DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS Proc. Conf. Prague, August 30 - September 3, 2004 Charles University, Prague (Czech Republic), 2005, 203 - 216 Equations and symmetries of generalized geodesies Boris Doubrov and Vojtech Zadnik Abstract. We are interested in equations of distinguished curves in general Cartan geometries. In this paper we present a way to construct equations for non-parameterized distinguished curves via symmetry algebras of model curves and Cartan's method of moving frame. We also discuss the correspondence, well-known in particular geometries, between maps preserving generalized geodesies of specific type and morphisms of the geometric structure. As examples we compute equations together with their symmetries for generalized geodesies in projective, projective contact and Lagrangean contact geometries. 1. Introduction For a Lie group G and a closed subgroup H C G, the Cartan geometry of type (G, H) on a smooth manifold M consists of the following data [13]: • a principal fiber bundle Q —> M with the structure group H; • Cartan connection u e with g to be the Lie algebra of G. Cartan geometry is called split if there is a (fixed) subalgebra n C g complementary to f) C g, the Lie algebra of H. The principal i^-bundle G —> G/H with the Maurer-Cartan form ujq e $11(G, g) is the flat (or homogeneous) model of Cartan geometries of type (G,H). Cartan geometries appeared first in the pioneer works of E. Cartan [2, 3] under the name of generalized spaces. One of his ideas was to generalize his moving frame method to submanifolds in Cartan geometries [4]. This works especially smoothly for curves, where the structure equations are automatically satisfied, and leads to the notion of distinguished curves. First, there are special types of curves in the homogeneous space G/H, namely, the orbits of one-parameter subgroups of G, known as homogeneous curves. This determines special classes of curves on all manifolds endowed with the structure of Cartan geometry of the same type via the notion of development [11, 15]. Explicitly, the curve on M is a distinguished curve if and only if it develops (at any point) into a curve of the form (1.1) hexp(tX)o = exp(t Adh(X))o for some h e H and X e g. In fact, any i^-invariant set C of curves in G/H, mapping 0 to the origin o = eH, leads to a well-defined set of curves on M which we call the distinguished curves of type C. This is the way how to distinct curves of different 2000 Mathematics Subject Classification. 34A26, 53B15, 53C30. Key words and phrases. Cartan geometry, moving frame, development, differential equations, symmetries. This paper is in final form and no version of it will be published elswhere. 203 204 boris doubrov and vojtěch žádník properties. Particular examples of such curves are geodesies in Euclidean, affine and projective geometries, conformal circles and null-geodesics in conformal geometries, chains in hypersurface CR geometries, and others. All these Cartan geometries are split and the mentioned types of distinguished curves can be specified, according to the notation above, by the condition X e A where A is a subset in n. In those cases one speaks about generalized geodesies of type A. One of the questions of this paper is whether the way from a given Cartan geometry to the family of distinguished curves of a specified type can be reversed, i.e., whether one can recover the whole geometry having just the family of distinguished curves. There are three major examples with affirmative and negative answers we have in mind. First, consider an affine geometry with geodesies as distinguished curves. Then it is well-known that there are many non-equivalent affine geometries having the same sets of geodesies considered as non-parameterized curves. So, we see that the affine geometry can not be recovered from its geodesies. On the other hand, any projective geometry is uniquely determined by its geodesies [2]. In particular, smooth map keeps the set of non-parameterized geodesies invariant if and only if it is a projective motion. Besides the projective geometries, the second well-known instance of the feature above to be satisfied is the conformal geometry; see [14] for the infinitesimal version of the later statement in the case of definite-signature conformal metric concerning conformal circles with distinguished parameters. However, using the techniques presented below, one can prove the same considering just non-parameterized conformal circles. Note that the same question in the case of indefinite conformal metric with null-geodesics as the specified type of distinguished curves is completely different and more or less trivial to answer. Let us recall here the converse statement, namely, that any morphism of Cartan geometry (Q, uj) respects the distinguished curves of any specified type, is trivially satisfied due to the equivalent definition of distinguished curves as projections of flow lines of constant vector fields in X(Q), cf. [13]. The main aim of this paper is to develop a method leading up to the system of differential equations, say £, which describe any specified class of distinguished curves. In fact, this is a modern presentation of Cartan's familiar ideas involved in [2, 3, 4]. The rest of this paper is devoted to making an intuition concerning the questions above. Concretely, we discuss the infinitesimal symmetries of £ which are essentially useful, however, one has to be a bit careful in this field. At any rate, there is no hope to get result of general character in this way but it serves just as a test of the conjecture above should be satisfied or not. We wish to come back to this topic elsewhere. Acknowledgments. Authors would like to thank Jan Slovak for the fruitful communication and a number of remarks and corrections. Most of tedious computations is done with the help of the computational system Maple including the Desolv package by John Carminati and Khai Vu [1]. Second author supported by the grant of MSMT CR #MSM14310009 and by the Junior Fellows program of the Erwin Schrodinger Institute (ESI), in different times. 2. Symmetry algebras and distinguished curves Here we summarize all basics, skipping those details which can be found in the referred literature. In this section we consider a fixed Cartan geometry of type (G, H) equations and symmetries of generalized geodesics 205 and distinguished curves specified by C, an LMnvariant set of homogeneous curves in G/H mapping 0 to the origin. One of the convenient tools describing basic properties of homogeneous curves is the symmetry algebra of a curve defined in [6]. In particular, it encodes the order of initial condition, which determines the homogeneous curve of the current type uniquely, as well as it helps to decide whether the admissible distinguished reparameterizations are projective or affine [5, 7]. This approach also leads to a handy criterion for a curve to be a distinguished curve of specified type, Corollary 1. Hence, with the help of the Cartan's moving frame method [4], one obtains the system £ of ordinary differential equations describing the curves in question. Such systems can be thought as a deformation of special invariant differential equations on a model space, which were well-studied for low-dimensional geometries, see [12]. Having equations of distinguished curves in hand, we use the progress of [12] in order to look for the infinitesimal symmetries of £. The main output is the system of partial differential equations, called determining equations, whose solutions are the infinitesimal symmetries. Occasionally, it happens that the infinitesimal symmetries of £ are in fact infinitesimal transformations of the geometric structure we began with. Of course, this must be encoded somehow in the shape of determining equations, which we demonstrate in Sections 3 and 4 in the cases of projective and Lagrangean contact geometries. 2.1. As above, let C be a set of homogeneous curves in G/H. Without any loss of generality, we assume C to be an orbit of the structure group H, i.e., any two curves of C are conjugated by an element of H. In fact, the study of distinguished curves goes in this direction in order to discern classes of curves of different behavior. 2.2. Fix a homogeneous curve Lit) = exp(tX)o, the representative of C = Cl-Following [6], compute the symmetry algebra symL C g as follows. For the sequence (2.1) f) = O0 5 Ol ^ o2 5 . . . of Lie subalgebras, defined recursively as Oj+i = {Y 6 o« : [Y,X] C (X) + 0«}, let r be the order that the sequence stabilizes from. Now we put the symmetry algebra of L to be the subalgebra symL = (X) + or of g. Equivalently, for any (non-parameterized) curve L c G/H, the symmetry algebra symL C g is defined as (2.2) symL = {X £ g : Rx(p) £ TpL for all p e L} where Rx denotes the vector field on G/H generated by X e q so that Rx(p) = |Qexp(tX)p. In fact, R: g —> X(G/H) is an (anti-)homomorphism of Lie algebras and any subalgebra of g gives rise to an integrable distribution on G/H. Obviously, for any p G L, R(symL)(p) C TpL and the curve L is homogeneous if and only if R(sym L)(p) = TpL. For later use, let us mention that under the usual identification T(G/H) = G Xh (fl/f)), via the Maurer-Cartan form ujq on G, the vector field Rx £ X(G/H) is written as (2.3) Rx(gH) = y,Adg-i(X) + i)l for any gH e G/H. Easily, homogeneous curves L\ and L2 coincide up to conjugation by an element of H (say h £ H) if and only if the symmetry algebras sym(Li) and sym(L2) are conjugated (via Ad^ e GL(g)). 206 boris doubrov and vojtěch žádník 2.3. Let (p: Q —> M, lo) be a Cartan geometry of type (G,H), and let c be a (parameterized) smooth curve on M. Denote by C C M the non-parameterized image of c. Then, by definition, c is a curve of type Cl if and only if it develops (at any point) into a curve which is conjugated to L by an element of H. Fixing a point x G C and following the notation of [15], this just means that there is an element u e p^1(x) c £ such that (devxc)(t) = lu,L(t)} c S^M = G/i^, where SM = G xH {G/H) is the Cartan's space of M. Here we use the following definition of development. For x = c(to), let c: / —> £ be any curve over C such that p{c(t)) = c(£o + *) and c(0) = u. Let further Y: / —> g be given as y(i) = cj(^c(t)). By the existence and uniqueness theorem for ordinary differential equations on Lie groups, there is a unique curve a: I —> G, such that a*tJG = ^ and a(0) = e, and then the development of c is given as (devxc)(t) = fu,a(t)oJ c SXM. Since ujq is the Maurer-Cartan form on G, then a*log = ^a is just the Darboux (or left logarithmic) derivative of a. Hence we say, that c is a curve of type Cl, for L(t) = exp(tX)o, if and only if there is a lift c of c in Q as above such that cj(jjc(t)) = X, especially, X = Sexp(tX) is constant. Concerning non-parameterized curves, the following Proposition appears. Proposition 1. Let C c M be an immersed 1-dimensional submanifold. Then the following two conditions are equivalent: (1) C admits a (local) parameterization that turns C into a distinguished curve of type Cl, (2) there is a (local) smooth section s: C —> Q of the projection Q —> M such that s*uj £ $11(C,q) takes values in symL C 0. proof. Consider c: / —> C, x = c(to) £ C, and c: I ^ Q over c as above such that dev^ c = fa, L\ with u = c(0) G p_1(x) and L(t) = exp(tX)o. Then, defining the section s: C —> £ by the prescription c(i) i—> c(t — to); we have got lm(s*uj) = X which belongs to symL by definition. Conversely, let c: / —> C be any parameterization of C such that c(0) = x. Let s: C —> be a section such that the assumption Im(s*w) C symL is satisfied, i.e., for c = s o c and y(i) = oj[-^c(t)) we have Im(y) c symL. Let further Sym(L) be a virtual subgroup in G (not necessarily closed) corresponding to the Lie algebra sym(L). Then from [6, Theorem 2] it follows that L, considered as unparameterized curve, (locally) coincides with the orbit of Sym(L) through o. Since Y(t) e sym(L) for all t G /, we see that a(t) e Sym(L), and thus a(t)o belongs to L. Let a: t i—> a(t)o. Show that a'(0) 7^ 0, i.e., a defines a parameterization of L in a neighborhood of the origin. Indeed, under the identification T(G/H) = G Xh (fl/fy) as in 2.2, the tangent vector field of a is written as gf(t) = la(t),Y(t) + fjj. Hence, evaluated in 0, it is equal to a'(0) = |[e, y(0) + fjj. But y(0) 0 fj, by definition, since c is transversal to the fibers of p: Q —> M. □ 2.4. On a coordinate neighborhood U of a point x £ M, any section s :U ^ Q of the projection £ —> M defines the Cartan gauge 6 = s*uj e $11(C7, g) which can only changes, under the change of section by a map h : U ^ H, according to the formula (2.4) 9 = Adh-i 9 + Sh. Any two Cartan gauges satisfying the condition above, on the intersection of domains, are called compatible and the relation 'to be compatible' is an equivalence relation equations and symmetries of generalized geodesics 207 on the set of local 1-forais {9a e $l1(C/a,g)}, see [13, Ch. 5] for details. We will work in this framework below and, in these terms, Proposition 1 can be formulated as follows. Corollary 1. Let C be a non-parameterized curve in M. Then the following two conditions are equivalent: (1) C is a curve of type Cl, (2) for any x G C, there is a neighborhood U 3 x and Cartan gauge 6 G nl(U,Q) such that lm(0|c) Q sym(L). Start with a Cartan gauge 6q = G\c £ ^1(C, g) along c (correctly, along c n u). The first necessary condition for the curve C to be a curve of type Cl is on the tangent space level. Namely, for some parameterization c : / —> C, the tangent vectors c has to be contained in the subset of TM which corresponds to the H-invariant set Adn(X) + fj C g/fj, provided the curve L representing the class Cl is generated by X. If this is the case, one can surely find a calibration h : U —> H such that Q\ = Adft-i #o + ^ restricted to C, takes values in (X) + fj. Considering c to be a curve of type Cl (up to reparameterization), we repeat this idea to build up a sequence of Cartan gauges 6{ whose restrictions to C take values in (X) + o^_i, i £ R, where the subalgebras a.j C fj are as in (2.1). Basically, this is the idea of moving frame, cf. [6]. Conversely, considering c to be a general curve on M, the question on Im(0j|c) C (X) + cij_i in each step yields some differential conditions on c which must be satisfied in order c to be a curve of type Cl up to ith order and up to reparameterization. Finally, the last step, corresponding to sym(L) = (X) + or, gives the system of differential equations El C Jr+1(R,M) we are interested in. In general, no all of the above constraints are differential equalities but often also inequalities. The typical instance of that case are the first-order conditions on chains in contact parabolic geometries, see Sections 4 and 5. However, we can always deal just with the final system of differential equations of order r + 1 keeping in mind that the initial conditions on the solution to be the right curve have to satisfy all the constraints up to order r. 2.5. Now, in the half-time, write r instead of r + 1 and consider the system El C Jr(R,M) of ordinary differential equations to be of the form Fu(t,x^) = 0, v = 1,...,N, where any Fv : Jr(R,M) —> R is a smooth function and x^ represents the derivatives of x = (x1,..., xm), m = dimM, up to order r. Let further £ e X(R x M) be a vector field on the space of independent and dependent variables, written as d d (2.5) Z{t,x)=xl){t,x)— + M is a morphism of Cartan geometry if and only if, for any Cartan gauge 6, the pullback f*6 is compatible with 6, i.e., there is a smooth map h : U —> H such that f*6 = Adh-i 6 + 5h, see 2.4. Note that this condition is satisfied for any Cartan gauge if and only if it is satisfied for one of them, by transitivity of the relation 'to be compatible'. For our purposes, we have to find the infinitesimal analogy of the above compatibility condition. An easy computations yields that vector field £ £ X(M) is an infinitesimal transformation of the Cartan geometry if and only if, for one (or, equivalently, any) Cartan gauge 8, there is a smooth map Y : U —> f) such that (2.6) Cj:6 = -adY9 + dY holds. Altogether, the question on whether has an infinitesimal symmetry £ of El to be an infinitesimal transformation of the Cartan geometry is equivalent to the question whether, for general £ satisfying the system of determining equations from 2.5, is there a smooth map Y: U —> fj such that the condition (2.6) is satisfied for a Cartan gauge 0 e $11(C7, g). In order to resolve the later question, one has first to look for such &Y:U —> f) that C^O = — ady 0 mod fj, because of dY contributes only to fj. The rest should be concluded (if the conjecture is true) by a game with coefficients of £ and their partial derivatives involving the identities which follow from assumption, i.e., that the system of determining equations is satisfied by £. Note that in the homogeneous model one also solves this problem by explicit solution of determining equations, as suggested in 2.5, which leads to the very visible description of the Lie algebra inf(El) of infinitesimal symmetries of El- Then one concludes by comparing the dimensions of Lie algebras in question, due to the inclusion q C mi (El) which is here by definition. Thus, computation of the symmetry algebra of El in the flat case can be considered as a test on whether the class of distinguished curves of type Cl does determine the Cartan geometry. If this test fails, i.e., the dimension of inf(££) is bigger than dimg, then the answer should be negative (the typical instance is the case of affine connections). However, even in the case when the test fails it is still possible (usually, due to some global arguments) that a map keeping the set of curves of type Cl stable is a transformation of the geometric structure. See Section 5 for example. Anyway, there are still more arguments required to establish correctly an answer to our question, so we wish to visit this problem elsewhere in a more conceptual way. Especially, in order to make the test and the hypothesis precise, two essential things are needed to clarify: first, which classes of distinguished curves should be considered as models and, second, which kind of global arguments can arise... equations and symmetries of generalized geodesics 209 In the next, we demonstrate the just presented techniques in three particular cases. In Section 3 we go carefully through the example of projective plane geometry in order to make the reader familiar with all the general notions above. Sections 4 and 5 represent essential points of the process in the case of 3-dimensional Lagrangean contact and projective contact geometry, respectively. Computation for all these geometries in higher dimensions is completely analogous, only a bit more longer. 3. Projective geometry 3.1. Projective 2-dimensional geometry is a split Cartan geometry of type (G, H) modeled by the projective plane MP2 with the principal group G = SL(3,M) of projective transformations and H C G, the stabilizer of some fixed point in MP2. On the infinitesimal level, we schematically write * * * 0 * * 0 * * trace = 0 (3.1) f, = with the natural choice of complementary subalgebra (3.2) 0 0 0 * 0 0 * 0 0 There are no distinguished directions and no distinguished types of generalized geodesies in projective geometries. In other words, any element of n lies in the i^-orbit of the vector (3.3) 0 0 0 1 0 0 0 0 0 £ n, and any generalized geodesic in G/H is a shift qLq of the curve Lq some g G G. Hence we set C = {qLq : g e H}. exp(£Xo)o, for 3.2. The symmetry algebra of the curve Lq is computed in two steps such that sym(L0) = {X0) + Oi, explicitly, (3.4) sym(Lo) * * * * * * 0 0* : trace = 0 The shape of the symmetry algebra involves the basic properties of generalized geodesic as suggested in introduction. In particular, in our case, non-parameterized geodesies are uniquely given by a direction in one point, which is a consequence of sym(Lo) = (Xq) + ai> i-e-> r = 1, and any such curve admits the projective class of distinguished parameters, which one concludes from the pair of Lie algebras ({X0) + Or, ar), following [5]. 3.3. For a coordinate system on U C M, any Cartan gauge 6 e $11(C7, g) can be calibrated by h : U —> H such that the n-part of 6 coincides with dx (due to 210 boris doubrov and vojtěch žádník surjectivity of the map H we write (3.5) GL(n) induced by Ad). Hence starting on this level, Zlk a\k alk Z2k a2k a2k. dxk where of course we sum over k = 1, 2 and the trace has to vanish. Consider a general curve c : / —> M and C C M to be the non-parameterized image of c. Then the pullback of 6q to the curve c(t) = (x1 (t), x2(t)) is (3.6) c*en a-k Si ft Zlk alk aik Z2k a2k a2 a2k. xkdt. An appropriate calibration leads to a compatible Cartan gauge Q\ along C with values in (Xq) + oq, i.e., with zero in the (3,l)-entry. In particular, for (3.7) 0 0' 1 0 4 i we really get, according to (2.4) and summing over k, £ = 1, 2, (3.8) akxk x1 a 12 if \alkix Zikjrxk 1 -k 1 xl • X X Z2kXk 1 • k a2kx 1 irir llk (ii)2 l2k a2k±-i ,X dt + 0 0 <2i. Of course, we have assumed x1 / 0, without any loss of generality. 3.4. At this moment, the equations for geodesies are read in the (3,2)-entry of c*6i in order to take values in the subalgebra sym(Lo) = (Xq) + a± C 0. Altogether, the system of ordinary differential equations describing (non-parameterized) geodesies is just one equation of second order, namely, (3.9) x^x1 — x2^1 kl=l CLf^j^X X CLouX X )x . Visibly, there is no contribution of functions z\\. and z2k into the equations above and, as an exercise, one can verify that each affine geodesic (with an arbitrary parameter) of any linear connection from the projective class is really solution of this system. 3.5. Consider a vector field £ G X(R x M), as in 2.5, with V = V'W- Then the determining differential equations for £ to be an infinitesimal symmetry of the system (3.9) are found to be (3.10) 0, (3.11) £=1 2 \ a2k) \ ~ V22)) equations and symmetries of generalized geodesics 211 + E £=1 + a for all i,j,k G {1,2} such that j ^ i and k ^ i. Lower indices after the comma denote the partial derivatives with respect to x = (x1, x2). In particular, the first set of equations reads as = 0, i.e., the functions ip1 and ip2 depend only on x. The second clear consequence of the system above is that ip = V'W maY be arbitrary, i.e., any reparameterization of a solution of (3.9) is again solution; just as one could anticipate. 3.6. Using the normal coordinates in the case of locally flat projective geometry, one begins with Cartan gauge " 0 0 0" dx1 0 0 dx2 0 0 (3.12) o which leads to an extra easy version of the geodesic equation £, (3.13) x2xx - x2x1 = 0, and the determining equations are (3.14) 1 2 n ^,22 = 11 = U> o 2 2 2^,12' ^,22 2^ 21) with ip1 and ip2 to be functions only of x. Now, let £ G X(M) be an infinitesimal point symmetry of (3.13), i.e., the system of partial differential equations (3.14) is satisfied, provided that £ = pl-^j- Write 0 instead of 0q and try to find a map Y : U —> f) such that C^9 = — ady 0 + 222^ respectively, and the rest of the matrix vanishes either trivially or as a consequence of (3.14). More concretely, the left-up corner vanishes because of vanishing of the whole trace and q^k \^jjk vanish, for any j, k e {1, 2}, as follows: f 333 2f 331 0, where i / j, and, for j / k, one gets ^,33^ Wjjk' hence f3 0 as well. 3.7. Alternative way available in the flat case is to solve the system of determining equations (3.14), as suggested in 2.6. Going this way, one can find the general solution of that system looks like (3.19) (io1(x1, x2) = ciix1)2 + c2x1x2 + C3X1 + C4X2 + C5, 2/ 1 2\ 12, / 2\2 , 1 , 2 1 f [X , X ) = C\X X + C2[X ) + CqX + CjX + Cg, for arbitrary constants q g R. Hence the dimension of inf(£) equals to 8 and so inf(£) = 0 by dimension reasons. In fact, G = SL(3,M) is the maximal possible symmetry group of second-order ordinary differential equation in two variables which is then necessarily equivalent to that in (3.13), cf. [8, 12]. Note that the opposite direction, i.e., the inclusion g C inf(£) which is trivial in general, can also be verified on this elementary level. In fact, the difference C^9 — ady 9 — dY in (3.17) vanishes if and only if all the relations in (3.14) are satisfied, i.e., £ is an infinitesimal symmetry of (3.13). 4. Lagrangean contact geometry 4.1. Lagrangean contact geometry in dimension 3 is a split Cartan geometry of type (G,H) modeled by the projectivization of the tangent space to MP2. The group G = SL(3, R) consists of all projective transformations naturally prolonged to PT(MP2), and H is the stabilizer of a fixed line in the tangent space at some fixed point. On the infinitesimal level, we have (4.1) * * * 0 * * 0 0* : trace = 0 with the complementary subalgebra n = nf © nf © n2 such that (4.2) "0 0 0" 1 f "0 0 0" * 0 0 0 0 0 0 0 0 J I 0 * 0 and n2 0 0 0 0 0 0 * 0 0 For simplicity we restrict our attention just to the essential points of the process in the flat case, though most of computations below can be done in general. There is a natural contact structure on G/H = PT(MP2) generated by tangent vectors to curves lifted from MP2. Via the identification T(G/H) = G Xh (fl/fy) as in 2.2 and n = g/rj, the contact distribution corresponds to the two-dimensional equations and symmetries of generalized geodesics 213 i^-invariant subspace in n defined by ni = nf © nf. There are several subsets in n invariant under the action of H which distinguish tangent vectors in T(G/H). Except those within m, there is just the complement n \ ni corresponding to vectors lying outside of the contact distribution. There are distinguished curves of the particular interest which emanate in the later directions, namely, the curves represented by Lo = exp(£Xo)o where Xq g xi2- These are called chains, in analogy with the Chern-Moser chains well-known from hypersurface CR geometries. From the symmetry algebra sym(Lo) below one can deduce they behave just like the classical chains, i.e., in any direction outside the contact distribution there is a unique unparameterized chain admitting a projective class of distinguished parameters. that (4.3) 4.2. Computing the symmetry algebra of Lo, we get sym(Lo) = (Xq) + oi so sym(L0) = * 0 * 0*0 * 0 * : trace = 0 4.3. Let us fix the local coordinates (x,y,z) on G/H = PT(MF2) so that (x,y) are affine coordinates on MP^ and z = ^. Then the contact structure on G/H is defined by the 1-form 7 = dy — zdx. Similarly to 3.3 and 3.6, we start with the Cartan gauge (4.4) 9n 0 0 0 dx 0 0 dy — zdx dz 0 and its pullback c*6q to the curve c(t) = (x(t),y(t), z(t)). Calibration 0 (4.5) h y—zx 1 0 y—zx 1 leads to a compatible Cartan gauge c*Q\ with values in (Xq) + rj. Of course, this is possible if and only if y — zx = 7(c) / 0, i.e., vector c is transverse to the contact distribution. 4.4. Now, the equations on chains are just written in (1, 2) and (2, 3) entries of the matrix c*0\, explicitly, we have got £ consisting of yx — yx = 0, (zy — zy) + z(xz — xz) = —Ixz1 (4-6) ... ... , ... ... 4.5. Skipping the explicit description and solution of determining equations, we just conclude that the system (4.6) is really invariant under reparameterizations and the Lie algebra inf(£) of all infinitesimal symmetries of £ is 8-dimensional and so coincides with q by dimension reasons. Altogether, we have got a good reason to believe that the set of chains allows to recover the initial Lagrangean contact geometry. 214 boris doubrov and vojtěch žádník 5. Projective contact geometry 5.1. Three-dimensional projective contact geometry is a split Cartan geometry of type (G, H) modeled by the projective space RP3 with the principal group G = Sp(4, R) c SL(4, R) acting transitively on RP3 via the restriction of the standard action of SL(4,R). Let H c G be the stabilizer of some fixed point, then RP3 = G/H. The natural contact structure on RP3 arises from the symplectic structure on R4 invariant with respect to the action of Sp(4,R). On the other hand, there is a natural flat projective connection on RP3 and these two structures are compatible in the following sense. If a straight line (i.e., a geodesic in the flat projective geometry) is tangent to the contact distribution at one point then it is a contact curve. See [9, 10] for more details and the compatibility of contact and projective structures in general case. An appropriate matrix representation leads to the following infinitesimal description, (5.1) a b c d x e f c y g -e -b z y —x —a in particular, dimg = 10. Hence, schematically, (5.2) * * * * 0 * * * 0 * * * 0 0 0 * with the complementary subalgebra n = ni © ri2 so that (5.3) ni 0 0 0 0 * 0 0 0 * 0 0 0 0 0 G 0 > and ri2 0 0 0 0 0 0 0 0 0 0 0 0 * 0 0 0 As in the Lagrangean contact case, we are focused especially on the flat model hereafter. There are only two distinct types of tangent vectors in T(G/H), namely, those lying inside and outside of the contact subbundle. Under the familiar identification T(G/H) = G Xh (fl/f)) and n = g/fj as before, the former case corresponds to the 2-dimensional subspace ni C n whilst the later one to the complement n \ ni. As in the Lagrangean contact case, we have also got the chains, the distinguished curves transversal to the contact distribution which are represented by any element of the subset ri2 C n in the same sense as before. Further, there is only one more type of generalized geodesies which can be represented by an arbitrary element of ni. Of course, these are the geodesies of the projective structure on the contact distribution and, in particular, they have the same properties as chains up to the initial condition to be directed within the contact subbundle. In general, projective contact structure induces the true projective structure whose geodesies are precisely the just discussed curves. equations and symmetries of generalized geodesics 215 5.2. Let Lo = exp(£Xo)o, Xq g ri2, be a chain. Again, symmetry algebra of Lq has the form sym(Lo) = (Xq) + oi, schematically written as ~* 0 0 * 0**0 0**0 * 0 0 * (5.4) sym(L0) 5.3. Appropriate local coordinates on G/H lead to the Cartan gauge (5.5) dz 0 0 dx 0 dy 0 ydx + xdy dy 0 0 0 -dx and, in these coordinates, the contact distribution is then given as the kernel of the 1-form 7 = dz — ydx + xdy. Consider the pullback c*6q to curve c(t) = (x(t),y(t), z(t)) and choose the calibration (5.6) -yx+xy 1 -yx+xy 0 0 0 1 0 z—yx+xy y z—yx+xy 1 which yields the compatible Cartan gauge c*9i with values in (Xq) + oo- Of course, we have considered z — yx + xy = 7(c) / 0. 5.4. The equations on chains are then read from (1,2) and (1,3) or, equivalently, from (2,4) and (3,4) entries of c*9\ in order to take values in sym(Lo). Hence, the system £ consists of two equations (yz - yz) + y(xy - xy) = 0, (5.7) , . [xz — xz) + x{xy — xy) = 0. 5.5. The determining equations of £ form a 3-variable analogy of those in (3.14), in particular, the system (5.7) is invariant under reparameterizations and the Lie algebra inf(£) has the maximal possible dimension, i.e., 15. Hence the system of differential equations £ is equivalent to the trivial one, [8], and the Lie algebra inf(£) of infinitesimal symmetries of £ coincides with sl(4, M), the Lie algebra of infinitesimal transformations of the true projective structure on G/H = MP3. Anyway, we still can claim that chain-preserving infinitesimal transformations coincide with g = sp(4, R), the Lie algebra of infinitesimal transformations of the projective contact structure. In order to prove this, it just remains to show that any chain-preserving transformation respects the contact distribution, but this is clear more or less by definition. The right reason of the former result, i.e., inf(£) = sl(4, M) d 0, is that there is never used the essential constraint z —yx+xy / 0 in the computation of infinitesimal symmetries of £. Omitting this inequality, one really recovers both the chains and the geodesies of type ni as solutions of £. Of course, this can be nicely presented using the same techniques as yet: choosing a suitable element X\ e m, computing the symmetry algebra of L\ = exp(£Xi)o, and fixing the same coordinates as in 5.3, one ends with the differential equation xy — xy = 0 provided that z — yx + xy = 0, 216 boris doubrov and vojtěch žádník i.e., the curve is tangent to the contact distribution. Now, it is an easy exercise to show that any solution of these two equations is also solution of (5.7). References [I] J. Carminati, K. Vu, Symbolic computation and differential equations: Lie symmetries, Journal of Symbolic Computation 29 (2000), no. 1, 95-116. [2] E. Cartan, Sur les varietes a connexion projective, Bull. Soc. Math. France, 52 (1924), 205-241. [3] E. Cartan, Les espaces a connexion conforme, Annates de la Societe Polonaise Math., 2 (1923), 171-221. [4] E. Cartan, The theory of finite continuous groups and differential geometry treated by the moving-frame method (Russian), Moscow Univ. Press, 1963. [5] B. Doubrov, Projective reparametrizations of homogeneous curves, Arch. Math. 41 (2005) 129- 133. [6] B. Doubrov, B. Kormakov, Classification of homogeneous submanifolds in homogeneous space, Lobachevskii J. Math. 3 (1999), 19-38. [7] M. Eastwood, J. Slovak, Preferred parameterisations on homogeneous curves, Comment. Math. Univ. Carolin. 45 (2004) 597-606. [8] M.E. Fels, The equivalence problem for systems of second-order ordinary differential equations, Proc. London Math. Soc. (3) 71 (1995), no. 1, 221-240. [9] D. Fox, Contact projective structures, arXiv:math.DG/0402332. [10] J. Harrison, Some problems in the invariant theory of parabolic geometries, Ph.D. thesis, University of Edinburgh, 1995. [II] L.K. Koch, Development and distinguished curves, preprint (1993). [12] P. Olver, Equivalence, invariants, and symmetry, Cambridge Univ. Press, 1995. [13] R.W. Sharpe, Differential geometry: Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics 166, Springer-Verlag, 1997. [14] K. Yano, The theory of Lie derivatives and its applications, North-Holland Publ., 1957. [15] V. Zadnik, Remarks on development of curves, Suppl. Rend. Circ. Mat. Palermo, Serie II, Suppl. 75 (2005) 347-356. V. Zadnik Masaryk University, Brno, Czech Republic International Erwin Schrodinger Institute for Mathematical Physics, Wien, Austria e-mail: zadnik@math.muni.cz B. Doubrov Belarussian State University, Minsk, Belarus e-mail: doubrov@islc.org