SECOND ORDER GEOMETRY OF SPACELIKE SURFACES IN DE SITTER 5-SPACE

The de Sitter space is known as a Lorentz space with positive constant curvature in the Minkowski space. A surface with a Riemannian metric is called a spacelike surface. In this work we investigate properties of the second order geometry of spacelike surfaces in de Sitter space S5 1 by using the action of GL(2,R)×SO(1, 2) on the system of conics defined by the second fundamental form. The main results are the classification of the second fundamental mapping and the description of the possible configurations of the LMN -ellipse. This ellipse gives information on the lightlike binormal directions and consequently about their associated asymptotic directions. 2010 Mathematics Subject Classification: 53A35, 53B30.


Introduction
We investigate properties of the second fundamental form, with respect to lightlike normals, of spacelike surfaces in de Sitter 5-space.In the Euclidean case, Little in [12] describes elements of the local second order geometry of surfaces, such as asymptotic directions and inflection points, in terms of invariants of a classical object: the curvature ellipse.The classification of the second fundamental form also appears as an useful device in the investigation of the singularities of the 5-web of the asymptotic lines of surfaces in R 5 , given by Romero-Fuster, Ruas, and Tari in [14].For surfaces in Minkowski space R 4  1 , Izumiya, Pei, and Romero-Fuster introduce the concept of curvature ellipse in [6] and study geometric properties of spacelike surfaces in Minkowski 4-space in terms of properties of this ellipse.In [5], they study properties of the curvature ellipse in spacelike surfaces in Minkowski (n + 1)-space, R n+1 1 , n ≥ 2 (using isothermal coordinates), and by using this setting they obtain geometric characterization for a spacelike surface M to be contained in hyperbolic n-space (de Sitter n-space or n-dimensional lightcone), n = 3, 4.This characterization is given in terms of the umbilicity of theses surfaces with respect to some normal field.Izumiya, Pei, and Romero-Fuster study spacelike surfaces in hyperbolic 4-space (see [7]).
Bayard and Sánchez-Bringas in [2], study the complete invariants of a quadratic map R 2 −→ R 2  1 , under the actions of SO(2) + × SO(1, 1) + , where SO(2) + and SO(1, 1) + are the connected components of the identity of the Euclidean and Lorentzian groups of R 2 and R 2  1 respectively.The invariants of these actions have been applied in the classification of the configurations of curvature ellipses of spacelike surfaces in Minkowski 4-space.
Spacelike submanifolds in de Sitter space have been investigated from the viewpoint of singularity theory in [1,8,9,10,11].In this paper we follow this approach to study properties of the second order geometry in the case of spacelike surfaces in de Sitter space S 5  1 .We focus on the classification of the orbits of the GL(2, R) × SO(1, 2) action on the system of conics defined by the second fundamental form.We define an ellipse in affine space that we call LMN -ellipse, that gives informations on the lightlike binormal directions and consequently about their associated asymptotic directions.The main results are the classification of the second fundamental mapping and the description of the possible configurations of the LMN -ellipse.The techniques depend on the choice of an orthonormal basis (n T 0 , n S 1 , n S 2 where n T 0 is timelike and n S 1 , n S 2 are spacelike vectors) for the normal 3-space at each point and on the existence of a 2-dimensional cone of lightlike normal vectors.These conditions also hold for spacelike surfaces in R 5  1 , therefore the results of this paper also hold in this case.
The paper is organized as follows.In Section 2, we give the basic concepts on extrinsic geometry of spacelike submanifolds in de Sitter n-space in R n+1 1 .In Section 3, we introduce some invariants (which are preserved by Lorentzian transformations) of the second order geometry of the spacelike surface in de Sitter 5-space.In Sections 4 and 5, we obtain the normal forms of the matrix α of the second fundamental form, under the action of GL(2, R) × SO (1,2), when the rank of α is 1 or 2, respectively.We also study the number of lightlike binormal and their associated asymptotic directions in each case.In Section 6, we obtain normal forms of α on 4 parameters, when rank α = 3.In Section 7, we obtain the equation of asymptotic directions at a non lightlike inflection and non conic point.This is a fourth order equation, then at each non conic and non lightlike inflection point there are at most 4 asymptotic directions associated to the lightlike binormals directions.In Section 8, when rank α = 3 we study the number of lightlike binormal directions and consequently the number of their associated asymptotic directions.We give some examples of spacelike surfaces and lightlike binormals and their associated asymptotic directions.Finally, in Appendix A we define trigonometric polynomials representing the equation of the binormal directions and discuss their solutions.

Preliminaries
In this section we review basic notions of spacelike surfaces in de Sitter space.Let R n+1 = {x = (x 0 , . . ., x n ) | x i ∈ R (i = 0, . . ., n)} be the (n + 1)-dimensional vector space.For any vectors x = (x 0 , . . ., x n ) and y = (y 0 , . . ., y n ) in R n+1 , the pseudo scalar product of x and y is defined by x, y = −x 0 y 0 + n i=1 x i y i .We call (R n+1 , , ) a Minkowski (n + 1)space and write R n+1 x is positive, negative, or equal to zero, respectively.The norm of the vector x is defined by We define de Sitter n-space by an open subset and X : U −→ R n+1 1 an embedding map.We write M = X(U ).We say that M is a spacelike surface if any tangent vector v = 0 at each point on X(U ) is always spacelike.

1
, where e 1 , e 2 generate the tangent space on M at each point and n T 0 and n S 1 , . . ., n S n−2 are respectively a timelike normal section and spacelike normal sections.
The first fundamental form on M is given by ds 2 = 2 i,j=1 g ij du i du j , where g ij = X ui , X uj .For i, j = 1, 2 and k = 0, . . ., n − 2, we denote .
We call h k ij the coefficients of the second fundamental form.The geometry of the second fundamental form of spacelike surfaces in Minkowski space has been investigated in [5].
We also denote by X : U −→ S 5 1 a parametrization of spacelike surface in de Sitter 5-space.For any point p ∈ X(U ) = M , each tangent vector v ∈ T p M \{0} is spacelike and orthogonal to the position vector p.In this case, we consider the second fundamental form in the direction ϑ ∈ S 1 as the projection of the normal curvature vector into N p M ⊂ T p S 5  1 .
Let n T 0 , n S 1 , n S 2 : U −→ N M be a timelike normal section and spacelike normal sections of M .Let S 4 0} be a sphere in the lightcone.We define a map e : U × S 1 −→ LC * 5 as e(u, θ) = n T 0 (u) + cos θn S 1 (u) + sin θn S 2 (u).
The coefficients of the second fundamental form of the embedding of the surface into S 5  1 with respect to the basis {n T 0 , n S 1 , n S 2 } are given by The second fundamental form with respect to a lightlike normal e(u, θ), II e(u,θ) : T p M −→ R, is given by X uiuj , e(u, θ) du i du j .
Let II e(u,θ) (du 1 , du 2 ) = L(u, θ) du 2 1 + 2M (u, θ) du 1 du 2 + N (u, θ) du 2 2 , then we have We call II e(u,θ) = ( L M M N ) the matrix of the second fundamental form in the direction e(u, θ).We remark that if we take the normalized lightlike normal direction ē(u, θ), the corresponding second fundamental form is expressed by II ē(u,θ) .We will use this last one but still denote by II e(u,θ) .
The discriminant of the binary differential equation (briefly, BDE) II e(u,θ) = 0 is given by The sign of the discriminant ∆ is invariant under change of parametrization of the surface.We say that a direction θ ∈ S 1 is a lightlike binormal at p = X(u) if ∆(u, θ) = 0, that is rank(II e(u,θ) ) ≤ 1.In this case we call a kernel direction of the second fundamental form II e(u,θ) by asymptotic direction associated to the lightlike binormal direction.Then for each θ such that ∆(u, θ) = 0, the associated asymptotic directions are the solutions of II e(u,θ) (du 1 , du 2 ) = 0, that is, We also say that the point p = X(u) is a lightlike inflection point if (L, M, N )(u, θ) = 0 for some θ ∈ S 1 , in this case any direction is asymptotic (rank(II e(u,θ) ) = 0).
To explain the relations of the geometry of the asymptotic directions on spacelike surface, let C = {(l, m, n) | m 2 − ln = 0} be a cone and E(u, θ) = A+cos θB+sin θC be the ellipse in the 3-space R 3 = {(l, m, n)}, where A = (a 0 , a 1 , a 2 ) t , B = (b 0 , b 1 , b 2 ) t , and C = (c 0 , c 1 , c 2 ) t .We call this ellipse as LMN -ellipse.The center A of the ellipse varies with the choice of normal frames.We say that the vector (l, m, n) is elliptic, hyperbolic, or parabolic respectively if its discriminant m 2 −ln is negative, positive, or equal to zero.Equivalently (l, m, n) is, respectively, inside, outside, or on the cone C. We remark that the determinant of the second fundamental form is LN −M 2 .Properties of lightlike binormal directions and lightlike inflection points are related to the configuration of the cone and the LMN -ellipse.We have the following characterizations.
We intend to classify the configurations of the LMN -ellipse in further sections to investigate the geometry of the asymptotic directions, associated to the lightlike binormals, on spacelike surfaces in de Sitter 5-space.
We can consider the one-parameter family of the height functions on M , H(u, θ) = X(u), e(u, θ) .If θ is a lightlike binormal direction at the point p then the height function h θ , obtained when we fix the parameter θ, has a singularity more degenerate than Morse at p. Moreover any direction in the kernel of the hessian of h θ at p is an associated asymptotic direction (see [7]).
From now on, for short, we omit the word "lightlike" in the expressions "lightlike binormal direction" and "lightlike inflection points".

Invariants associated to the second fundamental form
The aim of this section is to introduce some invariants of the second order geometry of the spacelike surface in de Sitter 5-space.First we define invariants of these surfaces, which are preserved by the Lorentzian transformations, that is, by the following actions (1) and ( 2).
(1) Change of parametrization, (2) Change of the normal vector fields, Φ : 1 } is the Lorentzian transformation group on Minkowski 3-space.As usual GL(2, R) will denote the general linear group on R 2 .Let α and k be defined as The families Φ and ψ of elements respectively in SO(1, 2) and GL(2, R) are parametrized by u.That is, for any u ∈ U , Φ := Φ(u) ∈ SO(1, 2), and ψ := ψ(u) ∈ GL(2, R).By computation, we have the following table It follows that det Ψ = (det ψ) 3 .Let (L, M, N ) = (a 0 + b 0 cos θ + c 0 sin θ, a 1 +b 1 cos θ+c 1 sin θ, a 2 +b 2 cos θ+c 2 sin θ), then we have another expression of the action (1): that we can also write by The action ( 2) is generated by the following actions where ± 1 and ± 2 are not necessarily to be the same signs.
For each u ∈ U , let I(u) be the first fundamental form of the spacelike surface X(U ), then k(u)/ det I(u) and rank α(u) are invariant under the actions ( 1) and ( 2).The maximal and minimal values of ∆(u, θ)/ det I(u) are also invariants.
The following proposition follows from the above discussion.
We now fix a point u ∈ U and concentrate to classify the configurations of LMN -ellipses.To do this we study the action of G = GL(2, R) × SO(1, 2) on the system of conics defined by α.
We may distinguish the configuration of the LMN -ellipse by the multiplicity of the solutions of the equation ∆(u, θ) = 0, 0 ≤ θ ≤ 2π.We fix u ∈ U and let ∆(u, θ) ≡ 0 and θ 1 , . . ., θ k be distinct solutions of ∆(u, θ) = 0 with multiplicities m 1 , . . ., m k , where m i is a number such that ))-type if at this point we have both, k binormal and k asymptotic directions, with same multiplicities m 1 , . . ., m k .In Section 8, we show that this always happens when the rank of α is 3.If there is no solution we say that α is 0-type then there is no asymptotic direction.We also say that a point p is of S 1 -type if ∆(u, θ) ≡ 0. We call p an inflection point of M if the LMN -ellipse goes through the origin for some θ, and in this case all directions are asymptotic.If not we say that p = X(u) is non-inflection point.We say that α is of S 1 -type if all lightlike normal directions are binormals and all tangent directions are asymptotic directions.We will see, in Section 4, that if the ellipse degenerates into a segment (or a point) outside the origin and on the cone then all the directions θ are binormals (∆(u, θ) ≡ 0) but there is only one associated asymptotic direction.
Let X(u) be a spacelike surface in de Sitter 5-space and α the matrix of coefficients of the second fundamental form at p = X(u) with respect to the lightlike normal direction e(u, θ).Following [13], we say that a point p ∈ X(U ) is of type M i , i = 1, 2, 3 if the rank of α is i.We now introduce some denominations for vectors and planes which are in the image of the second fundamental form in the cases rank α = 1 or 2.
(1) If rank α = 1, then there exists a vector x = (x 0 , x 1 , x 2 ) such that x = A, B, C , where v 1 , . . ., v n denotes the vector space generated by the vectors The sign of D1 is invariant under the action of G. Therefore we have three cases D1 > 0 (M 1 -elliptic case), D1 < 0 (M 1 -hyperbolic case), and D1 = 0 (M 1 -parabolic case).Respectively, the vector x is elliptic, hyperbolic, or parabolic.Equivalently, the vector x is inside, outside, or on the cone C.
(2) If rank α = 2, then there are two vectors x = (x 0 , x 1 , x 2 ) and y = (y 0 , y 1 , y 2 ) such that x, y = A, B, C .We write D2 by The sign of D2 is invariant under the action of G. Therefore we have the cases (see [12,4]): D2 > 0 (M 2 -elliptic case), D2 = 0 (M 2 -parabolic case), and D2 < 0 (M 2 -hyperbolic case).Equivalently, the plane x, y is elliptic, parabolic, or hyperbolic.The following geometric conditions hold: a plane by the origin is elliptic if and only if it intercepts the cone C only at the origin, it is hyperbolic if and only if it is transversal to the cone, and it is parabolic if and only if it is tangent to C (see [13]).
We assume that rank α ≥ 1. Changing coordinates if necessary, we can assume that A = (a 0 , a 1 , a 2 ) does not vanish.
Let (l, m, n) be a vector in the three space {(L, M, N )}.Let α = (A, B, C) t be the second fundamental form at u ∈ U .We say that α is A-elliptic or A-hyperbolic respectively if p is not an inflection point and the vector A is always elliptic or always hyperbolic under the action of G.We also say that α is A-parabolic if p is not an inflection point and α is neither A-elliptic nor A-hyperbolic.We define the open and closed elliptic discs, called α-discs, by 1}, and These are elliptic discs of center A, in the plane generated by the vectors B and C.They are regions bounded by the LMN -ellipse.There is a possibility that D α and D α degenerate to a segment or a point.By using these concepts we have the following lemmas that are important to classify α.Lemma 3.2.Let D α be the open α-disc defined as above.For any element W ∈ D α , there exists a transformation Φ such that W is parallel to A , where α = (A , B , C ) t := Φα.Therefore: Proof: Suppose that W = A+p B+q C. Then there exists (θ, t) ∈ S 1 ×R such that (cos θ, sin θ) = (p/ p 2 + q 2 , q/ p 2 + q 2 ) and sinh t/ cosh t = p 2 + q 2 .We may write Φ = Φ 1,t • Φ 2,θ .Therefore, This completes the proof.
Lemma 3.3 (Inertial property for the closed α-discs).Let α and α be equivalent under the action of G and W = A + p B + q C ∈ D α the image of W = A + p B + q C ∈ D α by this action.Then there exists a homeomorphism Θ, where (p , q ) = Θ(p, q), of the closed disc D 2 = {(p, q) | p 2 + q 2 ≤ 1} such that for any (p, q), W , and W have same sign and same rank.That is, sgn(det W ) = sgn(det W ) ∈ {0, ±1} and rank W = rank W .
Proof: It is sufficient to check that for each action Φ 1,t , Φ 2,θ , Φ 3,± ∈ SO(1, 2) and ψ ∈ Diff u (U ), there exists Θ that satisfies the statement.We omit the case of action Φ 3,± ∈ SO(1, 2).Given ψ ∈ Diff u (U ), we may put Θ = id D 2 , then we obtain W = D u ψ t W D u ψ with a regular matrix D u ψ.In this case, rank and index of W do not change.
The proof of the next result is analogous to the proof of Lemma 3.3.(5) If rank α = 3, we have no inflection point.Therefore α is A-parabolic if and only if D α includes some elliptic, parabolic, and hyperbolic points.
Proof: By applying the above lemmas, we have the results.

Rank one case
We consider the rank one case and obtain the normal forms of α.Using these normal forms, in this case, it is easy to study the number of binormals and asymptotic directions.It can occur finite number of asymptotic directions with infinite number of binormals or vice-versa; no asymptotic and no binormal directions; or infinite number of binormals associated to an unique asymptotic direction.(a) M 1 -elliptic: α is equivalent to some α 1,e .
Proof: We only prove the M 1 -parabolic case, the other two cases follow similarly.

Rank two case
In this section, we classify α when rank α = 2.According to the type of the plane generated by A, B, and C, and the type of α given in Section 3, we have the following result.The normal forms for α are displayed below and must have rank 2. (3) where c 2 < 0 for α e .
For the M 2 -parabolic case, that is, the plane A, B, C parabolic, we need the following lemma.M 2 -hyperbolic case: In this case, the plane A, B, C is hyperbolic, then there are two generators D, E such that D, E = A, B, C and we can assume det D > 0. We can get the simultaneous diagonal normal forms of D and E, that is, there is a regular matrix S ∈ SO(2) such that S t DS = ± ( 1 0 0 1 ) , and S t ES = λ1 0 0 λ2 .Therefore, we obtain which can be reduced to (a), (b), and (c) if A is respectively elliptic, parabolic (we assume a 0 = 0, a 2 = 0) or hyperbolic.
M 2 -parabolic case: In this case the plane A, B, C is parabolic, then there are hyperbolic and parabolic vectors D, E (i.e.det D < 0 and det E = 0) such that D, E = A, B, C .Set S ∈ GL (2) such that E = S t ES = ± ( 1 0 0 0 ) and S t DS = D .In this case, for any γ, δ ∈ R, det(γE + δD ) = (det S) 2 det(γE + δD) ≤ 0, and zero if and only if δ = 0.This means that the plane D , E is also parabolic, then it contains only one parabolic direction, which is E = ( 1 0 0 0 ).Moreover, it follows that it must contain D = ( 0 1 1 0 ).As A, B, C are linear combinations of D , E , we have Taking Φ 2,θ • α , for apropriate θ, we can get b 1 = 0.By Lemma 5.2, we may assume that A is hyperbolic, then it is not hard to show that α is equivalent to α p as in (3).
The M 2 -elliptic case follows similarly.
To complete the classification, we can now apply Proposition 3.5.Using the normal forms, it is easy to study the number of binormals and asymptotic directions.Observe that if E(u, θ) is a segment and the line passing through it does not contain the origin, then ∆(u, θ) ≡ 0 does not hold.Therefore, in this case, we have a finite number of binormals.
Theorem 5.3.The number of asymptotic directions (AD) and the types of lightlike binormal directions (BD), for each normal form of Proposition 5.1, are in Tables 3 to 6. Table 3. M 2 -hyperbolic.
Proof: The number of binormals is always finite then in the non-inflection case the asymptotic directions are of same type as the binormal directions (these types are defined in Section 3).We use the normal forms given in Proposition 5 and we obtain Tables 3 to 6 by straightforward calculations (see also Figures 3 to 10).(g) A-parabolic

Rank three case
As the group G = GL(2) × SO(1, 2) has dimension 7 and the space of 3 × 3 matrices α has dimension 9, the classification of G-orbits in this space has at least 2 parameters, but our aim is to classify as follows.We use similar arguments as in Sections 4 and 5, according to the A-elliptic, A-hyperbolic, and A-parabolic type of α, to obtain pre-normal forms for rank 3 matrices depending on 4 parameters.(4) In each case, the normal forms must have rank 3.
We shall denote α and α by α to simplify the notation, when it is necessary.
By Proposition 3.5, there are three types for α.We prove the M 3 and A-elliptic case, the other cases follow similarly.
In this case, we have the following simplified form: On the other hand, the number of the asymptotic directions corresponds to the number of the real solutions of ∆(u, θ) = 0, which will be discussed in further sections.
In the rank three case, it is more difficult to study the number of asymptotic directions, because we need to analyze the type of solutions of a polynomial of degree at most 4. Then we split this analysis in the next sections.

Equation of the asymptotic directions
In this section we obtain the equation of the asymptotic directions associated to the lightlike binormal directions on spacelike surface in de Sitter 5-space in terms of the coefficients of the second fundamental form.The discriminant ∆(u, θ) = M (u, θ) 2 − L(u, θ)N (u, θ) can be written as The lightlike direction e(u, θ) (more precisely, ē(u, θ)) is a binormal if and only if ∆(u, θ) = 0.This equation can be written as (6) (1 cos θ sin θ) If there is some θ which satisfies the above equation ( 6), then we find the asymptotic directions (cos ϕ, sin ϕ) associated to θ by solving the following equations: where L(u, θ), M (u, θ), N (u, θ) are the coefficients of the second fundamental form with respect to the lightlike direction e(u, θ).This equation can be written as We eliminate the parameter θ.By using a mathematical software (such as Maxima, Mathematica, or Maple), we obtain the following completed square trigonometric homogeneous polynomial: , and |c * i | are cofactor matrices of α and rank α ≥ 2. We consider, without loss of generality that p is of type M 3 .This is not restrictive as M 2 -points form a curve on a generic surface M .So the equation obtained at M 3 -points is also valid at M 2 -points by passing to the limit.Notice that when rank α = 1, T (u, ϕ) always vanishes.
With the previous calculations we prove the next result.
Theorem 7.1.There is at most four asymptotic directions passing through any non inflection point and non conic point, on a generic spacelike surface in S 5 1 .These directions are solutions of the implicit differential equation where the coefficients k i , i = 0, 1, 2, 3, 4 depend on the coefficients of the second fundamental form, and are given above.If T (u, ϕ) ≡ 0 at p = X(u) ∈ M , then p is an inflection point.

The types of asymptotic and binormal directions
We are interested in analyzing the number of lightlike binormal directions and their associated asymptotic directions.To do this it is easier to analyze the LMN -ellipse, as we justify in Section 2. This ellipse is described by using the coefficients of the second fundamental form, using the normal forms of α(u) from Sections 4, 5, 6.
Remember that a direction θ ∈ S 1 is a lightlike binormal at p = X(u) if ∆(u, θ) = 0.In this case we call a kernel direction of II e(u,θ) by asymptotic direction associated to the lightlike binormal.As before, we say simply binormal directions and asymptotic directions.For the cases where α has rank 1 or 2, we analyzed the binormals according to their (m 1 + m 2 + m 3 + m 4 )-type with m 1 + • • • + m 4 = 0, 2, 3, 4 or S 1 -type, in Sections 4 and 5.We have also studied the number of the asymptotic directions, in each case.
If rank α = 3, then T (u, ϕ) = 0 if and only if ϕ is an asymptotic direction.According to the number of real solutions of a polynomial of degree at most 4, the asymptotic directions are of the following types: (4), (3+1), (2+2), (2+1+1), (1+1+1+1), ( 2), (1+1), (0), or of a conic type (i.e. the LMN -ellipse E(u, θ) is on the cone, and all lightlike normal directions are binormals and all tangent directions are asymptotic).If T (u, ϕ) ≡ 0, then p = X(u) is a conic point.It is not an inflection point because E(u, θ) does not pass through the origin when the rank is 3.We can distinguish the cases by the solution types of the equations T (u, ϕ) = 0 or ∆(u, θ) = 0, because if rank α = 3, for each binormal θ, there is only one asymptotic direction with same multiplicity.
We now use the results of Appendix A to relate the solutions of ∆(u, θ) = d 1 (u)+d 2 (u) cos θ+d 3 (u) sin θ +d 4 (u) cos 2 θ +d 5 (u) sin θ cos θ+ d 6 (u) sin 2 θ = 0 with the roots of the trigonometric polynomial (a) The finite number of binormal directions and their multiplicities are given in Tables 7, 8, and 9. (b) The number of asymptotic directions (and their multiplicities) is the same number of binormal directions (and their multiplicities).All the possibilities for the finite number of asymptotic directions are described in Tables 7, 8, or 9. (c) At a conic point all normal lightlike directions are binormals and all tangent directions are asymptotics.
Proof: (a) To discuss the number of solutions of ∆(u, θ) = 0, as in Lemma 8.1, we change coordinates and in these new coordinates we obtain the associated t-polynomial F 2 (t), defined in Appendix A, where ∆(u, θ(t)) = 1 (1+t 2 ) 2 F 2 (t), with t = tan( θ 2 ).Since the degree of the t-polynomial F 2 (t) is at most four, we have the following classification.
For next tables, n means the multiplicity of binormal direction θ = π as the solution of ∆(u, θ) = 0, * means that we do not care about the value and N/A means no applicable.This finishes the proof.

Lemma 3 . 4 . 2 ) 3 ) 4 )
If the rank of α is 2 then the type of the plane A, B, C is an invariant.Using the last results we have the following proposition.Proposition 3.5.Let D α be an open α-disc defined as above, then we have:(1) α is A-hyperbolic (or A-elliptic) if and only if any element of D α is contained in the hyperbolic region (or elliptic region respectively).(Ifrank α = 2 and α is M 2 -parabolic, then α is A-parabolic or u is inflection point if and only if D α includes some parabolic point.(If rank α = 2 and A, B, C is M 2 -hyperbolic, then α is A-parabolicor u is inflection point if and only if D α includes some elliptic, parabolic, and hyperbolic points.(Ifrank α = 2 and A, B, C is M 2 -elliptic, then α is A-parabolicor u is inflection point if and only if D α includes some parabolic point p.If u is an inflection point, then p is at the boundary of the disc and if α is A-parabolic, then p is in the open disc.

Table 8 .
Table of types on F 2