RENDICONT1 DEL CIRCOLO MATEMAT1CO DI PALERMO Série II, Suppl. 72 (2004), pp. 219-230 DISTINGUISHED CURVES ON 6-DIMENSIONAL CR-MANIFOLDS OF CODIMENSION 2 VOJTECH ZADNIK ABSTRACT. 6-dimensional CR-manifolds of codimension 2 represent a very special and a very distinguished class of geometric structures among CR-structures of higher codimension. General properties of distinguished curves are studied for hyperbolic and elliptic cases in the framework of parabolic geometries. The notion of distinguished curves is developed for any Cartan geometry, see e.g. [4] or [7]. These curves generalize geodesies in affine geometries so they are called generalized geodesies too. Similarly to affine geodesies, the general properties of such curves are visible already in the homogeneous model of the Cartan geometry. If the Cartan geometry is parabolic we can say much more, especially with regard to the order of jet which determines a unique distinguished curve in any single point. The general approach of [1] is applied here to find out the essential properties of generalized geodesies in CR-geometries of codimension 2 on 6-dimensional manifolds. These are very distinguished cases of CR-structures of higher codimension since there is only a finite number of nonisomorphic homogeneous models in these dimensions. More precisely, there are three such models where two of them carry the structure of a parabolic Cartan geometry. The two cases are called hyperbolic and elliptic, respectively, and the third one presents some singular transition. This was discovered and further developed in [6]. In nearly all of other cases, there is either a continuum of nonisomorphic models or the classification is discrete but none of the models admits a structure of parabolic geometry. In sections 3 and 4 we present all types of generalized geodesies and their properties for hyperbolic and elliptic CR-geometries. In particular, there is an important class of curves generalizing chains, the invariant set of curves well known from CR-geometries of the hypersurface type. Compared to these geometries, the discussion will be a bit richer. Acknowledgements. The author thanks to Jan Slovak for very useful discussions on the topic and to the grant GACR Nr. 201/02/1390 for the support. Most of technical 2000 Mathematics Subject Classification: 32V05, 53A20. Key words and phrases: parabolic geometry, CR-structures, distinguished curves. The paper is in final form and no version of it will be submitted elsewhere. 2 2 0 VOJTECH ZADNIK computations was performed by the help of the computational system Maple © 1981- 2001 by Waterloo Maple Inc. 1. INTRODUCTION The modelling CR-manifolds are the so called CR-quadrics which appear as follows. Let M be a real submanifold in C^ such that the dimension of the maximal complex subspace TCR M C TXM does not depend on the base point x E M. Let us denote the complex dimension by n. If the dimension of M is 2n+k then M is called an embedded CR-manifold of codimension k. Locally, any such CR-manifold can be viewed as an embedded CR-manifold in C1 ** and expressed as a graph I m ^ ) = fj(z, z, Re(tv)) for j = 1,..., k where (z, w) = (z\,..., z-,, W\,..., Wk) are coordinates in C1 "1 "* and fj are real functions. Let us assume / is chosen in such a way that /(0) = 0 and df(0) = 0, i.e. M passes through the origin and T0M = {Jm(w) = 0}. Moreover, we may assume that T0 CR M = {w = 0}. After some manipulation, the above equations can be written as (1) lm(w) = h(z,z) + o(3) where h is a Hermitian form with values in Ck determined by the second derivatives of fj with respect to z and z in 0, i.e. the Levi form of M in the origin. All details can be found e.g. in [5]. The geometrical meaning of the last equation is that M is osculated by the quadric Q = {Im(w) = h(z,z)} up to second order in any fixed point. If the codimension k equals to 1 then there is a finite list of osculating quadrics which depends only on the signature of the Levi form. For general k, the list of classifying quadrics has mostly the cardinality of continuum. One of the few exceptions are 6-dimensional embedded CR-manifolds of codimension 2. In order to get the quadrics in that case it remains to classify all nondegenerated Hermitian forms on C2 with values in C2 . According to [5], any of the mentioned forms can be expressed in suitable coordinates in one of the following forms (2) h(z,z) = (\zl\\\z2\'>), (3) h(z,z) = (\zi\2 ,Re(zlz2))i (4) h(z,z) = (Re(ziz2),lm(ziz2)) and these three cases are called hyperbolic, parabolic, arid elliptic, respectively. Any embedded nondegenerated CR-manifold of codimension 2 in C4 is divided into disjunct sets of points with respect to the type of osculating quadrics. For any hyperbolic and elliptic point there is a neighbourhood of points of the same type while the parabolic points have not got this property. So both hyperbolic and elliptic points form open areas on M while parabolic points form some closed transitions. The last point to remind is that any embedded CR-manifold whose all points are either hyperbolic or elliptic carry the structure of a |2|-graded parabolic geometry, which was found in [6]. This is due to the fact that in both cases the automorphism group of the corresponding quadric is a semisimple Lie group and the stabilizer of some its point is a parabolic subgroup. DISTINGUISHED CURVES ON 6-DIMENSIONAL Cfi-MANIFOLDS OF CODIMENSION 2 221 This section introduces necessary notions and definitions. Further, we present some technical results of [1] which will provide a recipe for applications. 2.1. Definitions. Let p : Q -> M be the principal bundle of a Cartan geometry of type (G^P) and let u G ^(QiQ) be the Cartan connection. P is a closed subgroup in a Lie group G, p and Q are corresponding Lie algebras. The absolute parallelism u defines the so called constant vector fields u~x (X) on Q, for any X G fl, determined by the condition u(u~l (X)(u)) = X for all u G Q. Moreover, the same property provides a natural identification TM = Q xP ($/p) where the action of P on g/p is induced by the restricted adjoint action Ad : G -> GL(Q). The identification is given by the assignment {u, X + p} i-> Tp • u~l (X)(u). Let us suppose the subalgebra p has got a complementary subalgebra n in g. Then n is identified with g/p and via this identification the subalgebra n comes to be a P-module; the "truncated" adjoint action is denoted by Ad. Any complementary subalgebra n gives rise to a general connection on Q which is principal if and only if n is invariant with respect to the restricted Ad-action of P on g, in that case the Cartan geometry is called reductive. Generalized geodesies on M are defined as projections of flow lines of horizontal constant vector fields, i.e. as curves of the shape cu >x (t) = p(Fl" ( \u)) for any u e Q and X G n. Obviously, the tangent vector of curve cu >x in c(0) = p(u) is {rx,X} with respect to the above identification. Another representative of that vector defines another curve in general if the geometry is not reductive. The essential question appears in this context: What are all generalized geodesies with the common tangent vector in some point? The answer may depend on the type of the tangent vector if there are distinguished ones (as nullvectors in conformal geometries are). Distinguished directions of the same type in any tangent space are contained in a common orbit of action of the structure group P. They correspond to P-invariant subsets in n so the classification of generalized geodesies always depends on the classification of Pinvariant subsets in n. For any fixed subset A C n all curves cu >x with X G A form a subset among all generalized geodesies which is denoted by the symbol CA- Generalized geodesies of type CA emanate in those tangent directions which correspond to the P-orbit of the subset A in n. The homogeneous model of a Cartan geometry of type (G, P) consists of the principal bundle G -> G/P with the Maurer-Cartan form playing the role of the Cartan connection. Constant vector fields on G are the left invariant ones and their flows are left shifts of 1-parametric subgroups. So generalized geodesies of type CA on the homogeneous space G/P take the form c9 >x (t) = gexp(tX)-P for any g G G and X eACn. 2.2. Developments. The essential tool for our purposes is the notion of development of curves via the construction of the Cartan's space SM = Q xP G/P (with the obvious action of P on the homogeneous space G/P). The Cartan connection on Q induces a general connection on SM which allows to define for any curve c on the base 222 VOJTECH ZADNIK manifold M and any fixed point x G c its development into the fibre over x which is identified with G/P. This construction is often used to distinguish curves on M (or generally, on all manifolds endowed with a Cartan geometry of type (G, P)) by means of distinguished curves in the homogeneous model. The following statement holds, see [7]Lemma. Let p : Q —> M be a Carton geometry of type (G, P) with the Lie algebra decomposition 0 = n © p and let cu 'x be a generalized geodesic for u EC/ and X G n. Then the development ofcu 'x in x = p(u) is the curve {„, exp(tX)-P} C SXM. Hereby defined developments generalize the classical concept of the development of curves on manifolds with affine connection. In that case the homogeneous space G/P = Rm is globally identified with n so that the two actions of the structure group P = GL(m,R) coincide. Then the Cartan's space SM equals to the tangent bundle TM and a curve is an affine geodesic if and only if it develops into a straight line within the tangent space of any single point. 2.3. Technicalities. Due to the above correspondence of generalized geodesies on M and their developments in G/P we may focus only on generalized geodesies in the homogeneous space G/P going through the origin e-P. In other words, we are interested in curves of the form cb 'x (t) = bexp(tX)-P = exp(t Adf,X)-P with b G P and X G ACn. Further, let us assume the geometry in question is parabolic. Let G —> G/P be a homogeneous model of a parabolic geometry, i.e. G is a semisimple Lie group and P its parabolic subgroup. On the infinitesimal level, the subalgebra p C 0 defines a grading 0 = 0_* © • • • © 0* such that p = 0O © • • • © 0*. The subalgebra n = 0_jt © • • • © 0_i is a canonical complement to p, usually denoted by 0_. The subalgebra 0O is reductive and 0i © • • • © 0^ = p+ is nilpotent. Altogether, P is the semidirect product G0 x exp p+ where G0 is the subgroup of P (with the Lie algebra 0o) whose all elements respect the gradation of 0. Now any element b G P is uniquely written as b = b0expZ, with &0 G G0 and Z G p+, but it can be rewritten b = exp(Ad&0 Z)-&0 as well. Hence, generalized geodesies cb0exPz,x an(j cexp(Ad6o z),Adb0x ^Q^^fe DV tn e definition, so we will assume only curves of the form c expZ 'x hereafter. If we restrict to the curves of type CU, the subset A C 0_ must be G0-invariant for the above elimination to be valid. This convention will be kept in the rest of paper. Moreover, two curves cbl 'Xl and cb2 'Xi coincide if and only if curves ce 'Xl and c6 * b2 'x * do, so we will further suppose bi = e. Let us compare generalized geodesies ce 'x and cexpZ,y . The equation (1) exp(*K) = exp(* AdexpZ Y) • u(t) defines a curve u : R —> G, at least locally, which is analytic for arbitrary entries X, Y, and Z. The two curves coincide if and only if u takes values in P which is equivalent to all derivatives _^(0) are tangent to P. Similarly, the two curves have got a common r-jet in 0 if and only if the previous condition holds for all i < r. By technical reasons we prefer to differentiate the map Su instead of _, where Su : TR -» 0 is the so called left logarithmic derivative of u. The map Su is defined by Su = u*u where u is the Maurer-Cartan form on G. By definition, Su is determined by Tu so the following lemma certainly holds. DISTINGUISHED CURVES ON 6-DIMENSIONAL CK-MANIFOLDS OF CODIMENSION 2 223 Lemma. Curves ce >x and cexpZ,Y determine a common r-jet in 0 if and only if the derivatives (5u)W(0) belong to p for alii 1. (2) 5u(t) =X- Adu(t)-i AdexpzY, (3) (5u)M(t) = (-zdxySu(t). In particular, the condition 5u(0) 6 p is satisfied if and only if X = AdtxpZY, i.e. the curves have got the same tangent vector in the origin. Let us suppose k is the length of grading of $. An easy consequence of (3) is that the assumption (5u)^(0) G p for all t < k + 1 implies (5u)^(0) = 0 for all t > k + 1. So any two generalized geodesies share the same (k + 2)-jet in 0 if and only if they coincide. The number k + 2 only proves the finiteness of the order but the estimate is not sharp at all, as one can see in next sections. 2.4. Reparametrizations. The previous paragraph will serve us to find all generalized geodesies of a certain type CA which share a common tangent vector. However, some of those curves may parametrize the same unparametrized geodesic. In this paragraph we solve the question when two generalized geodesies coincide up to some reparametrization? In this context we have to rewrite the equation (1) as follows: (4) exp( Step by step, we search Z e p + and Y £ A such that the curves ce,x and cexpZ,Y coincide. At the same time we get the order of jet which decides the two curves are equal. The first condition 8u(0) E p restricts Z E p+ and Y E A to fulfil the equality Y = Ad~*pZ(X). For any |2|-graded parabolic geometry the latter condition means Y = X - [Zi,X2], where Z\ is the (h-part of Z and X2 the g_2-part of X. The other conditions (Su)^(0) E p further reduce possible Z E p+ for the two curves share a common (i + l)-jet. All such elements form a subset in p+ denoted by the symbol H,+1 . More precisely, for any r > 1 we put Br = {Z E p+ : JQC6,X = <70"cexpZ,y } where Y = A d ^ p f ) . In particular, Bl = {Z E p+ : Ad;^pZ(X) E A}. Whenever the condition (Su)^(0) E p implies (5u)(r+1 )(0) = 0 then generalized geodesies of a given type are uniquely determined by a jet of order r+1. This is equivalent to the condition B^1 = B3 to be true for all s > r + 1. (3) Now we are interested in the dimension of the set of parametrized generalized geodesies of type CA sharing the same tangent vector f. This set is denoted by C^ and its dimension does not depend on the vector of the type in question. In view of the above arguments, let f = {e, X} be the fixed vector and further let r be an order of jet which determines generalized geodesies with the given tangent vector uniquely. Obviously, for any Z E Br the curves ce,x and c expZ,Ad «*pz(x) coincide. If another representative of the vector {e,X} is chosen then the analogously defined subsets Bx C p+ are naturally identified with the initial ones. So the set C\ is parametrized by the quotient Bl /Br which is easy to describe. (4) Finally, we are interested in distinguished parametrizations of generalized geodesies of a given type. Lemma 2.4 allows us to find all reparametrizations which appear if two generalized geodesies parametrize the same curve. The function ip is guessed according to its behaviour in 0, so we must always check the guess is right, i.e. the equation (4) in 2.4 holds true for all t. Reparametrizations which appear in this way will be either projective or affine. On the other hand, for any generalized geodesic c9,x and any reparametrization C3 defined by (z, w) i-> (l,z,u>). The action of G on the cone factors to a transitive action on the CR-sphere Q where any element of G acts by a CR-automorphism. Conversely, the group of CR-automorphisms of Q is just the quotient of G by a noneffective kernel which is isomorphic to Z3. The stabilizer P of the origin is the Borel subgroup in the semisimple G. Altogether, Q can be identified with the quotient space G/P and the principal bundle G -> G/P represents a homogeneous model of 3-dimensional CR-geometries of the hypersurface type. In the above described representation of the principal group G = SU(2y 1), the Lie algebra g = su(2,1) looks like < I - -2iimy -s J : x, z/, z e C, a, b G R >. The parabolic subalgebra p consists of upper triangular matrices and the gradation 0 = g_2 © 0_i © 0o © Si © 02 is given by the five diagonals. With respect to the identification TQ = G xP 0_ from 2.1, the CR-subbundle is associated as TCR Q = G Xp 0_i. Now we can apply the process suggested in 2.5 to describe generalized geodesies in this case. First of all, there are two distinguished kinds of vectors in TQ. The first ones belong to the CR-subspace TCR Q, the second ones are transverse. The two classes of vectors correspond to the complementary P-invariant subsets 0_i and 0_\0_i in 0_. There are no other distinguished directions in TQ, i.e. no other nontrivial F-invariant subsets in 0_. The previous two subsets are clearly Go-invariant and there is another Go-invariant 226 VOJTECH ZADNIK subset in g_, the last component g_2 in the gradation of g_. Its complement in g_\g_i is also Co-invariant and represents the generic case. Curves determined by the subset g_2 are the well known Chern-Moser chains which form a CR-invariant class of curves with particularly nice properties. Especially, the P-orbit of g_2 is the entire g_ \ g_i so the chains emanate in all directions transversal to the CR-subspace. Let us discuss all mentioned types of generalized geodesies: (1) A = g_!. For an arbitrary Z G p+, the curves ce,x , cexpZ,y share the same tangent vector in origin if and only if Y = X. The curves have got a common 2-jet if and only if Z belongs to g2. The condition on 3-jet, i.e. (Su)"(0) G p, is satisfied if and only if Z = 0 which implies (Su)"(0) = 0. Altogether, curves of this type are determined by a 3-jet. Parametrized curves cexpZ, *(2), for any Z G p+, are different from each other and keep the common tangent vector f = {e,X} hence the set C^_x is parametrized by elements of p+, the dimension of which is 3. With the notation of 2.5, Bl = p+, £2 = g2, and£3 = .94 = .-- = 0. Let X = I x o o J G g_i. Looking for the admissible reparametrizations one can find that all curves from C|_t which coincide up to parametrization with ce 'x are of the form c exp (sZ ),x where s G R and Z = f o o ix). Reparametrizations which appear in this way are just the projective ones and (f"(0) = —2s(p'(0)2 \x\2 really takes all values. (2) A = g_2. The curves cc,x , cexpZ,y have got the same tangent vector in the origin if and only if Z G g2 and Y = X. They share a common 2-jet if and only if Z = 0. Then (Su)f (0) = 0 so the chains are determined by a 2-jet and the set C|_2 is parametrized by elements of g2 whose dimension is 1. Moreover, all curves from C|_2 parametrize the same unparametrized chain and the admissible reparametrizations are projective. For X = (o o o J fixed and Z = foooj G ^2 arbitrary, the value of (f"(0) is -2s(f'(0)2 a. Altogether, in any direction which does not belong to the CR-subspace of the tangent space there is a unique unparametrized chain endowed with a canonical projective structure. This is a very classical result which can be also found in [2] or [3]. (3) A = g_ \ (g_i U g_2). The curves ce,x , c**v%y share the same tangent vector if and only if Y = X - \Z\, X2] and Z = Z\ -f Z2 G p+ is arbitrary. They have got the same 2-jet if and only if Z = 0. Then Y = X and (Su)'(0) = 0 so the curves of this type are determined by a 2-jet and the set C^ is 3-dimensional, parametrized by all elements of p+ . The distinct difference against the above two cases is that there are no two curves in C\ which would coincide up to some reparametrization. It turns out the class of admissible reparametrizations on any curve of this type is only affine. For any tangent vector transversal to the CR-subspace there is, besides the chains, the 3-dimensional family of uniquely parametrized generalized geodesies of the generic type. 3.2. Hyperbolic quadric. The hyperbolic quadric H is given by the equality (2) in the first section, i.e. it is expressed as (1) lm(wl) = \zi\2 lm(w2) = \z2\2 DISTINGUISHED CURVES ON 6-DIMENSIONAL Cfl-MANIFOLDS OF CODIMENSION 2 227 with respect to the coordinates (z\, z2, wi, w2) of C4 . Obviously, the hyperbolic quadric is the direct product of two CR-spheres discussed in 3.1 which lie in the subspaces (zi,ivi) and (z2,w2), respectively. The action of the group G = SC/(2,1) x 577(2,1) on H is given by the product of the two actions of £7/(2,. 1) on each CR-sphere. The isotropy subgroup P of the origin is the product of two copies of the parabolic subgroup in the CR-sphere case. Obviously, G is semisimple and P parabolic. Any element of G acts on H by a CR-automorphism. Conversely, the group of automorphisms of H is isomorphic to the semidirect product (2) (SU(2,1)/Z3 x SU(2,1)/Z3) x Z2 where the group Z2 consists of the identity and an involutive automorphism which interchanges the two spheres in the product. The discussion on generalized geodesies is based only on results of 3.1 due to the product structure on H. Especially, the tangent bundle TH is a direct product of TL H and TR H, the left and the right subbundle, which correspond to the vanishing right and left part of g_ = g_ x g?, respectively. Vectors belonging to one of these subspaces are called singular. Let us suppose the principal group G of the geometry is the group (2). Then P = (B/Z3 x B/Z3) x Z2 where B is the Borel subgroup of SU(2,1), i.e. the parabolic subgroup from the CR-sphere case. All P-invariant and G0-invariant subsets in g_ = g_ x g_- are obtained as products of P and Go-invariant subsets in each slot up to their interchanging. Now the singular directions correspond to the P-invariant subset g_ U gR which is just the P-orbit of g_. Similarly, all P and Go-invariant subsets in g_ discussed below are written in the brief form, i.e. A x B C g_ x gR means its P and Go-orbit Ax BUB x A, respectively. Altogether, there are 5 kinds of tangent vectors in TH and 9 types of generalized geodesies on H. The corresponding Go-invariant subsets in g_ are -4i = {0} x 8 * , A* = g_! x g ^ \S, A7 = g_2 x fl*2\S, A2 = {0} x g*2, -45 = g_i x g_2\S, -48 = g_2 x g _ _ \ S , -4a = {0} x B-- , A6 = g_x x g__ \ S, A9 = g__ XQR _\S. The symbol g__ denotes the set of generic vectors g_ \ (g_x U g_2) in g_, alike for g__ and g , and S represents singular vectors, i.e. S = Ai U A2 U A3. Subsets Ai and A4 are P-invariant so they define distinguished directions in TH, namely they are singular and nonsingular vectors in TCR H, respectively. Further, the P-orbit of A2, denoted by P(-42), coincide with P(-43) = {0} x (gR \ g*x) and this set describes all singular directions which do not belong to TCR H. Next type of tangent vectors is given by P(A7) = P(As) = P(A9) = g_ \ g_i which corresponds to nonsingular directions not belonging to TCR H. The last type of distinguisfiedf directions correspond to the orbit P(A5) = P(A6) = Qi, x (-« \ - £ ) . Curves which emanate in singular directions are fully classified in 3A so there are 6 types of generalized geodesies emanating in nonsingular directions of TH left to be discussed. The product structure of H with the isolated action of the structure group on each slot leads directly to the following results compiled only from 3.1: 228 VOJTĚCH ŽÁDNÍK For A4, A5 , and Ae, one side in the product is isomorphic to g_L and so the corresponding generalized geodesies are determined by a 3-jet, otherwise a 2-jet is enough. The dimensions of families of generalized geodesies with the common tangent vector are obtained by summing the numbers from 3.1, i.e. dimC^ = 6, dimC^5 = 4, dimC*fl = 6, dimC^7 = 2, dimC^ = 4, and dimC^9 = 6. The only difficulties are to express explicitly all generalized geodesies with a common tangent vector parametrizing the same curve if the admissible reparametrizations are projective. In general, if at least one slot of a Go-invariant subset is Q__ then the admissible class of reparametrizations is affine. This is the case of ^6, -4s, and ^9. Otherwise the reparametrizations are projective. All details can be found in [8]. Among all nonsingular types of generalized geodesies there are three of them of a particular interest. The first one is given by the choice A4 = g_i where corresponding generalized geodesies emanate in generic directions of the CR-distribution TCR H. Curves of this type are determined by a 3-jet in one point and they admit the projective class of reparametrizations. The second distinguished choice is A7 = g_2, which defines chains in nonsingular directions as discussed in [6], and the last choice of Ag = g defines curves of the generic type. For any nonsingular direction which does not belong to T CR H there is a 1-dimensional family of unparametrized chains with the projective class of parametrizations and 6-dimensional family of uniquely parametrized curves of typec,49. 4. ELLIPTIC STRUCTURES The elliptic quadric E is expressed in section 1 by the equality (4). In coordinates (zi, z2) u>i, w2 ) of C 4 it is the graph of (1) Im(ivi) = Re(zif2 ) Im(iv2 ) = lm(ziz2 ). The group of automorphisms of the CR-structure on E is less visible than in the hyperbolic case. However, the group is isomorphic to (2) C7 = 5L(3,Q/Z3 xiZ2 as shown in [6]. Similarly to the hyperbolic case, Z3 represents a noneffective kernel in SX(3,C) and, on the infinitesimal level, Z2 provides the interchanging of the two components of T CR E, see below. Isotropy parabolic subgroup P of the origin is the semidirect product _?/Z3 x Z2 where B is the Borel subgroup of 5L(3,C) consisting of upper triangular matrices. The Lie algebra of G is g = £l(3,C), viewed as a real Lie algebra, and the corresponding parabolic subalgebra p defines the gradation of g according to the five diagonals. In addition, the subspace 0-i = { ( x ° £ ) '• xty eC> Cg_ defining the CR-distribution decomposes into g_i = g_x x gR x which induces a product structure on TCR E. At the same time, subspaces Q_X and QR X distinguish tangent vectors in the CR-subspace. More precisely, the minimal P-invariant subset in g_i containing $_x is g_x U g^. Its complement in g_x is P-invariant too and the complement of g_i in g_ is the last P-invariant subset in g_ defining directions transversal to the CR-distribution. There are 4 types of generalized geodesies on the elliptic quadric which correspond to the DISTINGUISHED CURVES ON 6-DIMENSIONAL CR-MANIFOLDS OF CODIMENSION 2 229 following Go-invariant subsets in fl_: 4i = fl-iUfl_\, -42=fl-l\(fl_lUfl?1), -43 = fl-2 , A4 = fl_\(fl_1Ufl_2x(flf:1Ufl?1)). The natural candidate on the post of A± is the complement of fl_2 in fl_ \ fl_i, i.e. the subset fl_\(fl_iUfl_2). However, the actual A\ arises by excluding the Go-invariant subset A = fl_2 x (fl£x U ^-i) from the latter set. The only reason of its excluding is that the class of geodesies of type CA coincide with the class of chains determined by A3 = fl_2; this is shown in [8]. Moreover, the P-orbit of both A3 and AA is fl_ \ fl_i, hence for any tangent vector transversal to TCR E there are generalized geodesies of these two types. (1) A\ = fl^Ufl^. Two curves ce,x and cexpZ,y share the same tangent vector in the origin if and only if Y = X and Z G p+ is arbitrary. Let be K = f i o o j _fl£i- The curves have got a common 2-jet if and only if Z belongs to_?2 = <(oo_J :w,z _ C >. Now already (Su)'(0) = 0 so the curves of this type are determined by a 2-jet. The set C\x is parametrized by all elements of p+/-?2 = < f o o o J :i/EC[ so its dimension is 2. All curves from C\x which parametrize the same curve as cC)X look like (f^i*2 )^ for any s G R and Z = f o o o J. Admissible reparametrizations are projective and * where s G R and Z = ( o o y_1 ]. Admissible reparametrizations \o o o / are the projective ones and