Contact Geometry of Curves

Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equ...

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Дата:	2009
Автор:	Vassiliou, P.J.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2009
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/149111
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Contact Geometry of Curves / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 30 назв. — англ.

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spelling	irk-123456789-1491112019-02-20T01:26:18Z Contact Geometry of Curves Vassiliou, P.J. Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M,g) is described. For the special case in which the isometries of (M,g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M. The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space H3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries. 2009 Article Contact Geometry of Curves / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 30 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53A35; 53A55; 58A15; 58A20; 58A30 http://dspace.nbuv.gov.ua/handle/123456789/149111 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M,g) is described. For the special case in which the isometries of (M,g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M. The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space H3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries.
format	Article
author	Vassiliou, P.J.
spellingShingle	Vassiliou, P.J. Contact Geometry of Curves Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Vassiliou, P.J.
author_sort	Vassiliou, P.J.
title	Contact Geometry of Curves
title_short	Contact Geometry of Curves
title_full	Contact Geometry of Curves
title_fullStr	Contact Geometry of Curves
title_full_unstemmed	Contact Geometry of Curves
title_sort	contact geometry of curves
publisher	Інститут математики НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/149111
citation_txt	Contact Geometry of Curves / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 30 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT vassilioupj contactgeometryofcurves
first_indexed	2025-07-12T21:22:36Z
last_indexed	2025-07-12T21:22:36Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 098, 27 pages Contact Geometry of Curves? Peter J. VASSILIOU Faculty of Information Sciences and Engineering, University of Canberra, 2601 Australia E-mail: peter.vassiliou@canberra.edu.au Received May 07, 2009, in final form October 16, 2009; Published online October 19, 2009 doi:10.3842/SIGMA.2009.098 Abstract. Cartan’s method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M, g) is described. For the special case in which the isometries of (M, g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M . The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré half-space H3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries. Key words: moving frames; Goursat normal forms; curves; Riemannian manifolds 2000 Mathematics Subject Classification: 53A35; 53A55; 58A15; 58A20; 58A30 1 Introduction The classical topic of immersed submanifolds in homogeneous spaces via repère mobile or mo- ving frames is discussed here in the simplest case, that of curves. Several authors have written on the method of repère mobile, over the years since Cartan’s works, such as [5]; these include S.S. Chern [6], J. Favard [8], P.A. Griffiths [12], G.R. Jensen [14], M.L. Green [11], R. Su- lanke [26], R. Sharpe [21] and M.E. Fels & P.J. Olver [9, 10]. Some of these authors have the goal of placing Cartan’s method on a firm theoretical foundation as well as extending its range of application beyond the classical realm. More recently, a reformulation of the method of moving frames, due to Fels and Olver [9, 10] has lead to renewed activity and a great many new applications and perspectives, have arisen (see [18] and references therein). Whereas Car- tan emphasised the construction of canonical Pfaffian systems whose integral manifolds are the Frenet frames along the submanifold, a much more direct approach is favoured in the Fels–Olver formulation and this has a number of significant advantages. However, in this paper, we shall reconsider the role of Pfaffian systems in the method of moving frames in the light of recent results in the geometry of jet spaces with the principal goal of making the contact geometry of curves more explicit and giving some indication about its possible applications. Another goal is to provide additional insight into the relationship between Cartan’s method of moving frames and the equivariant method of Fels and Olver1. ?This paper is a contribution to the Special Issue “Élie Cartan and Differential Geometry”. The full collection is available at http://www.emis.de/journals/SIGMA/Cartan.html 1A point we make herein is that the geometry of jet spaces provides a useful mediation between the two approaches. mailto:peter.vassiliou@canberra.edu.au http://dx.doi.org/10.3842/SIGMA.2009.098 http://www.emis.de/journals/SIGMA/Cartan.html 2 P.J. Vassiliou The considerations in this paper were inspired by a paper of Shadwick and Sluis [20], in which the authors observed that many of the Pfaffian systems derived by Cartan admit a Cartan pro- longation which is locally diffeomorphic to the contact distribution on jet space Jk(R,Rq), for some k and q, thereby explicitly adding contact geometry to Cartan’s method of moving frames. Another way to view the aims of this paper is the further development of the ideas in [20] in re- lation to moving frames for curves by exploring the application of a recent generalisation [27, 28] of the Goursat normal form from the theory of exterior differential systems [25, 3] allowing for the explicit determination of differential invariants and other invariant data in cases which have not been previously explored in detail. Of particular interest are curves in general Cartan geo- metries and in this paper we have focused on the Riemannian case and conjecture that similar results hold for other Cartan geometries. We show that given any n-dimensional Riemannian manifold (Mn, g) then the Pfaffian system whose integral manifolds determine the Frenet frames along curves in M has a Car- tan prolongation which can be identified with the contact system on jet space Jn(R,Rn−1). The explicit construction of the identification requires only differentiation. In case the isome- tries of (Mn, g) act transitively then the construction of the differential invariants that set- tles the equivalence problem for curves up to an isometry differs from the approach of Fels– Olver in that explicit a priori knowledge of the isometries or even the infinitesimal isometries is not required; as in the Fels–Olver method no integration is called for. The inputs for al- gorithm Riemannian curves consist only of the metric g and a realisation of the Lie group SO(n). Moreover, a contention of this paper is that the contact geometry of submanifolds to be described below should be a fundamental fact and lead to useful points of view that complement and enhance the geometric analysis of submanifolds by existing methods such as Cartan’s method of moving frames and the equivariant moving frames method of Fels and Olver. The content of this paper is as follows. After briefly recalling the method of moving frames, as practiced by Cartan, we study one of the simplest non-trivial examples: curves in 2-dimensional equi-affine geometry. It is then shown how the (classical) Goursat normal form applies to give the unique differential invariant and moving frame, explicitly. This familiar, illustrative example encapsulates the ideas proposed in this paper and is simple enough so that all de- tails can be given. An account of the generalised Goursat normal form [27, 28] is then given in the special case of total prolongations (uniform Goursat bundles) in preparation for the study of immersed curves in higher dimensional Cartan geometries. Section 4 illustrates the principles developed in the previous section by applying it to study curves in 3d-equi-affine geo- metry, computing the complete set of differential invariants via the generalised Goursat normal form. Section 5 is devoted to the contact geometry of curves in any Riemannian manifold and contains the main application of the paper. The general method is used to explicitly derive the differential invariants for curves in the Poincaré half-space H3 and for curves in a family of constant curvature 3-metrics. These invariants do not seem to have appeared in the literature before. The results demonstrate that the contact geometry of submanifolds can offer an alter- native path to invariant data for curves besides the Fels–Olver equivariant method and Cartan’s method which, in the latter case, relies so much on geometric insight and special tricks for the construction of the Frenet frames2. Finally, it should be mentioned that while this paper only explores the case of curves, the contact geometry of higher dimensional submanifolds could be similarly studied, commencing with the well known characterisation of contact systems in any jet space given in [2, 30]. 2In Cartan’s writings the distinction between the Frenet frames along a submanifold and the exterior differential system whose solutions are the Frenet frames is sometimes blurred. It’s best to keep these two notions quite separate since the construction of the latter is algorithmic while that of the former is not. Contact Geometry of Curves 3 2 Method of repère mobile applied to curves According to [14] the general problem treated by Cartan in [5] and elsewhere is that of the invariants of submanifolds in the homogeneous space of a Lie group G under the action of G. In this section, I will give a very brief description of the method of repère mobile, Cartan’s principal tool for addressing this type of problem. More complete discussions can be found in the references quoted above such as [8, 12, 14, 26]. The exposition given by Cartan in [4] is still well worth reading. Let G be a Lie group and H ⊂ G a closed subgroup. Then we have the H-principal bundle H ι // G π �� G/H of left cosets of H in G and we let M := G/H. Map π is the natural projection assigning a left- coset gH to each element of g ∈ G. There is a natural left-action of G on M : g ·zH = gzH, for all g ∈ G. Let x be a local coordinate system on M . Cartan typically began with a representation of G which could be “decomposed” into columns e1, e2, . . . , er of H ⊂ G and x, a column vector whose components are the coordinates x on M . Cartan defines differential 1-forms ωi, 1 ≤ i ≤ r by dx = r∑ i=1 ωi ⊗ ei. (1) The 1-forms ωi are semi-basic for π. Furthermore, we have 1-forms ωj i defined by dei = r∑ j=1 ωj i ⊗ ej . (2) The 1-forms ωi, ωj i , i, j = 1, . . . , r are the components of the Maurer–Cartan form ω on G; the integral submanifolds of the Pfaffian system ω1 = 0, ω2 = 0, . . . , ωr = 0 foliates G by the left cosets of H. Suppose f : T → M is an immersion of a manifold T into M . Then a moving frame is a local map F : T → G such that f = π ◦ F . That is, the moving frame assigns to each point t ∈ T a coset f(t) ∈ G/H. With this general set up, Cartan addresses the following problem for submanifolds of M . Let f1 : T1 → M and f2 : T2 → M be submanifolds. Find necessary and sufficient conditions, in the form of differential invariants, such that there is a local diffeomorphism µ : T1 → T2 and element g ∈ G such that f2 ◦ µ = g · f1. The ‘◦’ denotes function composition while ‘·’ continues to denote the left-action of G on M . A special case of this is the so-called fixed parametrisation problem where one takes T1 = T2 = T and µ is the identity on T . This congruence problem is the one that will be studied in this paper. In case the submanifolds of M are curves, Cartan begins by choosing a codimension 1 subset of the semibasic 1-forms and defines the Pfaffian system Ω : ω2 = 0, ω3 = 0, . . . , ωr = 0. 4 P.J. Vassiliou One studies the solutions of Ω since these project via π down to curves in G/H, which are the objects of interest. One way to do this is via the Cartan–Kähler theorem [3]. Accordingly, one computes the exterior derivatives of the ωj , j = 2, . . . , r and appends these “integrability conditions” to Ω forming the differential ideal Ω̄ with independence form ω1. This procedure allows one to prove existence of integral curves for Ω and provides information about the number of such integral curves. However, this makes no use of the special origin of the 1-forms in Ω, arising as they do from the Maurer–Cartan form ω on G. As a result of this one can go much further. From the structure equations of ω and the vanishing of the exterior derivatives dωi we deduce additional 1-form equations of the form ωj i − pj iω 1 = 0, for some functions pj i on G, which are appended to Ω as integrability conditions, thereby forming the new Pfaffian system Ω̄ : ω2 = 0, ω3 = 0, . . . , ωr = 0, ωj i − pj iω 1 = 0. In essence, the method of moving frames consists of using the fact that H acts on the fibres of G → G/H on the right inducing a transformation of the Maurer–Cartan form ω on G, [23, Chapter 7]. Indeed, the transformation (x, e1, . . . , er) 7→ (x, e1, . . . , er)h, ∀ h ∈ H (3) on G induces the transformation ω 7→ Ad(h−1)ω + h−1dh = ω̃ (4) on the Maurer–Cartan form ω. In turn, this induces a transformation on the functions pj i . To proceed further we recall the notion of a Cartan prolongation. Definition 1. Let I be a Pfaffian system on manifold M and p : M̂ → M a fibre bundle. A Pfaffian system Î on M̂ is said to be a Cartan prolongation of (M, I) if 1) p∗I ⊆ Î; 2) for every integral submanifold σ : S → M of I there is a unique integral submanifold σ̂ : S → M̂ of Î that projects to σ; that is, σ = p ◦ σ̂. We say that σ̂ is the Cartan lift of σ. If we choose to view (G×Rs, Ω̄), where the factor Rs carries the “parameters” pj i , as a Cartan prolongation of (G,Ω) then (3) induces a reduction of the trivial bundle G × Rs → G by normalising the coordinates pj i on the fibres to simple constants like 0 and ±1. Once Ω̄ has been normalised, the process begins again by taking exterior derivatives of the enlarged, normalised Pfaffian system arising from Ω̄. Each step selects a subgroup K ⊂ H. If the process terminates at K = {identity} of G then the resulting Pfaffian system arising from Ω̄ is canonical. The integral submanifolds of Ω̄ are the Frenet frames, F. Hereafter we shall label this canonical Pfaffian system by the symbol ΩF. The main assertion made in this paper is that the canonical Pfaffian system ΩF determining each Frenet frame along an immersed curve admits a Cartan prolongation Ω̂F on E := G× Rν for some ν, so that (E, Ω̂F) is locally diffeomorphic to a jet space (Jk(R,Rq),Ωk(R,Rq)) where Ωk(R,Rq) ⊂ T ∗Jk(R,Rq) is the contact sub-bundle. The coordinates on the Rν factor of E Contact Geometry of Curves 5 carry the differential invariants of the problem. Indeed, the integral manifolds of Ω̂F, say, Γ̂ : I → G× Rν project down to the Frenet lifts Γ of curves γ : I →M as in E π̂ �� G π �� I Γ̂ FF�� Γ =={{{{{{{{{ γ // G/H (5) where I ⊆ R is an interval. Hereafter, one of our goals is to give examples which demonstrate the assertion made above, namely that the Pfaffian system Ω̂F can be identified with a contact system. This identification can be constructed explicitly and provides explicit coordinate for- mulas for all the invariant data: differential invariants, Fels–Olver equivariant moving frames and invariant differential forms. In Section 5 we will prove that this procedure can be applied to curves in any Riemannian manifold and in that case it is algorithmic3. Importantly, one is not required to explicitly know the group action a priori. Before this we will work out some pedagogical examples. The first of these is sufficiently low dimensional so that all details can be given. 2.1 Curves in the equi-affine plane The goal in this subsection is to provide a simple illustration of the method of moving frames as described in the previous subsection. We will construct the Frenet frame F for a plane curve up to equi-affine transformations by constructing the canonical Pfaffian system ΩF and the appropriate Cartan prolongation Ω̂F, as described above. Here equi-affine transformations means the standard transitive action of the Lie group G = SL(n,R) n Rn on Rn. For plane curves we take n = 2; the action on R2 with local coordinates ξ1, ξ2 is[ ξ1 ξ2 ] 7→ A [ ξ1 ξ2 ] + [ x y ] , where A ∈ SL(2,R) and x, y ∈ R. We identify R2 with G/SL(2,R) where the elements of G are matrices of the form g = 1 0 0 x a b y c d  and ad− bc = 1. We call this homogeneous space the affine plane and denote it by A2. For local coordinates on A2 we take x, the first column of g ∈ G and e1, e2 are the next two columns of g. Equations (1), (2) give the semi-basic forms ω1 = βdx− bdy, ω2 = −cdx+ ady, and connection forms ω1 1 = βda− b dc, ω1 2 = 1 a ( βb da+ db− b2dc ) , ω2 1 = a dc− c da, 3In this paper a construction or procedure is said to be algorithmic if it can be performed only by differentiation and “algebraic operations”, which includes constructing the inverse of a local diffeomorphism. However, integra- tion is strickly excluded. 6 P.J. Vassiliou where β = (1 + bc)/a, where we have chosen a chart on SL(2,R) in which a 6= 0; note that ω1 1 + ω2 2 = 0. It is useful to record the structure equations dω1 = ω1 ∧ ω1 1 + ω2 ∧ ω1 2, dω2 = ω1 ∧ ω2 1 − ω2 ∧ ω1 1, dω1 1 = ω2 1 ∧ ω1 2, (6) dω1 2 = −2ω1 1 ∧ ω1 2, dω2 1 = 2ω1 1 ∧ ω2 1. Successive adapted frames are integral curves of certain Pfaffian systems which will be denoted by Ωi, i = 1, 2, . . .. The first adapted frames for curves in A2 are integral curves of the Pfaffian system Ω1, consisting of the single 1-form equation Ω1 : ω2 = 0 with independence form ω1. From structure equations (6), we obtain 0 = dω2 ≡ ω1 ∧ ω2 1 mod ω2 and hence to complete Ω1 to a differential ideal Ω̄1 we extend it by appending the 2-form equation ω2 1 ∧ ω1 = 0. This equation implies that there is a function p on G such that the 2-form equation can be replaced by ω2 1 − pω1 = 0, a kind of “first integral”. We extend Ω1 by this 1-form equation and rename the extended Pfaffian system Ω̄1 to get Ω̄1 : ω2 = 0, ω2 1 − pω1 = 0. Note that the reconstituted Ω̄1 is no longer a differential ideal. As discussed in Section 2, an element h ∈ H acts on the frame [x, e1, e2] over each point x ∈ G/H on the right inducing the transformation (4) on the Maurer–Cartan form on G. This, in turn induces a transformation on the function p. The subgroup H1 ⊂ H that leaves Ω1-invariant has representation1 0 0 0 a b 0 0 1/a  . We obtain ω̃1 = a−1ω1, ω̃2 = aω2, ω̃2 1 = a2ω2 1. Hence 0 = ω2 1 − pω1 = a−2ω̃2 1 − p a ω̃1 = a−2(ω̃2 1 − p a3ω̃1). The Pfaffian system Ω̄1 is transformed to ω̃2 = 0, ω̃2 1 − p a3ω̃1 = 0. That is, the function p undergoes the transformation p 7→ a3p. Accordingly, we can choose a so that ap3 = 1 and transform Ω̄1 to4 Ω2 : ω2 = 0, ω2 1 − ω1 = 0. 4We have made a tacit genericity assumption that p 6= 0. The case p = 0 must be considered separately, as in [14]. To simplify the exposition we shall continue to make such genericity assumptions in this paper. Contact Geometry of Curves 7 The integral submanifolds of Ω2, with independence form ω1 are the “second order frames” for curves in A2. The subgroup H2 ⊂ H1 ⊂ H that preserves the elements of Ω2 is1 0 0 0 1 b 0 0 1  . We must now extend Ω2 to a differential ideal by computing the exterior derivative of ω2 1 − ω1. From the structure equations we obtain 3ω1 1 ∧ω1 = 0. As before, there is a function q on G such that the 2-form equation can be replaced by ω1 1 − qω1 = 0 so that (the reconstituted) Ω̄2 is given by the 1-form equations Ω̄2 : ω2 = 0, ω2 1 − ω1 = 0, ω1 1 − qω1 = 0 (7) and is no longer a differential ideal. By performing a H2 change of frame the 1-forms in (7) become ω̃2 = ω2, ω̃2 1 − ω̃1 = ω2 1 − ω1, ω̃1 1 = ω1 1 − b ω2 1. Thus Ω2 is invariant under a H2 change of frame while 0 = ω1 1 − qω1 = ω̃1 1 + (b− q)ω̃1. We can chose b = q to obtain the 1-form equation ω̃1 1 = 0, and giving rise to the final adapted frame (dropping tildes) Ω3 : ω2 = 0, ω2 1 − ω1 = 0, ω1 1 = 0 which “reduces the isotropy group to the identity”. Thus, Ω3 is the Pfaffian system ΩF and its integral curves are the Frenet lifts F of curves in A2. Computing the exterior derivative of ω1 1 = 0 we obtain the 2-form equation ω1 2 ∧ ω1 = 0 and hence there is a function κ on G such that ω1 2−κω1 = 0. This time there is no freedom left in our choice of frame that enables κ to be transformed away. The function κ here is intrinsic. Hence, in this case, the Cartan prolongation we seek is the Pfaffian system ΩF augmented by the 1-form equation ω1 2 − κω1 = 0 Ω̂F : ω2 = 0, ω2 1 − ω1 = 0, ω1 1 = 0, ω1 2 − κω1 = 0, on G × Rκ. The integral curves of Ω̂F with independence form ω1 determine the unique equi- affine invariant for plane curves. Theorem 1. Let I ⊆ R be an interval and γi : I → A2 be two immersed curves in the equi-affine plane, each parametrised by equi-affine arc-length. Then there is an element g ∈ G such that γ2 = g · γ1 if and only if their Cartan lifts Γ̂i : I → G× R satisfy Γ̂∗1κ = Γ̂∗2κ (8) identically on I. Proof. Let us firstly recall that (G × R, Ω̂F) → (G,ΩF) is a Cartan prolongation and Γ̂i is a Cartan lift of Γi, i = 1, 2; see diagram (5). By definition, the Γi are the Frenet lifts of γi 8 P.J. Vassiliou and Γ̂i are Cartan lifts of γi. Finally Γ̂i are integral submanifolds of Ω̂F on G×R and consquently for i = 1, 2 we have Γ̂∗iω 2 = Γ̂∗iω 1 1 = 0, Γ̂∗iω 1 2 = ( Γ̂∗iκ ) ( Γ̂∗iω 1 ) , Γ̂∗iω 2 1 = Γ̂∗iω 1. (9) Since both curves are parametrised by equi-affine arc-length s we have Γ̂∗1ω 1 = Γ̂∗2ω 1 = ds. From this and from (9) we deduce that Γ∗1ΩMC = Γ∗2ΩMC, (10) where ΩMC is the Maurer–Cartan form on G. It follows from the standard theorem about maps into a Lie group [22, Chapter 10, Theorem 18] that there is a fixed element g ∈ G such that γ2 = g · γ1. Conversely, if γ2 = g · γ1 for some g ∈ G, then the Frenet lifts Γi of γi satisfy (10) and are integral submanifolds of ΩF. But since Ω̂F is a Cartan prolongation of ΩF, there are Car- tan lifts Γ̂i of Γi which are integral submanifolds of Ω̂F. Equation (8) follows from this and equation (10). � Remark 1. This theorem encapsulates the basic idea of this paper and is proposed as a model for the study of curves in any Cartan geometry. The relationship between Theorem 1 and diagram (5) should be clear. The idea now is that by the Goursat normal form ( G× R, Ω̂⊥ F ) is locally diffeomorphic to the jet bundle ( J4(R,R), C(4) 1 ) , where C(4) 1 is the contact sub-bundle of TJ4(R,R). That is, there is a local diffeomorphism φ : G×R → J4(R,R) such that φ∗Ω̂⊥ F = C(4) 1 . In fact, Theorem 1 proves that knowing the diffeomorphism φ explicitly constructs the unique invariant κ for plane equi-affine curves explicitly, namely the equi-affine curvature, as well as the equi-affine arc-length. Explaining this is the goal of the next subsection. 2.2 Goursat normal form The Goursat normal form is a local characterisation of the contact distribution on Jk(R,R) for all k ≥ 1, which we denote C(k) 1 . The original theorem is not due to Goursat who was its populariser. It appears the theorem is originally due, in some form, to E. von Weber but the statement of it I give below essentially arises from a 1914 work of Cartan. A good reference is [25]. This reference describes an interesting, relevant but largely forgotten work of Vessiot [29]. First we establish some notation and definitions. 2.2.1 The derived flag Suppose M is a smooth manifold and V ⊂ TM a smooth sub-bundle of its tangent bundle. The structure tensor is the homomorphism of vector bundles δ : Λ2V → TM/V defined by δ(X,Y ) = [X,Y ] mod V, for X,Y ∈ Γ(M,V). If δ has constant rank, we define the first derived bundle V(1) as the inverse image of δ(Λ2V) under the canonical projection TM → TM/V. Informally, V(1) = V + [V,V]. The derived bundles V(i) are defined inductively: V(i+1) = V(i) + [V(i),V(i)] Contact Geometry of Curves 9 assuming that at each iteration it defines a vector bundle, in which case we shall say that V is regular. For regular V, by dimension reasons, there will be a smallest k for which V(k+1) = V(k). This k is called the derived length of V and the whole sequence of sub-bundles V ⊂ V(1) ⊂ V(2) ⊂ · · · ⊂ V(k) the derived flag of V. We shall denote by V(∞) the smallest integrable sub-bundle containing V. 2.2.2 Cauchy bundles Let us define σ : V → Hom(V, TM/V) by σ(X)(Y ) = δ(X,Y ) Even if V is regular, the homomorphism σ need not have constant rank. If it does, let us write Char V for its kernel. The Jacobi identity shows that Char V is always integrable. It is called the Cauchy bundle or characteristic bundle of V. If V is regular and each V(i) has a Cauchy bundle then, we say that V is totally regular. Then by the derived type of V we shall mean the list {V(i),Char V(i)} of subundles. Theorem 2 (Goursat normal form). Let V ⊂ TM be a smooth, totally regular, rank 2 sub-bundle over smooth manifold M such that a) V(∞) = TM ; b) dimV(i+1) = dimV(i) + 1, while V(i) 6= TM . Then there is a generic subset M̂ ⊆M such that in a neigbourhood of each point of M̂ there are local coordinates x, z0, z1, z2, . . . zk such that V has local expression{ ∂x + k∑ j=1 zj∂zj−1 , ∂zk } , where k = dimM − 2. That is, V is locally equivalent to C(k) 1 on M̂ . A proof can be found in [25, pp. 157–159]. The proof of a much more general result in which the Goursat normal form is a special case is given in [28]. The significance of the latter is that an procedure is provided for constructing the local contact coordinates x, z0, . . . , zk. This is procedure Contact B on page 287 of [28] with ρk = 1 and ρ1 = ρ2 = · · · = ρk−1 = 0; the ρi are defined in Section 3. In this special case we have the following. Procedure Contact for the Goursat normal form INPUT : Sub-bundle V ⊂ TM of derived length k which satisfies the hypotheses of Theorem 2. a) Fix any first integral of Char V(k−1), denoted x, and any section Z of V such that Zx = 1. b) Define a distribution Πk as follows: Πl+1 = [Z,Πl], Π1 = Char V(1), 1 ≤ l ≤ k − 1. c) Let z0 be any invariant of Πk such that dx ∧ dz0 6= 0. d) Define functions z1, z2, . . . , zk by zj = Zzj−1, j = 1, . . . , k. OUTPUT : Functions x, z0, z1, . . . , zk are contact coordinates for V. The proof of correctness of this procedure is given in [28]. 10 P.J. Vassiliou 2.3 Equi-affine invariants & the Goursat normal form We use this procedure to construct the various invariant objects for this geometry. In fact we will construct the Frenet frames by constructing all the integral submanifolds of Ω̂F. So we set V := Ω̂⊥ F : V = { ∂ω1 + ∂ω2 1 + κ∂ω1 2 , ∂κ } . Note that we have adopted the usual convention of denoting the frame dual to ω1, ω2, ω1 1, ω1 2, ω2 1 by ∂ω1 , ∂ω2 , ∂ω1 1 , ∂ω1 2 , ∂ω2 1 . In local coordinates we have V = { X := a∂x + c∂y + b∂a + κa∂b + 1 + bc a ∂c, ∂κ } . Calculation verifies that the hypotheses of Theorem 2 are met and that the derived length of V is k = 4. Then step a) of Contact requires that we construct at least one invariant of Char V(3) = { ∂b, ∂κ, a∂a + c∂c } which has invariants x, y, a/c. Any one of these can be taken as the “independent variable”. Since x, y are local coordinates on G/SL(2,R), we shall choose x for this purpose. It then follows that we may take Z to be Z = 1 a X = ∂x + c a ∂y + b a ∂a + κ∂b + 1 + bc a2 ∂c. Step b) requires the construction of Π4. We get Π1 = {∂κ}, Π2 = {∂κ, ∂b}, Π3 = { ∂κ, ∂b, 1 a ∂a + c a2 ∂c } , Π4 = { ∂κ, ∂b, 1 a ∂a + c a2 ∂c, ∂c } . The invariants of Π4 are in fact x, y and hence by step c), we set z0 = y and construct zj = Zzj−1, 1 ≤ j ≤ 4. We get z1 = c a , z2 = 1 a3 , z3 = −3b a5 , z4 = 3 5b2 − a2κ a7 obtaining the equivalence φ : U ⊂ G× R → J4(R,R) defined by φ ( x, y, a, b, c, κ ) = ( x, y, ca−1, a−3,−3ba−5, 3(5b2 − a2κ)a−7 ) between V and the contact distribution on J4(R,R). The inverse of φ is φ−1 ( x, z0, z1, z2, z3, z4 ) = ( x, z0, z −1/3 2 ,−3−1z3z −5/3 2 , z1z −1/3 2 , 9−1(5z2 3 − 3z2z4)z −8/3 2 ) = (x, y, a, b, c, κ). Contact Geometry of Curves 11 Hence if we express the curve in A2 as a graph (x, f(x)) then we deduce from φ−1 that the equi-affine curvature is the well known expression κ := 5f ′′′(x)2 − 3f ′′(x)f ′′′′(x) 9f ′′(x)8/3 = 1 2 ( f ′′(x)−2/3 )′′ . We also obtain the unique G-invariant 1-form, the equi-affine arc length by pulling back ω1 by φ−1, (φ−1)∗ω1 = f ′′(x)1/3dx and the Frenet frame F =  1 0 0 x f ′′(x)−1/3 −3−1f ′′′(x)f ′′(x)−5/3 f(x) f ′(x)f ′′(x)−1/3 (f ′′(x)2 − 3−1f ′′′(x)f ′(x))f ′′(x)−5/3  of the curve (x, f(x)) by pulling back an arbitrary element g ∈ G by φ−1. Of course, we can express everything in terms an arbitrary immersion (x(t), y(t)) into A2, rather than as a graph. Remark 2. Note that the procedure we have just described for the invariant data of curves in A2 is not algorithmic; we had to solve differential equations to obtain the equivalence φ. In practice, however, we find that when the contact system is that of Jk(R,Rq) where q > 1, this integration can often be avoided. We will illustrate this for curves in A3 and prove, in Section 5, that it holds for curves in any Riemannian manifold of dimension greater than 2. 3 On the generalised Goursat normal form To carry out the programme of the previous section for curves immersed in manifolds of dimen- sion greater than two we must be able to characterise the contact distributions on jet spaces Jk(R,Rq), for all k, q ≥ 1; the case q = 1 being the Goursat normal form. In principle this generalisation should include partial prolongations of the contact distribution on J1(R,Rq) and such a characterisation exists – the generalised Goursat normal form [27, 28]. However, so far the full scope of this characterisation has not been required. It turns out that only total prolon- gations of the first order jet space are sufficient. Accordingly, we will only briefly review those parts of [27, 28] that are needed for the results to be described in this paper. 3.1 The singular variety For each x ∈M , let Sx = {v ∈ Vx\0 \| σ(v) has less than generic rank}. Then Sx is the zero set of homogeneous polynomials and so defines a subvariety of the pro- jectivisation PVx of Vx. We shall denote by Sing(V) the fibre bundle over M with fibre over x ∈M equal to Sx and we refer to it as the singular variety of V. For X ∈ V the matrix of the homomorphism σ(X) will be called the polar matrix of [X] ∈ PV. There is a map degV : PV → N well defined by degV([X]) = rank σ(X) for [X] ∈ PV. We shall call degV([X]) the degree of [X]. The singular variety Sing(V) is a diffeomorphism invariant in the sense that if V1, V2 are sub-bundles over M1, M2, respectively and there is 12 P.J. Vassiliou a diffeomorphism φ : M1 →M2 that identifies them, then Sing(V2) and Sing(φ∗V1) are equiva- lent as projective subvarieties of PV2. That is, for each x ∈ M1, there is an element of the projective linear group PGL(V2\|φ(x) ,R) that identifies Sing(V2)(φ(x)) and Sing(φ∗V1)(φ(x)). We hasten to point out that the computation of the singular variety for any given sub- bundle V ⊂ TM is algorithmic. One computes the determinantal variety of the polar matrix for generic [X]. 3.1.1 The singular variety in positive degree If X ∈ Char V then degV([X]) = 0. It is convenient to eliminate lines of degree zero and for this reason we pass to the quotient V̂ := V/Char V. We have structure tensor δ̂ : Λ2V̂ → T̂M/V̂, well defined by δ̂(X̂, Ŷ ) = π([X,Y ]) mod V̂, where T̂M = TM/Char V and π : TM → T̂M is the canonical projection. The notion of degree descends to this quotient giving a map degV̂ : PV̂ → N well defined by degV̂([X̂]) = rank σ̂(X̂) for [X̂] ∈ PV̂, where σ̂(X̂)(Ŷ ) = δ̂(X̂, Ŷ ) for Ŷ ∈ V̂. Note that all definitions go over mutatis mutandis when the structure tensor δ is replaced by δ̂. In particular, we have notions of polar matrix and singular variety, as before. However, if the singular variety of V̂ is not empty, then each point of PV̂ has degree one or more. 3.1.2 The resolvent bundle Suppose V ⊂ TM is totally regular of rank c+ q+1, q ≥ 2, c ≥ 0, dimM = c+2q+1. Suppose further that V satisfies a) dim Char V = c, V(1) = TM ; b) Σ̂\|x := Sing(V̂)\|x = PB̂\|x ≈ RPq−1, for each x ∈ M and some rank q sub-bundle B̂ ⊂ V̂. Then we call (V,PB̂) (or (V, Σ̂)) a Weber structure of rank q on M . Given a Weber structure (V,PB̂), let R(V) ⊂ V, denote the largest sub-bundle such that π ( R(V) ) = B̂. (11) We call the rank q + c bundle R(V) defined by (11) the resolvent bundle associated to the Weber structure (V, Σ̂). The bundle B̂ determined by the singular variety of V̂ will be called the singular sub-bundle of the Weber structure. A Weber structure will be said to be integrable if its resolvent bundle is integrable. We will see that the resolvent bundle is the key to the construction of an identification of a given differential system with a contact system, if such an identification exists; and hence the name. Contact Geometry of Curves 13 An integrable Weber structure descends to the quotient of M by the leaves of Char V to be the contact bundle on J1(R,Rq). Thus, the resolvent bundle and its concomitant Weber structure is a constructive characterisation of the contact bundle on the 1-jets J1(R,Rq). The term ‘Weber structure’ honours Eduard von Weber (1870–1934) who was the first to publish a proof of the Goursat normal form. For completeness we record the following properties of the resolvent bundle of a Weber structure. Proposition 1 ([27]). Let (V, Σ̂) be a Weber structure on M and B̂ its singular sub-bundle. If q ≥ 3, then the following are equivalent a) its resolvent bundle R(V) ⊂ V is integrable; b) each point of Σ̂ = Sing(V̂) has degree one; c) the structure tensor δ̂ of V̂ vanishes on B̂: δ̂(B̂, B̂) = 0. Proposition 2 ([27]). Let (V, Σ̂) be an integrable Weber structure on M . Then its resolvent bundle R(V) is the unique, maximal, integrable sub-bundle of V. Checking the integrability of the resolvent bundle is algorithmic. One computes the singular variety Sing(V̂) = PB̂. In turn, the singular bundle B̂ algorithmically determines R(V). Example 1. The canonical model of an integrable Weber structure is the contact distribution on J1(R,Rq) for q > 1, extended by Cauchy characteristics V = { ∂x + q∑ i=1 pi∂ui , ∂p1 , . . . , ∂pq , ∂z1 , ∂z2 , . . . , ∂zc } . The quotient V̂ = V/Char V has singular sub-bundle B̂ = { [∂p1 ] , . . . , [ ∂pq ] } and the resolvent bundle of integrable Weber structure (V,PB̂) is R(V) = { ∂p1 , . . . , ∂pq , ∂z1 , ∂z2 , . . . , ∂zc } . The invariants of the resolvent bundle are spanned by {x, u1, . . . , uq}. So the resolvent bundle provides a geometric characterisation of the “independent variable” x and the “dependent variab- les” ui, after which differentiation by a canonically defined total derivative operator leads to higher order jet coordinates. See Sections 3.2 and 4 of this paper for further details. See [27, 28] for the general theory with proofs and further examples. 3.2 The uniform generalised Goursat normal form We are now able to give a characterisation of the contact distribution on Jk(R,Rq), C(k) q for any k, q ≥ 1, generalising the Goursat normal form to the uniform case. Theorem 3 (generalised Goursat normal form – uniform case, [27, 28]). Let V ⊂ TM be a smooth, totally regular, sub-bundle of rank q + 1 and derived length k, some k, q > 0, over smooth manifold M such that a) V(∞) = TM ; b) dimV(i+1) = dimV(i) + q, while V(i) 6= TM ; c) Char V(i) ⊂ V(i−1), 1 ≤ i ≤ k − 1; dim Char V(j) = jq, 0 ≤ j ≤ k − 1; d) If q > 1 then V(k−1) admits an integrable Weber structure. Then there is a generic subset M̂ ⊆ M such that around each point of M̂ the distribution V is locally equivalent to C(k) q . 14 P.J. Vassiliou If q = 1 then c) follows from b) and the Weber structure is not defined in which case Theorem 3 reduces to the Goursat normal form, Theorem 2. We call any sub-bundle that satisfies the hypotheses of Theorem 3 a uniform Goursat bundle in which case the Theorem asserts that generically every uniform Goursat bundle is locally equivalent to the canonical one. A proof of Theorem 3 can be found in [27] as a special case of that paper’s Theorem 4.1. However, the latter covers a very much larger class of sub-bundles than uniform Goursat bundles and there is therefore a much simpler proof in this uniform case. However, for the purposes of this paper an important thing is the procedure for constructing contact coordinates in the uniform case which is a special case of procedure Contact A on page 286 of [28]5 with ρ1 = ρ2 = · · · = ρk=1 = 0, ρk = q > 1. Note that the collection of non-negative integers σ = 〈ρ1, ρ2, . . . , ρk〉 shall be called the signature of V and is a complete local invariant of Goursat bundles. In the interests of completeness we mention that the non-negative integers ρi are defined by ρj = dim Char V(j) − dim Char V(j) j−1, 1 ≤ j ≤ k − 1, ρk = dimV(k) − dimV(k−1), where Char V(j) j−1 = Char V(j)∩V(j−1). It is proved in [27, 28] that a sub-bundle V on manifold M is locally diffeomorphic to a partial prolongation of the contact system C(1) q on J1(R,Rq) with ρj variables of order j, if and only if (M, V) is a Goursat bundle of signature σ. In this paper, we need only consider total prolongations in which the only nonzero element of the signature is ρk, where k is the derived length of V. Procedure Contact for uniform generalised Goursat bundles, q > 1 INPUT : Uniform Goursat bundle V ⊂ TM of derived length k with q > 1. a) Construct the (integrable) resolvent bundle R(V(k−1)) and all its q + 1 first integrals. b) Fix any one of the first integrals from step a) denoted x, and any section Z of V such that Zx = 1. c) Denote the remaining q first integrals of R(V(k−1)) by zj 0, j = 1, 2, . . . , q. d) Define functions zj 1, z j 2, . . . , z j k by zj m = Zzj m−1, 1 ≤ m ≤ k, 1 ≤ j ≤ q. OUTPUT : Functions x, zj 0, z j 1, . . . , z j k, 1 ≤ j ≤ q are contact coordinates for V. Remark 3. Even though an integration problem is presented for solution in step a) above, in fact, in every example of curves in a Cartan geometry that I’ve seen no integration is re- quired because the resolvent bundle turns out to be the vertical bundle for the fibration π ◦ π̂ : E → G/H. So a complete set of invariants of R(V(k−1)) can be taken to be the components of any coordinate system on G/H. In Section 5 we will prove this for curves in any Riemannian manifold of dimension at least 3. 4 Space curves up to equi-affine transformations As an illustration of the generalised Goursat normal form and its relation to the geometry of curves we consider immersed curves in R3 up to the standard action of G := SL(3,R)nR3. The goal is to use repère mobile and Theorem 3 to construct all the invariant data for this situation. 5Unfortunately in reference [28] procedure Contact was called an algorithm. Manifestly, it does not qualify as an algorithm because in a certain step an integration is required. Contact Geometry of Curves 15 We discuss this example principally for illustration since we permit ourselves to begin with the explicit transitive action. However, our goal in this paper is to drop any reliance on an a priori knowledge of a group action. To that end a contention of this paper is that explicit invariant curve data can be obtained without integration or explicit knowledge of the group action in a significant special case, namely Riemannian geometry. This will be established in Section 5. A straightforward extension of the n = 2 case covered in Section 2.1 to the n = 3 case leads to the matrix group with elements g =  1 0 0 0 x a1 a2 a3 y a4 a5 a6 z a7 a8 a9  , where det g = 1. We parametrise an open subset of the group G by solving det g = 1 for a9. From equations (1), (2) or otherwise we deduce the left-invariant Maurer–Cartan form ω =  0 0 0 0 ω1 ω1 1 ω1 2 ω1 3 ω2 ω2 1 ω2 2 ω2 3 ω3 ω3 1 ω3 2 ω3 3  , where ω1 1 + ω2 2 + ω3 3 = 0. For a coframe on G we take the ordered list[ ω1, ω2, ω3, ω1 1, ω 2 1, ω 3 1, ω 1 2, ω 2 2, ω 3 2, ω 1 3, ω 2 3 ] whose dual frame we label V = [ v1, v2, . . . , v11 ] . The Lie algebra multiplication table for V is v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v1 0 0 0 −v1 −v2 −v3 0 0 0 0 0 v2 0 0 0 0 0 0 −v1 −v2 −v3 0 0 v3 0 0 0 v3 0 0 0 v3 0 −v1 −v2 v4 v1 0 −v3 0 −v5 −2v6 v7 0 −v9 2v10 v11 v5 v2 0 0 v5 0 0 v8 − v4 −v5 −v6 v11 0 v6 v3 0 0 2v6 0 0 v9 v6 0 −v4 −v5 v7 0 v1 0 −v7 v4 − v8 −v9 0 v7 0 0 v10 v8 0 v2 −v3 0 v5 −v6 −v7 0 −2v9 v10 2v11 v9 0 v3 0 v9 v6 0 0 2v9 0 −v7 −v8 v10 0 0 v1 −2v10 −v11 v4 0 −v10 v7 0 0 v11 0 0 v2 −v11 0 v5 −v10 −2v11 v8 0 0 By a procedure similar to the one carried out in the n = 2 case we arrive at the Pfaffian system ΩF (see [8] for details) whose integral curves are Frenet lifts of curves in G/SL(3,R) ΩF : ω2 = 0, ω3 = 0, ω3 1 = 0, ω2 1 − ω1 = 0, ω3 2 − ω1 = 0, ω1 1 = 0, ω2 2 = 0, ω2 3 − 3ω1 2 = 0. I want to show that a Cartan prolongation of ΩF is a contact system. We calculate that Ω⊥ F = { v1 + v5 + v9, v7 + 3v11, v10 } . The Cartan prolongation we shall consider is obtained from this: Ω̂⊥ F : { v1 + v5 + v9 + κ1(v7 + 3v11) + κ2v10, ∂κ1 , ∂κ2 } defined over G×R2. To apply the generalised Goursat normal form, Theorem 3, we work with this dual bundle V := Ω̂⊥ F and calculate 16 P.J. Vassiliou i V(i) Char V(i) 0 { v1 + v5 + v9 + κ1(v7 + 3v11) + κ2v10, ∂κ1 , ∂κ2 } {0} 1 V ⊕ { v7 + 3v11, v10 } {∂κ1 , ∂κ2} 2 V(1) ⊕ { v4 + 2v8, v7 − v11 } Char V(1) ⊕ {v7 + 3v11, v10} 3 V(2) ⊕ { v4 − 2v8, v1 + v5 − 5v9 } Char V(2) ⊕ {v4 + 2v8, v7 − v11} 4 V(3) ⊕ { v6, v1 − 3v5 } Char V(3) ⊕ {v4 − 2v8, v9} 5 T (G× R2) T (G× R2) So hypotheses a), b) and c) of Theorem 3 are satisfied with q = 2 and derived length k = 5. Since q > 1, it remains to check the singular variety of the quotient V(4)/Char V(4). From the table we see that Char V(4) = { v4, v7, v8, v9, v10, v11, ∂κ1 , ∂κ2 } and V(4) = { v1, v4, v5, v6, v7, v8, v9, v10, v11, ∂κ1 , ∂κ2 } and hence V̂(4) := V(4) / Char V(4) = { [v1], [v5], [v6] } . We obtain that Im δ̂4 = {[v2], [v3]} and the polar matrix of the point 〈a1[v1] + a2[v5] + a3[v6]〉 ∈ PV̂(4) is( −a2 a1 0 −a3 0 a1 ) which has unit rank if and only if a1 = 0. Hence the singular variety of V̂(4) is RP1 with singular bundle B̂ = { [v5], [v6] } . Consequently, the resolvent bundle in this case is R ( V(4) ) = {v4, v5, v6, v7, v8, v9, v10, v11} ⊕ {∂κ1 , ∂κ2} = sl(3,R)⊕ R2. The resolvent bundle is integrable, showing that V(4) admits an integrable Weber structure, fulfilling hypothesis d) of Theorem 3. We can therefore conclude that Ω̂⊥ F is locally equivalent to the contact distribution C(5) 2 on jet space J5(R,R2). For future reference, we note that we often abuse the term “derived type” by referring to the list of lists[ [ dimV(j), dim Char V(j) ] ]k j=0 as the derived type of V, where k is its the derived length. Thus for the example just treated we may say that its derived type is[ [3, 0], [5, 2], [7, 4], [9, 6], [11, 8], [13, 13] ] since it is really the dimensions of these bundles that settles the recognition problem. 4.1 Differential invariants via Contact We now apply procedure Contact to construct the two equi-affine invariants for space curves. In the previous subsection we demonstrated that Ω̂⊥ F is a uniform Goursat bundle. The first step in procedure Contact requires the q+ 1 = 2 + 1 = 3 invariants of the (integrable) resolvent Contact Geometry of Curves 17 bundle, R(V(4)). The vector fields spanning this bundle are all vertical for the projection G × R3 → R3 and, indeed, they frame its fibres. It follows that the functions annihilated by these vector fields are spanned by x, y, z – the coordinates of the homogeneous space G/SL(3,R) in which the curves are immersed. Note that no integration is required6. A local coordinate calculation verifies this claim. By step b) we are at liberty to take any one of these as the parameter along the curve. We take x for this purpose. Continuing to follow b) we fix a vector field Z ∈ V such that Zx = 1. At this point we must construct the vector fields that span V. This is straightforward since V is constructed from the left-invariant vector fields on G (as well as ∂κ1 , ∂κ2). We obtain v1 = a1∂x + a4∂y + a7∂z, v2 = a2∂x + a5∂y + a8∂z, v3 = a3∂x + a6∂y + a9∂z, v4 = a1∂a1 − a3∂a3 + a4∂a4 − a6∂a6 + a7∂a7 , v5 = a2∂a1 + a5∂a4 + a8∂a7 , v6 = a3∂a1 + a6∂a4 + a9∂a7 , v7 = a1∂a2 + a4∂a5 + a7∂a8 , v8 = a2∂a2 − a3∂a3 + a5∂a5 − a6∂a6 + a8∂a8 , v9 = a3∂a2 + a6∂a5 + a9∂a8 , v10 = a1∂a3 + a4∂a6 , v11 = a2∂a3 + a5∂a6 , v12 = ∂κ1 , v13 = ∂κ2 , where a9 = 1− a2a6a7 + a1a6a8 − a3a4a8 + a3a5a7 a1a5 − a4a2 . From these we obtain Z = 1 a1 ( v1 + v5 + v9 + κ1(v7 + 3v11) + κ2v10 ) ∈ V. Note that Z is the total differential operator in this example. Finally, we let z1 0 = y, z2 0 = z as in step c) and compute the higher order coordinates as in step d) by differentiation by Z. This construction provides the components of the local equivalence φ : G×R2 → J5(R,R2) that identifies V with the contact distribution C(5) 2 on J5(R,R2). Local inverse ψ of the map φ gives the explicit differential invariants κ1, κ2, for curves in R3 up to “equi-affine motions”: κ1 = ( 24z2 2w3w5 − 35z2 2w 2 4 − 60z2w2 3z4 + 60z2w3z3w4 − 24z2w2w3z5 + 70z2w4w2z4 − 24z2w2z3w5 + 60w2w3z3z4 − 60w2z 2 3w4 − 35w2 2z 2 4 + 24z5w2 2z3 )/ (w3z2 − w2z3)7/3, κ2 = ( − 18z3 2w3w4w5 + 25z3 2w 3 4 − 36z2 2w 3 3z5 + 90z2 2w 2 3z4w4 + 36z2 2w 2 3z3w5 − 90z2 2w3z3w 2 4 + 18z2 2w3w2z4w5 + 18z2 2w3w2w4z5 − 75z2 2w2w 2 4z4 + 18z2 2w2z3w4w5 − 90z2w2 3w2z 2 4 + 72z2w2 3z5w2z3 − 18z2w3w 2 2z4z5 − 72z2w3w2z 2 3w5 + 90z2w2z 2 3w 2 4 + 75z2w2 2w4z 2 4 − 18z2w2 2z3w4z5 − 18z2w2 2z3z4w5 + 90w3w 2 2z3z 2 4 − 36w3z5w 2 2z 2 3 − 25w3 2z 3 4 − 90w2 2z 2 3w4z4 + 18w3 2z3z4z5 + 36w2 2z 3 3w5 )/ (w3z2 − w2z3)7/2, where zi = f (i)(x), wj = g(j)(x) for arbitrary smooth functions f(x), g(x) such that g′′′(x)f ′′(x)− g′′(x)f ′′′(x) 6= 0. Thus, if the curve in question has parametrisation (x, f(x), g(x)) then the complete set of differential invariants is {κ1, κ2}. Pulling back the remaining nonzero elements of the Maurer– Cartan form on G by ψ give explicit expressions for invariant differential one-forms on J 5(R,R2). One can easily obtain the invariants in an arbitrary parametrisation of the curve or in terms of arclength parametrisation from these formulas but we won’t record these here7. 6See Section 5 for further comment on this aspect. 7Note that the inversion of φ is greatly assisted by the fact that the algebraic equations involved are guaranteed, by the proof of Theorem 3.3 in [27] to be block triangular. 18 P.J. Vassiliou 5 Curves in Riemannian manifolds The geometries discussed in previous sections of this paper and the one treated in [28] are all of Klein type and one may wonder how the method fares when curvature is introduced. Also, in our previous (illustrative) examples, we have permitted ourselves knowledge of the explicit transitive group action in order to construct the differential system V and we do not want to make this assumption in general. In this paper we will be content to establish a result for Riemannian geometry and illustrate this by two examples. However, we conjecture that similar results hold for other Cartan geometries. Let ωi, πi j be the components of the Cartan connection8 of an arbitrary Riemannian mani- fold (M, g), where dimM = n and ωi, 1 ≤ i ≤ n are semi-basic 1-forms for the projection F(M) → M ; F(M) is the orthonormal frame bundle over M . The structure equations for the coframe on F(M) are dωi = n∑ j=1 ωj ∧ πi j , dπi j − n∑ k=1 πi k ∧ πk j = 1 2 n∑ k,l=1 R i j k l ω k ∧ ωl, where all indices range from 1 to n and πi j + πj i = 0 for all i, j. Dually, the frame satisfies [∂ωi , ∂ωj ] = −1 2 n∑ k,l=1 Rk lij∂πk l , [∂ωi , ∂πl j ] = −δij∂ωl , [∂πi l , ∂ πj k ] = δkl∂πi j + δij∂πl k . (12) We now state the main result of this section. Theorem 4. Let n ≥ 3 and γ : I →M be an immersed curve in an n-dimensional Riemannian manifold (M, g), I ⊆ R. Let ωi, πi j be the components of the Cartan connection for (M, g) on the orthonormal frame bundle F(M) →M . Then 1. There is a unique integral submanifold of V = { ∂ω1 + κ1 0∂π2 1 + κ2 0∂π3 2 + · · ·+ κn−1 0 ∂πn n−1 + n−2∑ l=1 κ1 l ∂κ1 l−1 + n−3∑ l=1 κ2 l ∂κl−1 + n−4∑ l=1 κ3 l ∂κl−1 + · · ·+ κn−2 1 ∂κn−2 0 , ∂κ1 n−2 , ∂κ2 n−3 , . . . , ∂κn−1 0 } , (13) which projects to the Frenet lift of γ to the orthonormal frame bundle F(M). 2. The sub-bundle V is a uniform Goursat bundle with signature 〈 n−1︷︸︸︷ 0, 0, . . . , 0, n− 1 〉. Hence there is a local diffeomorphism φ which identifies it with the contact distribution C(n) n−1 on jet space Jn(R,Rn−1). 3. The local diffeomorphism φ can be constructed by differentiation and algebraic operations alone. 4. If the isometries of (M, g) act transitively, then the local diffeomorphism φ induces lo- cal coordinate formulas for the complete invariants κ1 0, κ 2 0, . . . , κ n−1 0 of the curve γ up to isometries of (M, g). 8See Sharpe [21] for an account of Klein–Cartan geometries and Cartan connections; see also [23, Chapter 7]. Contact Geometry of Curves 19 Proof. The proofs of 1 and 4 are similar to the proof of Theorem 1. The proof of 2 is complicated to write down in all generality; it is more enlightening to prove it in a sufficiently non-trivial case. We therefore write out the detail proof in the case n = 6, after which the general case will be clear. The distribution in question, for n = 6, is V = { ∂ω1 + 5∑ l=1 κl 0∂πl l+1 + 4∑ l=1 κ1 l ∂κ1 l−1 + 3∑ l=1 κ2 l ∂κ2 l−1 + 2∑ l=1 κ3 l ∂κ3 l−1 + κ4 1∂κ4 0 , ∂κ1 4 , ∂κ2 3 , ∂κ3 2 , ∂κ4 1 , ∂κ5 0 } . We are required to prove that V ' C(6) 5 . Let us arrange some of the frame elements into subsets as follows p1 : { ∂π5 6 } , p2 : { ∂π4 5 , ∂π4 6 } , p3 : { ∂π3 4 , ∂π3 5 , ∂π3 6 } , p4 : { ∂π2 3 , ∂π2 4 , ∂π2 5 , ∂π2 6 } , p5 : { ∂π1 2 , ∂π1 3 , ∂π1 4 , ∂π1 5 , ∂π1 6 } . It is easy to show from (12) that h` = ⊕̀ j=1 pj ⊆ so(6), 1 ≤ ` ≤ 5, determine a flag of Lie subalgebras. Using this we compute that for each s in the range 1 ≤ s ≤ 6, the quotients V̂s := V(s)/V(s−1) have basis representatives V̂(1) = { ∂κ1 3 , ∂κ2 2 , ∂κ3 1 , ∂κ4 0 , ∂π5 6 } , V̂(2) = { ∂κ1 2 , ∂κ2 1 , ∂κ3 0 , ∂π4 5 , ∂π4 6 } , V̂(3) = { ∂κ1 1 , ∂κ2 0 , ∂π3 4 , ∂π3 5 , ∂π3 6 } , V̂(4) = { ∂κ1 0 , ∂π2 3 , ∂π2 4 , ∂π2 5 , ∂π2 6 } , V̂(5) = { ∂π1 2 , ∂π1 3 , ∂π1 4 , ∂π1 5 , ∂π1 6 } , V̂(6) = { ∂ω2 , ∂ω3 , ∂ω4 , ∂ω5 , ∂ω6 } . The Cauchy systems have bases Char V(1) = { ∂κ1 4 , ∂κ2 3 , ∂κ3 2 , ∂κ4 1 , ∂κ5 0 } , Char V(2) = { ∂κ1 3 , ∂κ2 2 , ∂κ3 1 , ∂κ4 0 , ∂π5 6 } ⊕ Char V(1), Char V(3) = { ∂κ1 2 , ∂κ2 1 , ∂κ3 0 , ∂π4 5 , ∂π4 6 } ⊕ Char V(2), Char V(4) = { ∂κ1 1 , ∂κ2 0 , ∂π3 4 , ∂π3 5 , ∂π3 6 } ⊕ Char V(3), Char V(5) = { ∂κ1 0 , ∂π2 3 , ∂π2 4 , ∂π2 5 , ∂π2 6 } ⊕ Char V(4) and hence V(5)/Char V(5) = {[ ∂ω1 ] , [ ∂π1 2 ] , [ ∂π1 3 ] , [ ∂π1 4 ] , [ ∂π1 5 ] , [ ∂π1 6 ]} . Structure equations (12) show that the singular sub-bundle is B̂ = {[ ∂π1 2 ] , [ ∂π1 3 ] , [ ∂π1 4 ] , [ ∂π1 5 ] , [ ∂π1 6 ]} and therefore the resolvent bundle is given by R ( V(5) ) = { ∂πi j } ⊕ { ∂κa b } = so(6)⊕ R15, 20 P.J. Vassiliou where a, b, i, j range over all possible values. These calculations show that that the derived type of V is [[6, 0], [11, 5], [16, 10], [21, 15], [26, 20], [31, 25], [36, 36]] from which one deduce’s that the signature of V is 〈0, 0, 0, 0, 0, 5〉. Since R(V(5)) is integrable and all other hypotheses of Theorem 3 are satisfied with q = 5 and k = 6, we have shown that V is locally equivalent to the contact system C(6) 5 on J6(R,R5). To prove 3, we invoke Theorem 4.2 of [28] which shows that to construct the local equiva- lence φ identifying V with C(6) 5 one only requires a complete set of invariants of the resolvent bundle R(V(5)). However, it is elementary to see that no element of R(V(5)) has components tangent to M ; on the other hand R(V(5)) spans the tangent spaces of the fibres over M . Hence, any coordinate system on M provides the needed invariants and no integration need be per- formed. � In case the Riemannian manifold M does not have a transitive isometry group then V may simply be regarded as a control system on M with controls κ1 n−2, κ 2 n−3, . . . , κ n−1 0 and with all other coordinates on E playing the role of state variables (outputs, in the language of control theory). The arc-length parametrisation along the curve plays the role of time in this control theoretic interpretation. The fact that V is a Goursat bundle then proves that it is differentially flat, a type of control system currently under investigation in the control community. In fact it is currently a significant open problem in control theory to geometrically characterise all differentially flat control systems. One other way to state Theorem 4 is to say that the “natural” framing of curves in any Riemannian manifold is differentially flat. If M does possess a transitive isometry group G then the functions κ1 0, . . . , κ n−1 0 form a com- plete set of curve invariants up to the action of G. One may wonder whether Theorem 4 is a genuine advance in the theory and/or practice of moving frames either in the sense of Cartan or in the sense of Fels–Olver. The contention of this paper is that it does represent an advance in both the theory and the practice of moving frames. As to the theory, I argue that endowing curves in a Cartan geometry with an explicit contact structure is significant given the fundamental role that contact structures play in ge- ometry and differential equations. As to the practice of the method of moving frames, in the case of Riemannian manifolds (M, g) we have given a framing for curves in M as solutions of a differential system Ω̂F for any metric g and shown that it can be explicitly identified with the contact system on a jet space Jk(R,Rq) for some k, q. In case the isometry group of (M, g) acts transitively, this identification delivers the complete set of invariant data including the differential curve invariants. We formulate this construction as an algorithm. Algorithm Riemannian curves INPUT : Riemannian manifold, (M, g), dimension n ≥ 3. a) Construct a parametrisation of SO(n) and orthonormal coframe ω on M . b) Lift ω to the orthonormal frame bundle over M and build the Cartan connection Θ for (M, g). This involves linear algebra. c) From Θ, build the differential system V defined in Theorem 4. d) Apply procedure Contact using, as invariants of the resolvent bundle, any coordinate system on M . e) Construct diffeomorphism φ by procedure Contact A. Contact Geometry of Curves 21 OUTPUT : Local diffeomorphism φ identifying V with contact distribution C(n) n−1. In case the isometry group of (M, g) acts transitively, obtain complete, explicit invariant data for curves in M , following the inversion of φ. Note that Riemannian curves is an algorithm in as much as steps a)–e) do not require any integration to be performed. All the steps involved are algebraic. 5.1 Curves in the Poincaré half-space The aim of this subsection is to apply the previous theorem to construct explicit expressions for curvature and torsion for curves in the Poincaré half-space H3, with Riemannian metric g = dx2 + dy2 + dz2 z2 . In principle we could approach this by putting coordinates on the Lie group SO(3, 1) and then consider curves in the homogeneous space SO(3, 1)/SO(3), as we did in the case of equi-affine space curves. However, this leads to unwieldy expressions which are difficult to handle, even with the help of a computer. Instead, we will use the method of equivalence to construct the Cartan connection for g and then use this to build the canonical Pfaffian system ΩF for the Frenet frame of a generic curve in H3. In fact, we will construct the dual vector field distribution V = Ω⊥ F . Begin by lifting the 1-forms θ1 = dx z , θ2 = dy z , θ3 = dz z to the orthonormal frame bundle F(H3) over H3 byω1 ω2 ω3  = O θ1 θ2 θ3  , where O = cos b cos c− sin a sin b sin c − cos a sin c sin a cos b sin c+ sin b cos c cos b sin c+ sin a sin b cos c cos a cos c sin b sin c− sin a cos b cos c − cos a sin b sin a cos a cos b  parametrises SO(3). The fibres of the orthonormal frame bundle F(H3) → H3 are diffeomorphic to SO(3) whose Maurer–Cartan form is the so(3)-valued 1-form Ω =  0 ω1 2 ω1 3 ω2 1 0 ω2 3 ω3 1 ω3 2 0  , where ω1 2 = −dc− sin a db, ω1 3 = sin c da+ cos a cos c db, ω2 3 = sin c cos a db− cos c da, and ωj i + ωi j = 0. As is usual in the method of equivalence, we compute the structure equations of the semi-basic forms obtaining dω1 = ω1 2 ∧ ω2 + ω1 3 ∧ ω3 + (sin c sin b− cos c sin a cos b)ω1 ∧ ω2 + cos a cos b ω1 ∧ ω3, 22 P.J. Vassiliou dω2 = ω2 1 ∧ ω1 + ω2 3 ∧ ω3 − (cos c sin b+ sin a cos b sin c)ω1 ∧ ω2 + cos a cos b ω2 ∧ ω3, dω3 = ω3 1 ∧ ω1 + ω3 2 ∧ ω2 − (sin b cos c+ sin a cos b sin c)ω1 ∧ ω3 − (sin b sin c− sin a cos b cos c)ω2 ∧ ω3. All torsion can be absorbed by redefining connection forms π1 2 = ω1 2 + (sin c sin b− sin a cos b cos c)ω1 − (sin a cos b sin c+ sin b cos c)ω2, π1 3 = ω1 3 + cos a cos bω1 − (sin a cos b sin c+ sin b cos c)ω3, π2 3 = ω2 3 + cos a cos bω2 − (sin b sin c− sin a cos b cos c)ω3, and we obtain structure equations dω1 = π1 2 ∧ ω2 + π1 3 ∧ ω3, dω2 = −π1 2 ∧ ω1 + π2 3 ∧ ω3, dω3 = −π1 3 ∧ ω1 − π2 3 ∧ ω2. The remaining structure equations are dπ1 2 = −π1 3 ∧ π2 3 + ω1 ∧ ω2, dπ1 3 = π1 2 ∧ π2 3 + ω1 ∧ ω3, dπ2 3 = −π1 2 ∧ π1 3 + ω2 ∧ ω3. These structure equations are those of the Maurer–Cartan form on SO(3, 1). Moreover, we are now able to define the e(3)-valued Cartan connection Ω̂ =  0 0 0 0 ω1 0 π1 2 π1 3 ω2 π2 1 0 π2 3 ω3 π3 1 π3 2 0  , with curvature dΩ̂ + Ω̂ ∧ Ω̂ =  0 0 0 0 0 0 ω1 ∧ ω2 ω1 ∧ ω3 0 ω2 ∧ ω1 0 ω2 ∧ ω3 0 ω3 ∧ ω1 ω3 ∧ ω2 0  for metric g. Setting n = 3 in (13), and adopting the usual notation κ1 0 = κ, κ2 0 = τ , κ1 1 = κ1, we study the integral submanifolds of V = { ∂ω1 + κ∂π1 2 + τ∂π2 3 + κ1∂κ, ∂κ1 , ∂τ } . It is easy to check that V has derived type [[3, 0], [5, 2], [7, 4], [9, 9]]. It can be checked that ρ1 = ρ2 = 0 and hence the signature of V is 〈0, 0, 2〉. This is the signature of contact system C(3) 2 . To complete the check that V is diffeomorphic to C(3) 2 , we compute the resolvent bundle and check its integrability. We find that Char V(2) = {∂π2 3 , ∂κ, ∂κ1 , ∂τ}, and V(2)/Char V(2) = {[ ∂ω1 ] , [ ∂π1 2 ] , [ ∂π1 3 ]} , whose structure is [∂ω1 , ∂π1 2 ] ≡ ∂ω2 , [∂ω1 , ∂π1 3 ] ≡ ∂ω3 , [∂π1 2 , ∂π1 3 ] ≡ 0 mod Char V(2). Contact Geometry of Curves 23 Hence, the singular bundle is B̂ = {[∂π1 2 ], [∂π1 3 ]} and consequently the resolvent bundle is R(V(2)) = { ∂π1 2 , ∂π1 3 , ∂π2 3 , ∂κ, ∂κ1 , ∂τ } = so(3)⊕ R3, which is clearly integrable. An easy calculation in local coordinates verifies that the invariants of R(V(2)) are indeed the coordinates x, y, z on the base of the orthonormal frame bundle F(H3) → H3. According to the Theorem we take one of x, y, z as the independent variable. If y is taken for this purpose, then we can take D = ∂ω1 + κ∂π1 2 + τ∂π2 3 + κ1∂κ as the direction of the total differential operator since 0 6= D (y) = −z cos a sin c = µ. Then the operator of total differentiation is Z = µ−1 ( ∂ω1 + κ∂π1 2 + τ∂π2 3 + κ1∂κ ) . Setting u = x, v = z, the remaining contact coordinates for V are then provided by differentiation u1 = Z u, v1 = Z v, u2 = Z u1, v2 = Z v1, u3 = Z u2, v3 = Z v2. The map φ (x, y, z, a, b, c, κ, κ1, τ) 7→ (y, u, v, u1, v1, u2, v2, u3, v3) pushes V forward to C(3) 2 , which, in contact coordinates has the form{ ∂y + u1∂u + v1∂v + u2∂u1 + v2∂v1 + u3∂u2 + v3∂v2 , ∂u3 , ∂v3 } . A local inverse of φ is easily constructed and provides all the invariant data for curves in H3. In particular we deduce explicit expressions for curvature and torsion for curves γ(y)=(u(y), y, v(y)), where for instance u1, v2, denote derivatives du/dy, d2v/dy2, etc. We obtain κ = − ( u6 1 + 2u4 1v 2 1 + 2u4 1vv2 + 3u4 1 − 2u3 1vu2v1 + v4 1u 2 1 + 2v2 1u 2 1vv2 + 4u2 1vv2 + u2 1v 2v2 2 + 4v2 1u 2 1 + 3u2 1 − 2u1v 2u2v1v2 − 2u1vu2v 3 1 − 2u1v1u2v + 1 + v4 1 + v2u2 2v 2 1 + 2v2 1vv2 + 2vv2 + v2 2v 2 + v2u2 2 + 2v2 1 )1/2/( v2 1 + u2 2 + 1 )3/2 , τ = ( 3u2 2u1 + 3u2v1v2 + vu2v3 − u3u 2 1 − u3v 2 1 − u3vv2 − u3 ) v2 /( u2 2v 2v2 1 + u2 2v 2 − 2u2v 2v1u1v2 − 2u2vv1u1 − 2v3 1u1u2v − 2u3 1v1u2v + u2 1v 2v2 2 + v2v2 2 + 4u2 1vv2 + 2v2 1vv2 + 2vv2 + 2u4 1vv2 + 2v2 1u 2 1vv2 + 1 + 2v2 1 + v4 1 + 4u2 1v 2 1 + 3u2 1 + 2u4 1v 2 1 + v4 1u 2 1 + 3u4 1 + u6 1 ) . Once again, the algebraic system that is presented for solution in this task is guaranteed to have a block triangular structure. We remark that semi-circular arcs parallel to the y − z plane γ(y) = ( C1, y, √ C2 2 − y2 ) , C2 > 0, −C2 < y < C2 are geodesics in H3; accordingly it can be checked that κ(γ(y)) = τ(γ(y)) ≡ 0. 24 P.J. Vassiliou 5.2 Curves in constant curvature Riemannian 3-manifolds Let λ be any nonzero real number. Here we point out that the construction of the previous subsection can be carried out for the three dimensional Riemannian manifold (M(λ), gλ) with metric gλ = dx2 + dy2 + dz2( 1 + λ 4 (x2 + y2 + z2) )2 of constant curvature λ, where M(λ) is an open subset of R3. Exactly the same calculation as before but with θ1 = dx Λ , θ2 = dy Λ , θ3 = dz Λ , where Λ = 1 + λ 4 (x2 + y2 + z2) gives rise to the corresponding Cartan connection Ω̂ =  0 0 0 0 ω1 0 π1 2 π1 3 ω2 π2 1 0 π2 3 ω3 π3 1 π3 2 0  , with curvature dΩ̂ + Ω̂ ∧ Ω̂ = −λ  0 0 0 0 0 0 ω1 ∧ ω2 ω1 ∧ ω3 0 ω2 ∧ ω1 0 ω2 ∧ ω3 0 ω3 ∧ ω1 ω3 ∧ ω2 0  . for the metric gλ. Exactly the same calculation as the one carried out for the Poincaré half-space gives rise to the curvature and torsion for curves in (M(λ), gλ). We get κλ = λ √ (yu1 − u)2 + (yv1 − v)2 + (vu1 − uv1)2 2 √ u2 1 + v2 1 + 1 , τλ = − ( 4 + λ(y2 + v2 + u2) )2( uv3 − vu3 + y(v1u3 − v3u1) ) 8λ ( 1 + u2 1 + v2 1 )( (yu1 − u)2 + (yv1 − v)2 + (vu1 − uv1)2 ) . The point to note here as in the previous example is that we are not required to know the explicit formulas for the action of the isometries on M(λ) before the invariants and the moving frame can be computed. Only infinitesimal data is required, in the form of the Cartan connection and then no integration need be performed. Remark 4. Interestingly, τλ is not a continuous function of λ at λ = 0, while lim λ→0 κλ is not the curvature of the curve in the corresponding limiting metric lim λ→0 gλ, which is Euclidean. 6 Closing remarks In this paper we have demonstrated that curves in various geometries can be endowed with a contact geometry by combining Cartan’s classical construction of moving frames with the generalised Goursat normal form. In particular we have shown that curves in any Riemannian manifold can be endowed with a contact geometry regardless of the nature of its isometry group or curvature tensor. Contact Geometry of Curves 25 We have also been concerned with the problem of explicitly computing differential invariants of curves immersed in spaces equipped with a transitive action of a Lie group. If this Lie group action is explicitly known and not too complicated then the method of choice for computing the differential invariants and other geometric data is the normalisation of the group action as in the equivariant moving frames method of Fels and Olver, described in [9, 10]. This method is very general, has a simple and elegant theoretical foundation and presents as simple a computational task as could be hoped for. However, if the explicit group action is not known or it is known but too complicated to work with and if the goal is explicit expressions for differential invariants then the Fels–Olver method can’t readily be used to compute invariants explicitly9. In this case, we have shown that for Riemannian manifolds (M, g), the contact geometry can be fruitfully used to derive differential invariants and this requires as input data only the metric g and a rea- lisation of the structure group SO(n). With this data the Cartan connection for (M, g) can be constructed by linear algebra and differentiation. Subsequently, Theorem 4 provides an algo- rithm, Riemannian curves, for the curve invariants and, if required the Fels–Olver equivariant moving frame. The symbolic computational aspects of procedure Contact and algorithm Riemannian curves presented in this paper should be mentioned briefly. The Maple package DifferentialGeometry is ideally suited to the computation of all the relevant bundles and determining the derived type of any sub-bundle V ⊂ TM over manifold M . For instance, the two Riemannian examples presented in Section 5, take only a few minutes to complete commencing only with the met- ric and realisation of matrix group SO(3). Furthermore, the construction of differential curve invariants requires the inverse of the local diffeomorphism φ produced by procedure Contact. The proof of correctness of Contact in [28] shows that this algebraic problem will be block triangular. Thus the procedures discussed in this paper have quite good computational fea- tures. However, of much greater significance stands the proposition that the Frenet frames along a curve and hence the curve itself can be endowed with a contact geometry. This should have significance not only for the geometry of curves but also for Cartan’s method of moving frames as well as for the equivariant moving frames method of Fels and Olver. This is because contact systems are fundamental geometric objects and play a central role in differential geometry and differential equations. In fact, the construction of any contact system out of the components of a Cartan connection can in many ways replace or complement the step by step construction of moving frames championed by Cartan. What is more, a characterisation of contact systems in arbitrary jet spaces in the spirit of the Goursat normal form is known [2, 30] and could be applied to study the geometry of submanifolds of dimension p > 1 in general geometries as we have done here in the case p = 1. This raises the interesting question of the extent to which the results of this paper can be extended to homogeneous spaces in general and how they are connected to existing theory such as [5, 6, 8, 12, 14, 11, 26, 1, 21, 9, 10, 24]. In this respect it should be mentioned that the Fels–Olver theory of moving frames has appli- cation well beyond the explicit calculation of differential invariants and moving frames. Much can be accomplished within the theory even without this explicit knowledge; see Mansfield [16] for details. What we hope to have achieved in this paper is the presention of evidence sup- porting the proposition that it is useful to enrich the philosophy and practice of moving frames by exploring its links with contact geometry. Finally, we mention that an intriguing question is the relationship between the contact geo- metry of curves as explained here and the integrable motion of curves in various ambient mani- folds [13, 7, 17, 19]. 9However, it is sometimes possible to simplify the task of normalising the group action by using the known action of a subgroup, a result due to Kogan, [15]. 26 P.J. Vassiliou Acknowledgements I am indebted to the anonymous referees for insightful comments and for corrections which greatly improved the paper. Any remaining errors are mine. References [1] Alekseev D.V., Vinogradov A.M., Lychagin V.V., Geometry I, Encycl. Math. Sci., Vol. 28, Spinger-Verlag, Berlin, 1991. 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J. 12 (1983), 27–40. http://arxiv.org/abs/math.DG/0404377 http://arxiv.org/abs/math.DG/0406234 1 Introduction 2 Method of repère mobile applied to curves 2.1 Curves in the equi-affine plane 2.2 Goursat normal form 2.2.1 The derived flag 2.2.2 Cauchy bundles 2.3 Equi-affine invariants & the Goursat normal form 3 On the generalised Goursat normal form 3.1 The singular variety 3.1.1 The singular variety in positive degree 3.1.2 The resolvent bundle 3.2 The uniform generalised Goursat normal form 4 Space curves up to equi-affine transformations 4.1 Differential invariants via Contact 5 Curves in Riemannian manifolds 5.1 Curves in the Poincaré half-space 5.2 Curves in constant curvature Riemannian 3-manifolds 6 Closing remarks References

Contact Geometry of Curves

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