν ) and II ( e , e ) = 0in the first case, and II ( e , e ) = − ~H, II ( e , e ) = ~H and II ( e , e ) = − εU Φ ( ν )in the second case, where ε = ±
1. Using that dG = II ( e , . ) ∧ e + e ∧ II ( e , . ) , we then computethe matrix of δ := dG ∧ dG in e , e : it is of the form (cid:18) cc (cid:19) in the first case and (cid:18) c c (cid:19) in the second case. Recalling (41), we have det g δ = ∆ = 0 , from which we get c = 0 , that is δ = 0in both cases: since G is moreover assumed to be regular, the surface necessarily belongs to ahyperplane (see [12, Theorem 1.3], in the Euclidian context). This hyperplane is degenerate: thisis clear if ~H = 0 since ~H is then a non-zero lightlike vector, normal to the surface and belongingto the hyperplane; if now ~H = 0 , then (51) reads II = ± U Φ ( ν ) ˜ E i , which gives K = −h II ( e , e ) , II ( e , e ) i + | II ( e , e ) | = ±| U Φ ( ν ) | , and, since Φ II = 0 and K = 0 , the vector U Φ ( ν ) is non-zero, lightlike, normal to the surface andnecessarily belongs to the hyperplane since it is in the range of II ; the hyperplane is thus alsodegenerate in that case.The converse statement readily follows from (48). Remark . According to Theorem 5.4 and the previous sections, the numerical invariants of aLorentzian surface in R , with regular Gauss map and whose classical invariants | ~H | , K, K N and∆ all vanish are the invariants given by (20) if Φ II = 0 and the invariants given by (22) if Φ II = 0and f II = 0 (the quasi-umbilic case); there is no invariant if f II = 0 (the umbilic case). Remark . It is straightforward to check that the quasi-umbilic surface ψ ( s, t ) = γ ( s ) + tT ( s )with γ ( s ) = ( a ( s ) , − a ( s ) , b ( s ) , − b ( s ))25nd T ( s ) = ( f ( s ) , f ( s ) , g ( s ) , g ( s )) , where a, b, f and g are real functions of the variable s such that a ′ f + b ′ g = 0 , f ′ g − g ′ f = 0and b ′′ a ′ − a ′′ b ′ = 0 , (52)is such that | ~H | = K = K N = ∆ = 0 , has regular Gauss map and does not belong to anyhyperplane. If we assume that( b ′′ a ′ − a ′′ b ′ )( s ) = 0 and ( b ′′ a ′ − a ′′ b ′ )( s ) = 0 for s = s instead of (52), we obtain a surface such that | ~H | = K = K N = ∆ = 0 and with regular Gaussmap, which is umbilic at ψ ( s ,
0) and quasi-umbilic at ψ ( s, t ) , s = s . Acknowledgement:
Sections 1, 2, 3 and 4.1 of this paper is part of V. Patty’s PhD thesis; hethanks CONACYT for support. The third author was partially supported by PAPIIT-DGAPA-UNAM grant No. 117714.
References [1] H. Anciaux, P. Bayard,
On the normal affine Gauss map of a submanifold in R m , in prepa-ration.[2] M. Asayama, S. Izumiya, A. Tamaoki and H. Yildrim, Slant geometry of spacelike hypersur-faces in hyperbolic space and de Sitter space , Rev. Math. Iberoam. 28:2 (2012) 371-400.[3] P. Bayard, M.-A. Lawn, J. Roth,
Spinorial representation of surfaces into 4-dimensional spaceforms , Global Analysis and Geometry 44:4 (2013) 433-453.[4] P. Bayard, F. S´anchez-Bringas,
Geometric invariants and principal configurations on spacelikesurfaces immersed in R , , Proc. Roy. Soc. Edinburgh Sect. A 140:6 (2010) 1141-1160.[5] P. Bayard, F. S´anchez-Bringas,
Invariants and quasi-umbilicity of timelike surfaces inMinkowski space R , , J. Geom. Phys. 62:7 (2012) 1697-1713.[6] L. Chen and S. Izumiya,
Singularities of Anti deSitter torus Gauss maps , Bull. Braz. Math.Soc., New Series 41:1 (2010) 37-61.[7] J. Clelland,
Totally quasi-umbilic timelike surfaces in R , , Asian J. Math. 16:2 (2012) 189-208.[8] S. Costa, S. Moraes and M. C. Romero Fuster,
Curvature ellipses and geometric contacts ofSurfaces Immersed in IR n , n ≥
5, Differential Geom. Appl. 27:3 (2009) 442-454.[9] Th. Friedrich,
Dirac Operators in Riemannian Geometry , Graduate Studies in Mathematics25 (2000) AMS.[10] S.K. Hong,
Totally umbilic Lorentzian surfaces embedded in L n , Bull. Korean Math. Soc. 34:1(1997) 9-17.[11] L. F. Mello, Mean directionally curved lines on surfaces immersed in R , Pub. Mat. 47 (2003)415-440. 2612] J. A. Little, On the singularities of submanifolds of higher dimensional Euclidean spaces ,Annali Mat. Pura et Appl., 83:4A (1969) 261-336.[13] V. Patty,
Representaci´on espinorial de superficies Lorentzianas en R , , Tesis de Doctorado,Posgrado Conjunto UNAM-UMSNH, in preparation.[14] F. Tari,