Transformation Groups, Vol. 9, No. 2, 2004, pp. 143-166 ©Birkhäuser Boston (2004) ON DISTINGUISHED CURVES IN PARABOLIC GEOMETRIES ANDREAS CAP Institut für Mathematik Universität Wien Strudlhofgasse 4 and International Erwin Schrödinger Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Wien, Austria andreas.cap@esi.ac.at JAN SLOVÁK Department of Algebra and Geometry Faculty of Science Masaryk University Janáčkovo nám. 2a 662 95 Brno, Czech Republic slovak@math.muni.cz VOJTĚCH ŽÁDNÍK Department of Algebra and Geometry Faculty of Science Masaryk University Janáčkovo nám. 2a 662 95 Brno, Czech Republic zadnik@math.muni.cz Abstract. All parabolic geometries, i.e., Cartan geometries with homogeneous model a real generalized flag manifold, admit highly interesting classes of distinguished curves. The geodesies of a projective class of connections on a manifold, conformal circles on conformal Riemannian manifolds, and Chern-Moser chains on CR-manifolds of hypersurface type are typical examples. We show that such distinguished curves are always determined by a finite jet in one point, and study the properties of such jets. We also discuss the question when distinguished curves agree up to reparametrization and discuss the distinguished parametrizations in this case. We give a complete description of all distinguished curves for some examples of parabolic geometries. Elie Cartan's idea of 'generalized spaces' as curved analogs of Felix Klein's geometries (i.e., homogeneous spaces) is a well understood geometrical concept, which, for a Lie subgroup P C G, generalizes the Maurer-Cartan form on the total space of the principal P-bundle G —> G/P to Cartan connections on principal P-bundles, see e.g., the introductory book [17]. The concept of parabolic geometries refers to those cases where P is a parabolic subgroup in a (real or complex) semisimple Lie group G. In [9], C. Feffer-man initiated a program to exploit the representation theory of parabolic subgroups in semisimple Lie groups in order to understand invariants of geometric structures like CR- Received September 15, 2003. Accepted January 26, 2004. 144 ANDREAS ČAP, JAN SLOVÁK AND VOJTĚCH ŽÁDNÍK geometries, projective geometries, or conformal Riemannian geometries. This approach has proved to be extremely powerful. First, all parabolic geometries can be described in terms of weaker analogies of classical G-structures on smooth manifolds and, similarly to the examples mentioned above, all such structures give rise to canonical normal Cartan connections, [19, 14, 3]. In fact, these constructions express Cartan's method of equivalence using the language of the modern representation theory and natural cohomological reasoning. The existence of the Cartan connection provides an effective calculus to deal with invariant objects, see e.g., [5] and the references therein. To a large extent, the understanding of the general (curved) geometries can be reduced to properties of the homogeneous model, and thus to purely algebraic questions. The goal of this paper is to use this approach in order to understand invariantly defined systems of distinguished curves for parabolic geometries, which we call (generalized) geodesies. After recalling basic concepts of parabolic geometries, geodesies are introduced and discussed along the lines of the classical approach in affine geometry, which uses the development of curves. This approach may be found in a similar context in [17] and [13]. In this way, many aspects of the study of the curves are reduced to the case of the homogeneous model. Thus the original 'smooth' question on curved manifolds can be transformed to an 'algebraic' problem, which is discussed in Section 2. In particular, we obtain estimates on the order of jets necessary to determine a geodesic, and this approach also leads to an algebraic description of all jets of geodesies in a point. The third section is devoted to the study of possible reparametrizations in the class of geodesies. Specializing the general results to 1 [-graded Lie algebras, we obtain generalizations of some well-known results on conformal, projective, and quaternionic geometries (see e.g., [1]). The final section provides further refinements for specific classes of curves, see in particular Theorems 4.2 and 4.3. Acknowledgments. Part of the work was done during a stay of the second author at the University of Adelaide under an ARC financial support, and his discussions with Michael Eastwood were most helpful and illuminating. The first author supported by project P15747 of the FWF. The second and third authors acknowledge the support from GACR, Grant Nr. 201/02/1390. 1. General concepts 1.1. Parabolic geometries Let us briefly recall the basic facts, more details can be found in [4] or [17], and the references therein. Let G be a real semisimple Lie group with Lie algebra g, and PcGa parabolic subgroup with Lie algebra p. A (real) parabolic geometry (Q, ui) of type (G, P) is a principal bundle Q with structure group P over a manifold M, equipped with a smooth one-form ui € Ql(Q,Q), which satisfies (1) ll)((z)(u) = Z for all u € Q and fundamental fields (z, Z € p C fl, i.e., ui reproduces the generators of fundamental vector fields, (2) (rh)*ui = Ad(&-1) o ui for all b € P, i.e., ui is P-equivariant with respect to the adjoint representation, and (3) w|tuc? : TUQ —> Q is a linear isomorphism for all u € Q, i.e., ui is an absolute parallelism on Q. The curvature of a parabolic geometry (Q, ui) is the horizontal two-form K € &2(G, fl) ON DISTINGUISHED CURVES IN PARABOLIC GEOMETRIES 145 defined by the structure equations K = duj + i.e., K(£,r)) = dh)(£,,rj) + u(r])}. Clearly, the Maurer-Cartan form ui on the principal fiber bundle G —> G/P is a parabolic geometry and the structure equations say that this geometry is flat, i.e., its curvature vanishes identically. (G —> G/P, uS) is called the homogeneous model for parabolic geometries of type (G, P). Morphisms between Cartan geometries (Q,uS) and (Q',uj') are principal fiber bundle morphisms p : Q —> Q' such that p*ui' = ui. It is quite elementary to prove that a geometry is locally isomorphic to its homogeneous model if and only if its curvature vanishes identically, see [17]. Each smooth (left) action of the structure group P on a smooth manifold S leads to a functor S on the category of Cartan geometries of type (G, P). The value of S on (Q, uS) is the associated fiber bundle Q Xp S with respect to the action of P, while a morphism p : (Q, uS) —> (Q', ui') induces the fiber bundle morphism p x p ids :GxpS—>Q'xpS. We call these bundles natural bundles. Moreover, this construction is functorial in the smooth action entry because each equivariant mapping a : S —> 5" induces the fiber bundle mapping idg x pa : QxpS^QxpS'. Thus we have a bifunctor on Cartan geometries and smooth left actions with values in fiber bundles. In particular, linear representations of P lead to functors valued in vector bundles and their linear morphisms, and the bifunctoriality of the construction extends all natural constructions like pairings, decompositions, and tensor products of representations to the natural bundles. Of course, all this is the obvious restriction of the usual functorial constructions over all principal fiber bundles to the category of Cartan geometries. A central example, which also illustrates the role of the Cartan connection, is given by the representation of P on g/p induced by the adjoint representation. This leads to the functor Q x p g/p, and via the Cartan connection ui this associated bundle can be identified with the tangent bundle TM. Indeed, since ui defines an absolute parallelism, there are the corresponding 'constant' vector fields ui~l{X) € X(Q) for all l£g, defined by uj(uj~1(X)(u)) = X for all u £ Q. Denoting by \u,X + p] the class in Q xp g/p of («, X + p) € Q x g/p and by 7r : Q —> M the bundle projection, one immediately verifies that \u,X + p] I—> Ttt(uj~1(X)(u)) defines the claimed isomorphism. For any parabolic subalgebra p C g, there is a grading 0 ... 0 g^ of g such that P = floffi- • -ffiflfc, and p+ =fliffi.. -ffiflfc is the nilradical of g, see [20, 3]. In particular, this implies that go is a reductive Levi component for p. Hence we obtain an identification n = fl_fc®...®0_i with g/p, which is an isomorphism of P-modules if we endow n with the 'truncated' adjoint action Ad. Via the Killing form, one further obtains an identification of n* with p+, which induces the identification of the cotangent bundle T*M with Q Xp n*. Thus all tensor bundles over M are identified with the natural bundles coming from tensor products of the representations n and n*. Moreover, the right hand ends g1 = gt © ... © g^ define a P-invariant filtration of g. Hence we obtain natural subbundles T%M C TM for all i < 0. The resulting filtration TM = T~kM D T~k+1M D ... D T~xM D 0 is the most importing object underlying a parabolic geometry. This filtration is trivial for 11 [-graded algebras and we call such parabolic geometries irreducible. 146 ANDREAS ČAP, JAN SLOVÁK AND VOJTĚCH ŽÁDNÍK A very special case of the construction of natural bundles is the choice S = G with the left action of P on G given by the group multiplication. This leads to the principal fiber bundle Q = Q x p G with the principal action given by the usual right multiplication in G and the canonical inclusion Q C Q, u i—> [u, e], where e € G is the unit element. Now, the Cartan connection uj extends uniquely to a G-equivariant one-form uj € f21(5, fl) reproducing the fundamental vector fields. One easily verifies that uj is a principal connection on Q. Whenever we have a left action of P on some manifold S which is the restriction of a left action of G, then we may view the natural bundle Q Xp S also as Q Xq S. Hence on any natural bundle of this type, there is a canonical connection induced by uj. Of course, if we consider restrictions of G-representations to P, then the resulting natural vector bundles, which are usually called tractor bundles, are equipped with canonical linear connections. 1.2. Development of curves The notion of the development of curves is related to a particular instance of natural bundles associated to restrictions of G-actions to P, namely the case of the canonical left action on G/P. The resulting space S = Q Xp G/P = Q Xq G/P is called Cartan's space over the underlying manifold M of the Cartan geometry in question. Of course, S —> M is a fiber bundle with typical fiber G/P, and from 1.1 we know that the parabolic geometry induces a canonical connection on this fiber bundle. Another remarkable fact about S is that for the point o = eP € G/P, and a point x € M, all points u € Q with tt(u) = x lead to the same class O(x) = [u, o] € Q xp G/P. Hence we obtain a canonical smooth section O of S —> M for every parabolic geometry (Q —> M, uj) of type (G, P). Moreover, the vertical tangent bundle VS can be identified with the associated bundle Q xPT(G/P). Since the basepoint o € G/P is a fix point for the action of P, we see that the restriction of VS to the image O(M) of the canonical section is the associated bundle Q xpT0(G/'P). Since T0(G/P) is canonically isomorphic with g/p and Qxp (fl/p) is naturally isomorphic to TM, we get a canonical isomorphism VS\o(x) — TM. Thus we may view the Cartan's space S as a nonlinear version of the tangent bundle in which the geometry in question is encoded by means of the local parallel transport of the induced connection. This point of view goes back to Cartan, and it was developed further in an abstract way in the second half of the 20th century (see e.g., [11]). This canonical parallel transport provides a straightforward generalization of the classical concept of the development of curves. By composing with O, a curve c : / —> M with / = (a, b) C R may be also viewed as a parametrized curve in S. Fixing to € / we find a neighborhood J of to in I on which the parallel transport along c : / —> M is well defined. Given s € J, we may follow the curve O o c from to to s and then follow the parallel transport backward for time t0 — s to return to the fiber over t0. More formally, we define a smooth curve dev(c, to) from an open neighborhood of 0 in R to c(to + s — t) with the initial point 0(c(s)). This curve is called the development of c at to- For a point u € Q over c(to), there is a unique curve c(t) in G/P mapping 0 6 R to o 6 G/P such that dev(c, to)(t) = [u, c(t)]. Any other choice for the point in Q has the form u-b for b € P, and for that choice the curve changes to £b-i o c. Hence we conclude that each choice of a P-invariant class C of curves which map 0 6 M to o 6 G/P leads to a distinguished class of curves on all manifolds endowed ON DISTINGUISHED CURVES IN PARABOLIC GEOMETRIES 147 with a Cartan geometry of type (G, P). We say that a curve c on M is a distinguished curve of type C at a point c(to) € M, if for some (and thus any) point u € Q the curve c constructed above lies in C. The natural choices for such sets C of curves, of course come from one-parameter subgroups in G: For a subset A C g, we can define a class Ca as {t i—> &exp(tX)P l£4,iG P}. So we take the one-parametric subgroups with generators in A, allow them to be shifted by left multiplications with elements of P, and project the resulting curves to G/P. Of course, for X € p this always leads to the constant curve o, so we may assume A n p = 0. On the other hand, if we want to have curves in all directions in the class Ca, then we have to assume that the restriction of the projection g —> g/p to A is surjective. The most obvious choice for A which satisfies this requirement is A = n. It should be noted that for X € g \ p the curve t i—> bexp(tX)P does not lie in Cn in general. Following the case of affine geometry and since we are mainly interested in having sets of distinguished curves which are as small as possible, we shall always assume iCnin the sequel. The parabolic subgroup P C G always has a canonical closed subgroup Go which corresponds to the Lie subalgebra flo C p. This group turns out to be reductive, and it can be characterized as the subgroup of those elements in G, whose adjoint action preserves the grading of g. In particular, the subspace n is stable under the adjoint action of Go- Now for & € Go and X € n, we of course have &exp(tX) = exp(i Adf, X)b, and thus bexp(tX)P = exp(t Adi, X)P. Thus it is natural to restrict attention to Go-invariant subsets A C n, and the corresponding distinguished curves are called (generalized) geodesies of type Ca- We often do not mention the type if A = n. The generalized geodesies of type Ca are easily described explicitly by means of the constant vector fields uj~1(X). Let us consider the projection c(t) of the flow line Fl" ^X\u) € Q to the manifold M. From the construction of the principal connection uj on Q one immediately concludes that the horizontal vectors for Hi in points u € Q are uj-1(X)(u) - (x(u) for all X € n. Thus, the curve t h-> Fl^1(X)(u)-exp(-tX) must be the horizontal lift of c to Q. Now, the induced parallel transport of an element [u, exptX] € S along c is given at time s by [Fl" ^(u),exp(£ — s)Xj and it reaches exactly the point 0(c(t)) in the canonical embedding of M into S at time s = t. But this exactly means that for each X € n the curve t i—> [u, exp tXj is the development of the projection of the flow line through u of the constant vector field ui~l(X) € X(Q). Since the allowed developments for curves in Ca have the form t i—> [u, exp tXj for u € Q and X £ A, we have proved the first part of: 1.3. Proposition. Let (p : Q —> M,uS) be a parabolic geometry of type (G,P) and let A C n be a Go-invariant subset. (1) The geodesies of type Ca on M are exactly the projections of flow lines of the constant vector fields ui~l(X) € X(Q) with X € A. (2) Let (p' : Q' —> M',ui') be another parabolic geometry of type (G,P), let ip : Q —> Q' be a morphism of parabolic geometries covering ipo : M —> M', and let c : I —> M be a smooth curve. Then c is a geodesic of type Ca if and only if Lp0oc : I ^ M' is a geodesic of type CA ■ Proof. The curve c in M is a geodesies if an only if c(t) = p o Fl" ^ (uj for some 148 ANDREAS ČAP, JAN SLOVÁK AND VOJTĚCH ŽÁDNÍK X € A and u € Q. Since ip*ui' = ui, we get p' o F\f~1(x) ( p(Fl" ^X\u)), which is well defined on some neighborhood U C n of 0. Choosing U sufficiently small, this becomes a diffeomorphism onto its image, thus giving rise to local coordinates on M. These are called normal coordinates for the Cartan geometry in question. Of course, in the setting of (1), we recover exactly the usual normal coordinates for affine connections on manifolds in this way. More information and a characterization of the normal coordinates can be found in [4]. We may rephrase our definition in terms of normal coordinates as follows: The geodesies of type Ca are those curves which are linearly parametrized straight lines through the origin with directions in A C n in some normal coordinates. Again, this generalizes the standard facts on affine connections. 1.4. Example. Let us mention four well-known examples of distinguished curves in parabolic geometries. (1) G = SL(m + 1,R), P is the stabilizer of a line in Rm+1. Normal parabolic geometries of type (G, P) are classical projective structures on m-dimensional manifolds. Generalized geodesies (of type C„) are exactly the geodesies of all connections in the projective class. They are determined by their 2-jet at one point as parametrized curves, but already determined by their direction in one point as unparametrized curves. (2) G = SL(m + 1,H), P is the stabilizer of a quaternionic line. This choice leads to almost quaternionic geometries (the complex version of which is dealt with in [1]). Again generalized geodesies are determined by their 2-jet at one point, but they form more complicated systems of curves than in the projective case, see [1]. (3) G = 0(p+l, q+l), P is the stabilizer of a null line. This leads to conformal pseudo-Riemannian geometries of signature (p, q). Here the (generalized) geodesies are the well-known conformal circles, which owe their name to the fact that for the homogeneous ON DISTINGUISHED CURVES IN PARABOLIC GEOMETRIES 149 model with signature (n, 0) one obtains all circles on the sphere. For general signatures, the geodesies in null directions, which behave similarly to the projective case, form an interesting subclass. (4) G = SU(p + P the stabilizer of a (complex) null line. This Hermitian analog of (3) leads to nondegenerate CR-structures of hypersurface type with signature (p, q). Here the Lie algebra is 2-graded and the geodesies of type Cg_2 are the well-known Chern-Moser chains. 2. Jets of distinguished curves 2.1. The bundles of C^-velocities Let us recall the natural bundles T£ of rth order fc-dimensional velocities on all smooth manifolds. By definition, TjTM = Jq (Rfc, M); so this is the bundle of r-jets of parametrized fc-dimensional (singular) submanifolds in M. In particular, r-jets of curves are elements in T[M. The action of all diffeomorphisms of M on T£M is defined by jet composition. Let us consider a category of Cartan geometries of fixed type (G, P) and a class of generalized geodesies Ca, for a Go-invariant subset A of n. Then the jets of distinguished curves of type Ca form a natural subbundle T£ C T[ on parabolic geometries of type (G, P). Clearly, T£ is a well defined functor, see Proposition 1.3(2) above, however its values are not smooth bundles in general, see the examples below. In the cases with Go-invariant subsets A C n we call the latter functors the bundle of rth order velocities of geodesies of type Ca- Our next goal is to prove that there always is a finite order r for which the entire geodesic is completely determined by a single value in T£ . 2.2. Jets of curves on G/P Using Cartan's space S, the development of curves defines a bijection between smooth curves c : / —> M defined on some neighborhood / of 0 € M such that c(0) = xq, and smooth curves to G/P which map 0 to o = eP. Of course, this bijection is compatible with taking jets in xq, i.e., two curves have the same £-]et in xq if and only if the corresponding curves in G/P have the same £-]et in o. By definition, this bijection also respects generalized geodesies of any type. Thus, to prove that geodesies of some type Ca are determined by some jet at one point, it suffices to consider the homogeneous model G/P and the point o. We start by considering A = n (which of course provides an estimate for any A C n). Thus, we have to study the curves cb,x (t) = bexp(tX)P, with be P and X e n, cf. 1.2. Since &exp(tX) = exp(i Adi, X)b we see that cb>x (t) = exp(i Adf, -X)P. For any two curves c(t) and d(t) in G, there is a uniquely determined curve u(t) in G such that c(t) = d(t)-u(t). The projections of c(t) and d(t) to G/P coincide if and only if u(t) € P for all t. Thus the curves cbl>Xl and cb2>X2 coincide if and only if the uniquely determined curve u such that exp(i Ad6l Xi) = exp(t Adb2 X2)-u(t) (1) has values in P. Since exp is analytic, the curve u must be analytic too, and hence it has values in P if and only if all derivatives uW(0) = ^r|0w are tangent to P. To formulate this precisely, we use left logarithmic derivative 5u : R —> g of the curve u : R —> G, see e.g., [10, p. 39]. In fact 5u : TR = Ixl^g, 5u(t) = T\u{t)-i o Ttu, but we shall 150 ANDREAS CAP, JAN SLOVAK AND VOJTECH ZADNIK identify the linear map 5u(t, ) : R —> g with its value at the unit 1 € TtR. Since knowing 5u is equivalent to knowing Tu, the following lemma is a simple observation. Lemma. For each order k € N we have JQCbl>Xl = JQCb2>X2 if and only if the derivatives (<5«)W(0) lie in p for all i < k - 1. 2.3. Some technicalities In order to compute the derivatives of 5u from formula 2.2(1), we can use the Leibniz rule for the left logarithmic derivative, 5(f-g)(x) = 5g(x) + Adg{x)-i 5f(x), see [10, p. 39], so it remains to compute the left logarithmic derivative of the curve t I—> exptX. For later use, we shall compute this expression with an arbitrary curve Y : R —> g instead of the line tX. By definition, the logarithmic derivative 5(fog) of the composition of two smooth maps / : M —» G, g : N ^ M is given by 5(fog) = (Sf)oTg. Thus, the key ingredient is the formula for <5(exp) : Tg —> g. The proof of this formula for the right logarithmic derivative in [10, p. 39] can be easily adapted to our case, leading to <5(exP)(y) = £r=o(^ad(->T- This proves: Lemma. Let Y : R —> g be a smooth curve with derivative Y' : R —> g. Then S(eXpoY)(t)=J2j^Yy^(-Y(t)r-Y'(t). The first terms in the formula for 5(expY(t)) read as Y'(t) - i[y(t),Y'(t)] + i[r(t), [Y(t),Y'(t)}} + .... Notice that if Y has values in n, then also Y' has values in n, and compatibility of the grading of g with the Lie bracket implies that at most k of these terms may be non-zero for |fc|-graded g. Thus, for example, 5(expY(t)) = Y'(t), if k = 1, 5(expY(t)) = Y'(t) - ±[Y(t),Y'(t)], if k = 2, 6(expY(t)) = Y'(t) - l[Y(t),Y'(t)] + [Y(t),Y'(t)]], if k = 3. On the other hand, if Y(t) = tp(t)Y for some fixed Y € 0 and a smooth function ip, then [Y(t),Y'(t)] = 0 and hence we always get 6(eXp 1, (Su)^(t) = (ad(-Adbl Xi))J(<5u(f)). Proof. Let us start with the first order derivative, so we have to prove (5u)'(t) = [<5«(i), Adbl Xi]. To do this, we have to compute the derivative of t 1—> Adu^-i : R —> GL(fl). Clearly, ft{t ^ Adu{t)-i) = (TAdoTv)(u'(t)), where v is the inversion in G and Ttu = u'(t). First, we will express Tgv and Tg Ad in general. From pg o v o \g = v we have Tg-ipg o Tgv o Te\g = Teu, thus Tgv = —Tepg-i o TgXg-i. Similarly, Ad oAg = Adg o Ad implies Tg Ad oTeXg = Adg oTe Ad, so Tg Ad = Adg o ad oTg\g-i. Altogether, -^Adu(t)-i = (Adu(t)-i oadoT\u(t)) o (-Tpu(t)-i oT\u(t)-i)(u'(t)). Since Adg = Te(\g o pg-i) and 5u(t) = T\u^-i o u'(t) the latter expression equals (-Adu(t)-i oadoAdu(t))((5u(t)). Thus, (Su)'(t) = Adu(t)-i [Adu(t) 5u(t), Adb2 X2] = [5u(t), Adu(t)-i Adb2 X2] and substituting Adu^-\ Adb2 X2 = Adbl X\ — 5u(t) from 2.3(2) the claim follows. Now, let i > 1 and assume that the formula is valid for all orders less then i. Then (5u)^(t) = ^|t(ad(-AdblX)J-1<5M(t)) and since (ad(— Adbl X))x~l is a linear map and we have computed (5u(t))' already, we arrive at (5u)^(t) = ad(- Adbl X)l-1{5u{t))' = ad(- Adbl X)l5u(t), which is the required formula. □ Let us notice that we have also derived the more general formula for the derivative of Adu(t)-i Y(t) with Y : R —> n. From the proof above we conclude ^|t(Ad„(t)-iy(t)) = Ad„(t)-i y'(t) - [5u(t),Adu{t)-i Y(t)]. (1) As a simple consequence of this lemma, we can prove that any geodesic is determined by a finite jet at one point: 2.5. Proposition. Let g be a \k\-graded Lie algebra, and let A C n be any Go-invariant subset. If two geodesies of type Ca have the same (k + 2)-jet at one point, then they coincide. 152 ANDREAS ČAP, JAN SLOVÁK AND VOJTĚCH ŽÁDNÍK Proof. As we have noticed in 2.2 it suffices to consider A = n, and we can complete the proof by showing that two curves cbl>Xl and cb2>X2 coincide if they have the same (fc + 2)-jet in 0. Denoting by u : R —> G the curve determined by equation 2.2(1), Lemma 2.4 tells us that (<5«)W(0) = (ad(— Ad^ Xi))z(<5«(0)). By Lemma 2.2, the assumption on the (fc + 2)-jet in 0 implies that ad(-Ad6l X1)i(Su(0)) £ p for all i < k + 1. Since &i £ P, we may hit this element with Ad^1, and the result remains in p. Putting X = X\ £ n and Z = Adb-i <5«(0) e p we conclude that ad(-X)J(Z) £ p for all i = 1,..., k + 1. Since Z £ p = go © ... ffi Qk and —X £ n = Q-k ffi • • • ffi fl-i, compatibility of the bracket with the grading implies that ad(—X)Z(Z) € 0 ... 0 flfc_i. Putting z = k + 1, we see that ad(— X)k+1(Z) has to lie both in n and in p, so it must be zero. This implies that (5uY(0) = 0 £ p for all £ > k + 1, and thus cbl'Xl = cb2'X2 and the claim follows. □ Let us remark at this point that the estimate r = k + 2 on the jet needed to pin down a geodesic is not at all sharp and we will improve it heavily depending on a particular choice of the class of geodesies. 2.6. Distinguished curves in a given direction The most natural way to approach the problem of distinguished curves is usually to fix a point x £ M and a tangent vector £ £ TXM, and look for geodesies emanating from x in direction £. Given a Go-invariant subset A £ n, the basic question then is how many geodesies of type Ca pass through x in direction £. Of course, it may happen that there are no such geodesies. As before, one may restrict the discussion to the point o in the homogeneous model G/P. Since the above question is perfectly geometric, the answer for a tangent vector £ £ Ta(G/P) = fl/p will only depend on the P-orbit of £. Clearly, there is at least one geodesic of type Ca in direction X, if the image of A in g/p meets the P-orbit of £. Otherwise put, if X £ n C fl is the unique element such that £ = X + p, then there is at least one geodesic of type Ca in direction £ if Adb(X) £ A for some b £ P. Second, suppose that A, B C n are Go-invariant subsets, and that for each X £ A there is an element b £ P such that Adf, X £ B, and vice versa. (Of course, this is a very restrictive condition, since we are using Ad^, which does not leave n invariant, but it happens in interesting cases.) Then this gives rise to a bijection between the sets Ca and Cb of curves in G/P, and consequently, geodesies of type Ca coincide with geodesies of type CB- Fix a Go-invariant subset A C n and an element X £ A, and consider the tangent vector £ = X + p £ Ta(G/P). Clearly, ce'X(t) = exp(tX)P is a geodesic of type CA in direction X, and any other geodesic of that type can be written as cb,Y with b £ P and Y £ A. It is a general fact, see [3, 2.10], that there are unique elements bg £ Go and Z £ p+ such that b = bg exp(Z) = exp(Adf,0 Z)b$. From the definition of distinguished curves, we conclude that cb0expZ,Y _ cexp(Ad(,0 Z),AdbQ Y and Adb0 Y £ A. Hence any geodesic of type Ca may be written as cexp^)'y for Z £ p+ and Y £ A. Hence we conclude that the set of geodesies of type Ca in direction £ = X+p can be equivalently described as {ceMZ),Y ! Zep+jY£ AAd^w Y = X}. Passing to a general curved geometry via developments as before, we obtain ON DISTINGUISHED CURVES IN PARABOLIC GEOMETRIES 153 Proposition. Let (p : Q —> M,uS) be a Cartan geometry of type (G,P), x € M a point, £ € TXM a tangent vector, and let A C n be a Go-invariant subset. Then there is a geodesic of type Ca through x in direction £ if and only if there are elements u € p~1(x) C Q and X € A such that £ = Tup-uj~l(X). Moreover, for any such pair (u,X), one obtains a bisection between the set of geodesies of type Ca through x in direction £ and the set {ce*v(z)>Y Z e p+,Y e A, AdCXF(z) Y = X} of curves in G/P. This bisection is compatible with finite jets in 0 in the obvious sense. Finally note that the curves ce*p(zi),Yi and ce*p(z2),Y2 j-^yg the same i?-jet in 0 respectively coincide if and only if the same is true for ce'Yl and cexp(Zl) exp(z2),Y2^ an(j we can write exp(Zi)-1 exp(Z2) as exp(Z) for some Z € p+. Hence we conclude that if for some I and each X € A we can show that any curve cexp(^)'y with Y € A which has the same i?-jet in 0 as ce,x must actually equal ce,x, then this implies that any geodesic of type Ca is uniquely determined by its i?-jet at a single point. 2.7. The |l|-graded case For irreducible parabolic geometries we easily reach a complete description. So we assume g = fl-i ffi flo © fli and A = n. The main simplification in the |l|-graded case comes from the fact that in this case p+ acts trivially on g/p, so the P action on this quotient factorizes over Go- In particular, for Z € p+ = gi and Y € n = we get AdCXF(y) Y = Y, so in view of Proposition 2.6 it remains to compare the curves ce'X and cexp(2)'X with Z € gi. For the corresponding curve u, we obviously get Su(0) = —[Z,X] — \ [Z, [Z,X]\. For the two curves having the same two-jet in 0, we must have (6u)'(p) = -[X1,6u(p)] = [Xi,[Z,Xi]] + i[Xi,[Z, [Z,Xi]]] ep, and thus [XU[Z,X{\] = 0. But this implies [Xu [Z, [Z, Xi]]] = [Z, [Xu [Z, Xi]]] = 0, and so (<5«)W(0) = 0 for all i > 2. Thus, we have proved: Proposition. Each generalized geodesic in an irreducible parabolic geometry is uniquely determined by its 2-jet at one point. 2.8. The distinguished jets Using the procedures above, one may compute explicitly the jets of all geodesies of type Ca- For the sake of simplicity, we shall restrict ourselves again to the case of |1|-graded Lie algebras. Thus, the value in T'l{G/P) over the origin will always determine a geodesic completely, and we shall compute explicitly the algebraic description of the standard fibers of T'cA- Understanding the higher jets of geodesies is an interesting problem, however the computations grow quickly out of hand. Let us describe all distinguished curves in normal coordinates through the origin, i.e., we have to represent each geodesic in the form 11—> exp(Y(t))P for a smooth curve Y : R —> with Y(0) = 0. This means that rather than with formula 2.2(1), we have to deal with exp(Y(t))-u(t) = exp(tAdexpZX) for Z £ gi and X e AC g_1. 154 ANDREAS ČAP, JAN SLOVÁK AND VOJTĚCH ŽÁDNÍK Using the results in 2.3 and formula 2.4(1), straightforward computations yield 5u(t) = X + [Z,X] + \[Z, [Z,X]] - Adu(t)-i(y'(t)), (5u)'(t)= [X + [Z,X] + \[Z, [Z,X]],Adu{trlY'(t)] - Adu{t)-! Y"(t). The requirement (<5«)W(0) € p, for i = 0,1 immediately implies r'(o) = x, Y"(0) = [X, [X,Z]]. Now it is easy to describe the standard fiber of T$ as follows. The standard fiber of T( is the smooth manifold Jq(R, 0-i)o> which is naturally identified with g_i x Q-\. Hence the standard fiber of T$ is a subset in g_i x fl_i, which we have computed to be S= {[[x&z]]] \ xeA,zeQl}. Recall that A is assumed to be Go-invariant, but not necessarily a linear subspace. A good example in which it is not a subspace is given by the null cone in W+q in the setting of Example 1.4(3). In that case, [X, [Z,X]] happens to be a multiple of X for each Z, which corresponds to the fact that geodesies in null directions are conformally invariant up to parametrization. For every parabolic geometry of type (G, P), there is the standard embedding i : P —> G^j = inv Jg (Rm, Wn)o, see e.g., [15, 17]. Further, the action of the structure group G^ on Jq (R, Rr™)o transforms to the action on g_i x fl_i, whose restriction to the subgroup i(P) keeps the subset S invariant because the set Ca of all geodesies is P-invariant. In fact, the action of Go obviously is the product of the adjoint actions on g_i x fl_i, while the action of P+ = expgi comes by the very definition of the curves from the left shift by the elements expl^, W € fli- Since Q\ is an abelian subalgebra, the action by exp W is given by exp^-[^',] = [Y"+[Y^'[Y^W]]]- Hence we obtain an alternative description of the standard fiber as the P-orbit of the Go-invariant subspace A x {0} C g_i x 3. Reparametrizations In this section we shall generalize our basic question to: When are two distinguished curves equal up to a change of parametrization? Thus we shall discuss the nonparamet-rized geodesies together with their preferred parametrizations. 3.1. Technicalities In order to deal with this question, we have to modify our basic equation 2.2(1). The answer is positive if and only if there exist mappings u : R —> P and ip : R —> R such that exp(cp(t) Adbl Xi) =exp(t Adb2X2)-u(t), (1) where ip is a local reparametrization, i.e., we require 1 and at every tel, with the notation as above i (6u)W = ^+1)X1 + EM)" (E cjia(^)r... (^«))°«) (adXl)fc((J1))ai... (^s^)as. Now, the differentiation of this formula and substitution from 3.1(3) means that we perform the last derivative on one of the 92's in the individual terms in the formula, or we attach a new ip to the existing terms which is differentiated only once. But this is exactly how all splittings of i + 1 (distinguishable) hits of k (indistinguishable) targets are obtained from the answers to the same question for i derivatives and k or k — 1 targets. Either the last hit has been to some existing one among k targets, i.e., we use the answer with i hits and k targets, or we have had to introduce a new target which was hit once, i.e., we used the answer with i hits and k — 1 targets. It is probably hard to deduce general results for all parabolic geometries and all classes of distinguished curves from this formula, but let us see how to use it in more specific situations. 156 ANDREAS ČAP, JAN SLOVÁK AND VOJTĚCH ŽÁDNÍK 3.3. Irreducible parabolic geometries We are going to give a complete answer to our question for |l|-graded algebras g. In order to decide when two distinguished paths cbl'Xl, cb2,X2 parametrize the same curve we have to compute explicitly the consequences of (<5«)W(0) G p in relation to the necessary and sufficient conditions for the solution of the given problem. At the same time we shall get a complete and explicit description of the reparametrizations. Lemma. With the notation as above, 5u(0) € p if and only if p'(0)X1=X2. (1) If 5u(0) £ p, then (<5«)'(0) € p if and only if ^Xl = [Xu[XuZ]], (2) and if i > 2 and (Su)^(0) € p for all j < i, then (<5«)W(0) € p if and only if ^+D(0) = ^^Qfr, for alii > 2. (3) Proof. Since our algebra g is |l|-graded, all iterated adjoint actions by X\ on <5«(0) vanish if the order is more than two. Thus only terms with k < 2 in Lemma 3.2 may survive and the general formula for i > 1 reads (5up(0) = v3(i+1)(0)Xi - v3(i)(0)[Xi)(Ju(0)] + Indeed, this can be either proved by inserting into the general formula from Lemma 3.2 or directly by induction. Next, recall 5u(0) = (*) = where A = V'(0) - |£$V(0), B = p(0), C = D = 1. In particular, the solution with R of the latter form are again geodesies if and only if there is Z € gi such that [X, [X, Z]] = X. Proof. It remains to prove the second statement. Obviously we may restrict ourselves to the case when p(0) = 0. Then each p satisfies all conditions from Lemma 3.3, provided there is a suitable Z for (2). □ Reparametrizations of the above type are called projective, see [2], where they are obtained as solutions of the Schwartzian differential equation p'" = | . Corollary. Suppose that g is \ l\-graded. Then the curves ce,Xl and cexpz,x'2 parametrize the same unparametrized geodesic if and only if there are a ^ 0 and b such that X2 = aXi and [X2, [X2, Z]] = bX\. This is equivalent to the existence of the projective local reparametrization p which is uniquely determined by the initial condition p{0) = 0, p'(Q) = a, and p" (0) = b. 158 ANDREAS ČAP, JAN SLOVÁK AND VOJTĚCH ŽÁDNÍK 3.5. Example. In the following examples we use the obvious fact that in the case of a l|-grading, elements of P of the form exp(Z) for Z € fli act trivially on Ta(G/P) = g/p. Then the P-action on this space factorizes over Go- (1) Conformal Riemannian structures correspond to G = 0(p +1,5+1) and the parabolic subgroup P as in 1.4(3). In an appropriate matrix representation, the grading of the Lie algebra g has the form fl-1 = {[! J.J] \^^P+g}^o = {[lll]\Aeo(P,q),aeR}, fll = ([oo-jVj \zew+q*\. I Lo 0 0 J > Here J is the matrix defining the standard pseudo-metric of signature (p, q) on M.p+V = 0-1- A direct calculation shows that [X, [X,Z]} = -2Z(X)X - \ \X\\2JZ\ where ||X||2 = XfJX and Z(X) = ZX is a real number. Obviously, the space g_i splits into three different orbits of the action of Go according to the sign of ||X||2. The orbit of null-vectors is of particular interest, since [X, [X,Z]] = —2Z(X)X in that case. This just means that all distinguished curves with the common tangent null-vector differ by a reparametrization; this recovers the classical result that the null geodesies of the metrics in the conformal class together with the class of projective parametrizations are invariants of the conformal structure. Of course, these curves will have their tangent vectors null in all their points. For all tangent vectors which are not null, the second derivative may be chosen arbitrarily. So that the standard fiber S in 2.8 has arbitrary entries in the bottom row if X is not null, but only multiples of X if X is null. On the other hand, there always is an element Z € gi such that [[Z, X], X] = X, so all geodesies carry a natural projective structure. (2) Almost Grassmannian structures. In this case, G = SL(n+m, R) and the parabolic subgroup P is the stabilizer of R™ C R™+r™, so it consists of block upper triangular matrices with two blocks of sizes n and m. On the infinitesimal level, i-i={[I§] |XeR-}, fl0 = {[i°B] |tr(A)+tr(B) = 0}, si = {[8 o] \zeRnm}. First, it is easy to see that the subgroup Go consists of block diagonal matrices, and its action on g_i is given by X n TXS-1, (S,T) € Go- Thus two elements of g_i lie in the same Go-orbit if and only if they have the same rank. Further, the computation of the iterated bracket yields [X, [X, Z]] = —2XZX. In particular, the choice of the pseudoinverse matrix Z = X^ provides always a multiple of X, and so all generalized geodesies enjoy the distinguished projective structure. If the rank of X is one, then we may choose X to be the matrix with the left upper element x\\ = 1 and all other 0. Then [X, [X,Z]] equals to z\\X for all Z and so this behavior must be shared by all matrices of rank one. Thus, the directions corresponding to rank one matrices behave like null directions in pseudo-conformal geometries. The other extreme is that X has maximal rank. Then one gets a lot of freedom in the available second derivatives of the ON DISTINGUISHED CURVES IN PARABOLIC GEOMETRIES 159 curves. The case that all elements of g_i are possible second derivatives occurs only if m = n and X has rank n. (3) Projective structures are the special case n = 1 of Example (2) above. In this case, the rank of X ^ 0 is always one. More explicitly, the product ZX is a real number, so the bracket [X, [X, Z]] is always a multiple of X. From this it follows that all unparametrized distinguished curves are determined by the direction at a given point. This agrees with the classical definition of a projective structure as a class of affine connections sharing the same unparametrized geodesies. All such connections are parametrized by smooth one-forms on the base manifold and they correspond to the Weyl connections defined in [4]. 4. More refinements In this section we improve the estimates on the jet at a point needed to pin down a geodesic for geodesies of certain types. The most general result is Theorem 4.3 but since the proofs of these results are a bit technical, we prefer to discuss two simpler special cases first. 4.1. Curves tangent to T_1M Let M be any manifold equipped with a parabolic geometry of some fixed type (G, P). A (generalized) geodesies with development of the form cb>x emanates in a direction in T~lM if and only if X € fl_i. Thus we are dealing with distinguished curves of type Cg_1 and from Proposition 1.3 we see that they will be tangent to the distribution T~lM at all points. To discuss geodesies of type C§_1, by Proposition 2.6 we have to fix X € fl_i and study the curves cexp(z)>y for Z e p+ = fli ®.. -S)Qk and Y e fl_i such that Ad(exp(Z))(Y) = X. Since Y € fl-i we get Ad(exp( Z))(Y) = Y for any Z, so we have to consider all curves of the form cexp(z)>x with Z € p+. By [3, 2.10] we get a nicer presentation of exp(Z). Namely, there are unique elements Zt € gt for i = 1,..., k such that exp(Z) = exp(Zi) • • -exp(Zfc). Since Ad(exp(VK)) = ead(w-r) for each W € fl, we get AdexpZ x = T,iu...,ik ra(adzi)J1 • • • (*dz*YkX- Moreover, since X € g_i a summand in the right-hand side lies in Qg if and only if ix +2i2 + ... + kik = £+ 1. We need another observation for the proof: Suppose that Y € Q is any element. The Jacobi identity reads as adx ° ady = ad^x,Y] + a(^Y ° adx- Inductively, this implies that ad^ o ady can be written as a linear combination of terms of the form adad^(-y^ o ad^ with 0 < i, j and i + j = n. In particular, if ad^1 (Y) = 0 for some £ > 0, then for each n > £ there is a linear map

x. Given Zi,..., Zk define W := Ad(exp(Zi) • • • exp(Zk))(X) — X € p. From the above discussion we see that W = EJ1,...,Jfe ^(adZ^ • • • (adZky*X, (1) where the sum is over all ..., i^) such that 0 < «i +2«2 + • • -+kik < fc + 1. Considering the curve u(t) associated to ce'X and cexp(zi)---exP(zk),x ^ equation 2.2(1), we see from 2.3 that <5«(0) = —W and Lemma 2.4 implies that (<5«){j)(°) = adx(w)-Consequently by Lemma 2.2 proving the result boils down to showing that adlx(W) € p for alH < k implies ad^- (W) € p for all ieN. For each £ = 1,..., k define W[ to be the sum of those terms in the expression (1) for W for which all ij with j > £ are zero, and put W'l = W — W[. In particular, we have W£ = 0, i.e., = W. Claim. // adx(W) € p for all i < £, then for each j < £ we have ad^1 (Zj) = 0, and for each n > £ we get adx{W't) € p. We prove this claim by induction on £. If £ = 1, we know that adx(W) € p. Looking at formula (1) for W and taking into account that X € g_i we see that adx{W) € p implies (and is actually equivalent to) [X, [Zi,X]\ = 0 and thus to ad'x(Zi) = 0. Hence it remains to show that adx(W{) € p for all n > 1. By definition, W[ = Y!l=l h adk x-Thus adx(W[) G p is equivalent to ad^ o adxZl X = 0 for i < n. From above we know that adx{Zi) = 0 implies that ad^ o ad.^ = <£>oad^_1, so inductively we conclude that ad^ o adZl = tp o ad^_z+1 o adZl for some linear map tp and by assumption n — i + 1 > 0. Hence applying this element to X we get tp o adx~l+2(Zi) which vanishes since n — i + 2 > 2. This completes the proof of the case £ = 1. Assume inductively that £ > 1 and we have proved the result for £ — 1. Given that adx(W) € p for all i < £, we by induction conclude that ad:'x1(Zj) = 0 for j = 1, ...,£— 1. Moreover, we know by induction that ad^(VK) € p implies ad^f(l4y^1) € p. By definition of the only term in adx{W'^_l) which does not automatically lie in p is adx([Ze, X]), so we conclude that ad^1^^) = 0. Hence it remains to show that adx{W[) € p for all n > £. Since we know by induction that ad^(T47£/_1) € p, it suffices to consider adx(W^ — W^-^). Now from the expression (1) for W we conclude that W't ~ W[_x = Y,iu...,it TO(ad^i)21 • • • (adZty 0 and i\ + 2*2 + ... + £ie < k + I. Obviously, adx(W^ — € p is equivalent to vanishing of ad^ oad^ o - ■ ■ o a(T£ X for all multi-indices (ii,..., if) such that ii + 2*2 + ... + £it < n. Since ad^1 (Z,-) = 0, we see from above that ad™ o adZj =

Inductively we conclude that for m > jij we get ad^J o adlz. = ip o ad^ JZ3 for some linear map ip. Thus we conclude that ad^ o adlZi o ... o ad^f = tp o adx n 2l'2 "' ^ o ad^f, ON DISTINGUISHED CURVES IN PARABOLIC GEOMETRIES 161 and by assumption n — i\ — 2«2 — ... — £{ig — 1) > £. Thus applying the right hand side to X, we obtain adrx(Ze), and by construction r > £ + 1, so this vanishes. Hence the proof of the claim is complete. But taking the claim in the case £ = k, we see that adx(W) € p for all z < k implies that ad^(W^) € p for all n > k. Since we have observed above that Wk = W, this completes the proof. □ 4.2. The case A = g_fe The other extreme class of geodesies on a manifold M equipped with a parabolic geometry of type (G,P) with |fc|-graded g is provided by the generalized geodesies of type Cg_k. Of course, for a point x € M and a tangent vector £ € TXM one must have £ € TXM \ T~k+1M in order to have a nontrivial geodesic of type Cg_k in direction £. On the other hand, this condition is not sufficient for such a geodesic, and the directions of these geodesies usually form a smaller cone in each tangent space. An important special case is parabolic contact geometries, i.e., those geometries corresponding to |2|-gradings, such that g_2 has dimension one and the bracket g_i xg_i —> g_2 is nondegenerate. These geometries always have an underlying contact structure. In these cases geodesies of type Cg_2 always exist for all directions in TM \ T~lM. A very well-known instance of this type of generalized geodesies is provided by the Chern-Moser chains on hypersurface type CR-structures. A slightly more general example of this type was studied for 6-dimensional CR-structures of codimension 2, in [16]. Let us recall that reparametrizations of the form ip(t) = ^fqrf| with A ^ 0 and AD — BC = 1 are called projective. Theorem. Each generalized geodesic of type Cg_k in a parabolic geometry of type (G, P) corresponding to a \k\-grading on g is uniquely determined by its 2-jet at a single point. Moreover, if two of such curves coincide up to parametrization, then this reparametriza-tion is projective. Conversely, given a generalized geodesic of type Cg_k corresponding to (u, X) € Q x g-k, every projective change of parametrization defines a geodesic of the same type if and only if there exists a Z € g^ such that [X, [X, Z]] = X. Proof. From 2.6 and 4.1 we know that for each X € g_k we have to compare ce>x to all curves of the form cb,Y with b = exp(Z\) ■ ■ ■ exp(Zk) for Z{ € (jj, Y € g~k and Ad(&)(y) = X. The last condition immediately implies that Y = X. Expanding W = Ad(b)(X) — X as in equation 4.1(1), we conclude that if this expression has trivial component in g_k+i, then [Z\,X] = 0. Hence we may omit all terms in the expansion for which i\ is the only nonzero index. Vanishing of the component in g-k+2 then implies \Zi,X\ = 0, so we may omit terms in which only i\ and 22 are nonzero. Inductively, we get [Z^, X] = 0 for all £ = I,... ,k — I. Hence we conclude that 5u(0) = -[Zk,X]-\[[Zk, [Zk,X]]. Now (<5u)'(0) € p implies [X, [Zk,X]] = 0 and so (<5u)'(0) = 0 exactly as in 2.7. Concerning reparametrizations, we may adapt the proofs of Lemma 3.3 and Proposition 3.4 along the same lines. Using the notation from there, the condition 5u(0) € p implies X'2 = tp'(fi)Xi and moreover [Zg, X2] = 0 for all £ < k — 1, inductively as above, and this is the only difference to the |l|-graded case. Further, (<5«)'(0) G p if and only if ip"(0)Xi = ip'(0)2[X2, [X2,Zk]\ and we finish the proof exactly as in the |l|-graded case. □ 162 ANDREAS ČAP, JAN SLOVÁK AND VOJTĚCH ŽÁDNÍK More generally, let us consider generalized geodesies of type Cg_ with arbitrary j. Geodesies of this type are always curves with tangents in T_JM and they emanate from a given point in M in certain directions in T_JM \ T_J+1M. 4.3. Theorem. Each generalized geodesic of type Cg_. in a parabolic geometry of type (G, P) with a \k\-graded g, 1 < j < k is uniquely determined by its r-jet at a single point provided that rj>k + l. Proof. This is a combination of the proofs of Theorem 4.2 and of Proposition 4.1 with minor generalizations, so we just outline the basic steps: For X € fl_j we have to compare ce'x to cb'Y for b = exp(Zi) • • • exp(Zk) with Z{ e fli and Y e Q-j and Ad(b)(Y) = X. This immediately implies Y = X, and we put W = Ad(b)(X) — X and expand this as in 4.1(1). The proof boils down to showing that &dx(W) € p for i < r implies the same result for all i. As in 4.2, W € p implies that [Zg, X] = 0 for all £ < j, so in the notation of the proof of Proposition 4.1 we obtain Wj_1 = 0. The analog of the claim in the proof of Proposition 4.1 is that if &dx(W) € p for all i < £, then for each s < £ and m < (s + we get a,dsx1(Zm) = 0, and further ad7x(W'-^+l^_l) € p for all n > £. This is proved by induction using the same arguments as in 4.1. For £ = r — 1, we obtain jr > k + 1, and as in 4.1, W'k = W, and we conclude that &dx(W) € p for alii < r implies the same property for all i, as required. □ The following two examples expose the diversity of the possible behavior of various classes of distinguished curves in specific parabolic geometries. All claims may be checked by direct computations following the results above and their more detailed version may be also found in [21]. 4.4. Example. Let us briefly illustrate the general results in the simplest cases of parabolic contact structures, so we are dealing with |2|-gradings such that g_2 is one-dimensional and the bracket g_i x g_i —> g_2 is nondegenerate. As we have mentioned in 4.2, we get in each direction outside the contact subbundle geodesies of type Cg_2 which generalize the Chern-Moser chains for CR-structure. From 4.2 we know that they are determined by their two-jet in a point as parametrized curves, and it follows that they are uniquely determined by their direction in one point up to parametrization, by dimension reasons. Moreover, each such geodesic carries a natural projective structure of distinguished parametrizations. Apart of these types of generalized geodesies, there are several other possibilities for nonequivalent types of geodesies as we may already observe at the simplest example of G being a real form of SL(3, C) and P the Borel subgroup. (1) G = SL(3,R). The corresponding geometries are the Lagrangian contact structures on 3-dimensional manifolds, i.e., three dimensional contact structures endowed with a decomposition of the contact subbundle into a direct sum of two line subbundles, cf. [18]. Geometrically, there are four different classes of tangent vectors. First, we have vectors tangent to one of the two subbundles (two classes); then there are the remaining vectors in the contact subbundle, and finally those outside of the contact subbundle. The subgroup P consists of all elements of G which are upper triangular, so on the Lie algebra level, we obtain n as the subalgebra of strictly lower triangular matrices, with the two entries directly below the main diagonal corresponding to g_i and the entry in ON DISTINGUISHED CURVES IN PARABOLIC GEOMETRIES 163 the lower left corner corresponding to Q-2- The action of the subgroup Go rescales each entry of a matrix in n by a nonzero factor, so the Go-orbits in n are determined simply by the nonzero entries of a matrix. First, there are two canonical invariant subspaces in g_i which correspond to the Lagrange subspaces of the contact distribution. They are A\ = { [* o ol | and A2 = ooo >, respectively, where the star denotes a nonzero entry. Generalized geodesies of these types exist exactly in directions tangent to one of the two line subbundles, so the two classes are disjoint but have the same properties. In both cases they behave like null-geodesics in conformal geometry, i.e., each such curve is determined by its 2-jet at one point, and with a given tangent vector there is a 1-dimensional fam- [o*ol ily of parametrized generalized geodesies determined by elements of the form I § § § J and |^o o * j G p+, respectively. Moreover, all curves from this family coincide up to a projective reparametrization. For 4 = g_i we get directions in the contact distribution. From 4.1 we know that such curves are determined by their 3-jet at one point. There is a 3-dimensional family of parametrized generalized geodesies (corresponding to all elements in p+) sharing a given tangent vector, which is not tangent to one of the two line subbundles. Admissible reparametrizations are the projective ones, so the dimension of the space of unparametrized generalized geodesies with the common direction in T~lM but outside of the Lagrange subspaces is two. Now we discuss the curves emanating in directions which do not belong to the contact distribution. For A = g_2 we obtain the analog of CR-chains as described in Theorem 4.2. Besides these chains, there are another curves going in all directions except those in the contact distribution; this class of curves corresponds to the generic choice of A = < * o o \ >. Curves of this type are determined by a 2-jet and to any tangent vector there is a 3-dimensional family of generalized geodesies. This set is parametrized by elements of p+. In contrast to the previous cases, there are no two curves with the common tangent vector, which would be the same up to a reparametrization. So here only affine reparametrizations are allowed. The two Go-orbits in n, which have not yet been mentioned are < * o o \ > and ooo >. Any element of either of these can by mapped to g_2 by some Adb with b € P, and vice versa. Hence from 2.6 we know that these lead to the same curves as A = Q-2, and thus the discussion is complete. (2) G = SU(2,1). The corresponding geometries are nondegenerated strictly pseudo-convex 3-dimensional CR-structures. In contrast to the Lagrangian contact structures, there is no distinguished Go-invariant subset in Q-\, so the discussion is similar as above, but easier, so we skip the details. 4.5. Example. Let us finish the paper with the discussion of generalized geodesies in the so-called x-x-dot geometries (the name comes from the shape of the Dynkin diagram with crosses describing the corresponding parabolic subgroup in sl(4, C)). Such structures appear as correspondence spaces in classical twistor theory, and they are 164 ANDREAS ČAP, JAN SLOVÁK AND VOJTĚCH ŽÁDNÍK related to the geometric theory of ODE's. Let us consider the group G = SL(4, R) with the parabolic subgroup P which may be indicated as P J o * * * I \ 0 0 * * ( y lo o * *\ ) The following discussion may be also understood as a block-wise generalization of the discussion of the matrices in the example 4.4(1) which we shall call the 'x-x' case. The examples with more 'dots' in the Dynkin diagram and just two crosses over the first two nods on the left will behave quite similarly to the x-x-dot case. The Lie algebra g_ is described by block matrices of the form g_ where the blocks Xi, X\ generate the subalgebra g_i and X2 belongs to g_2. The r o zi z21 truncated adjoint action of an element exp o o zx\ € P+ is given by the formula Lo o o J r o o on n 0 0 L x2 Xt o J o oo x1 + Z1(X2) 0 0 X2 X1-z1X2 0 In accordance with the x-x case, there are two distinguished Go-invariant subspaces in g_i corresponding to the blocks x\ and X\, respectively. The generalized geodesies emanating in the appropriate directions of the distribution T~lM have the same properties as above. In particular, curves of this type are determined by a 2-jet but as unparametrized curves they are given by a direction. Parametrized geodesies of this type with the common tangent vector form a 1-dimensional family parametrized by the elements of the form ||^o and {[002l]| resPectively, where K = ||ooZiJ I Zi(Xi) = 0 j, briefly written as K = {Zi(Xi) = 0}. In the latter case, what really affects the 2-jet is the value Z\[X\) instead of Z\, which is why the quotient appears. Generalized geodesies with the generic directions in T~lM are determined by a 3-jet and to any tangent vector there is a 3-dimensional family of (projectively) parametrized geodesies described by elements of p+/K, where K = {zi = 0, Zi(Xi) = 0, Zi(X\) = 0}. The only contrast with the x-x case appears in the directions not belonging to T~lM. The analogy of chains, i.e., the curves from Cg_2, does not exhaust all directions out of 0 o ol^ Zi(X2) 0 0 K C fl_ X2 -ziX2 0 J J (at each point) according to the orbit of g_2 with respect to the truncated adjoint action of P. Obviously, the complement is formed by all elements of g_ such that vectors X\ and X'2 are linearly independent; this set is Go-invariant. Now, the discussion splits into two branches where the first one follows the x-x case, but the second one brings something new. Let us start with the directions given by chains. First, it is easy to verify that the sets of curves given by the invariant subsets A-i = < o o o \ } and Ao = \ a=i o o \ \ B J 1 U x2 aX2 o J J 1 U x2 o o J J are the same and both of these choices coincide with chains defined by A = g_2. Of course, all chains depend on 2-jets in one point. For any tangent vector of this type there is a 1-dimensional family of parametrized chains, described by the elements of g2/{Z2(X2) = 0}, all parameterizing the same curve. Besides the chains, there is a 3-dimensional family of generalized geodesies emanating in the same directions as chains from a given point, defined by the subset ---5J """"""j -™ ~B-2' """" ~.-"«^ the distribution T~XM but only a 4-dimensional 'cylinder' | ' ON DISTINGUISHED CURVES IN PARABOLIC GEOMETRIES 165 A = J ^1 o]}' This family is parametrized by the quotient p+/K, where K = {zi = 0,^1(^2) = 0, ^2(^2) = 0}. Curves of this type are also determined by a 2-jet and the admissible reparametrizations are affine. Finally, we fix a tangent vector which does not belong to T~XM and is not tangent to a chain. By analogy to the previous case, there are two disjunct classes of generalized geodesies emanating in such directions, but having rather different properties than above. The first class corresponds to the invariant subset A = {[HI]}, where X\ and X2 are supposed to be linearly independent (we assume this in the rest of the example). Curves of this type are determined by a 2-jet; they allow projective reparametrizations, and to the given tangent vector there is a 3-dimensional family of parametrized geodesies described by elements of the form j |^o 0 z\ J | Zi(X2) = 0 j. The last distinguished class of curves corresponds to the generic choice of A = {[|i JJ]}-Again, curves of this type are determined by a 2-jet and allow the projective class of reparametrizations. 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