PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 24 – 27, 2002, Wilmington, NC, USA pp. 246–255

GEOMETRIC EQUIVALENCE ON NONHOLONOMIC THREE-MANIFOLDS

Kurt Ehlers Truckee Meadows Community College 7000 Dandini Blvd. Reno, NV 89512, USA

Abstract. We apply Cartan’s method of equivalence to the case of nonholonomic geometry on three-dimensional contact manifolds. Our main result is to derive the differential invariants for these structures and give geometric interpretations. We show that the symmetry group of such a structure has dimension at most four. Our motivation is to study the geometry associated with classical mechanical systems with nonholonomic constraints.

1. Introduction. Let {M, ds2 = h · , · i, E} be a smooth three-dimensional mani- fold endowed with a Riemannian metric ds2 and contact distribution E. A contact distribution on a 3-dimensional manifold is a smooth rank 2 vector subbundle of TM that is completely non-integrable. Locally E is defined by the Pfaffian equation η = 0 where η satisfies the nonintegrability condition dη ∧η 6= 0. The nonholonomic geodesic equations are obtained by computing the acceleration ∇c˙c˙ of a path c(t) and orthogonally projecting the result onto E. Our motivation for studying such a structure is a free particle moving in M, nonholonomically constrained to E, with kinetic energy 2T = h · , · i. The main question that we address in this paper is the following. Given two 2 2 nonholonomic contact structures {M, dsM , EM } and {N, dsN , EN }, is there a (local) diffeomorphism f : M → N preserving the nonholonomic geometry in the sense that nonholonomic geodesics in M are carried to nonholonomic geodesics in N? We answer this question by applying Cartan’s method of equivalence to uncover the differential invariants for such a structure. The analysis closely follows that for the subRiemannian (vakonomic) geometry as studied by Hughen [7] however we start with a smaller structure group and this leads to a richer variety of non- equivalent nonholonomic structures. In subRiemannianR p geometry one defines the length of a path c joining x to y to be `(c) = hc,˙ c˙idt. The distance from x to y is d(x, y) = inf (`(c)) taken over all paths tangent to E joining x to y. A path realizing this distance is a subRiemannian geodesic. In subRiemannian geometry the metric need not be defined off of E. It should be emphasized these geometries are not equivalent unless the distribution is integrable (see [11]). We have organized the paper as follows. In section 2 we sketch the derivation of the initial G−structure B0 for nonholonomic geometry on an n-dimensional man- n n ifold M . B0 is a subbundle of the coframe bundle over M . Sections of B0 are coframes adapted to the nonholonomic structure. Full details of the derivation can be found in [3] or [8]. Our original question of whether there exists a geometry pre- serving diffeomorphism is restated in terms of equivalence between G−structures.

1991 Mathematics Subject Classification. Primary: 70G45,53C10,37J60; Secondary: 58A17. Key words and phrases. Nonholonomic mechanics, equivalence method, G-structures.

246 GEOMETRIC EQUIVALENCE ON NONHOLONOMIC THREE-MANIFOLDS 247

In section 3 we restrict our attention to contact structures on 3-dimensional man- ifolds and apply the method of equivalence to obtain a subbundle B1 ⊂ B0 that is endowed with a canonical coframe and derive a basic set of differential invari- ants. In section 4 we interpret the invariants and apply the framing lemma to give a complete classification of nonholonomic structures on contact 3-manifolds with 4-dimensional symmetry.

2. Initial G-Structure. In this section we sketch the derivation of the initial G- structure for nonholonomic geometry on an n-dimensional Riemannian manifold M n endowed with a completely nonintegrable distribution E of constant rank m. For convenience, we adopt the following index notation: Capital Roman letters run from 1 to n, small Roman letters run from 1 to m, and Greek letters run from m+1 to n. Summation over repeated indices is to be understood unless otherwise stated. Let {X1, ..., Xn} be a local frame for TM. A frame for which {X1, ..., Xm} spans E is said to be adapted to the distribution. Unless otherwise stated, all frames will be assumed to be adapted to the distribution. E is totally nonholonomic if and only if span{Xi, [Xi,Xj], [Xk, [Xi,Xj]], ...} = TM for any adapted frame. Chow’s theorem then implies that any two points in M can be joined by a path tangent to E. The frame bundle F ∗M is a right principle GL(n)-bundle with projection τ : ∗ −1 ∗ F M → M and right action defined by Rg(η) = g η where η is a section of F M and g ∈ GL(n). It will be convenient to regard a section of F ∗M, evaluated at a n point m, to be a linear isomorphism from TmM into R represented by a column vector of one-forms so that it can be multiplied by a matrix on the left in the usual ∗ way. The initial G-structure B0 will be a subbundle of F M with structure group G0 ⊂ Gl(n). Sections of B0 will be coframes adapted to the geometric structure in a sense that we now describe. A coframe (η1, ..., ηn)tr is adapted to the distribution and metric if the ηα annihi- 2 1 1 m m late E and the metric restricted to the distribution is dsE = η ⊗ η + ··· + η ⊗ η . (We use ”tr” to denote transpose.) Let I denote the ideal in Λ∗(M) generated by the ηα. The derived ideal of I, denoted I0, is generated by the ηα for which dηα ∈ I. If we set I(0) = I and I(k+1) = (I(k))0 then we have a decreasing filtra- tion I = I(0) ⊃ I(1) ⊃ · · · ⊃ 0. The filtration terminating with the zero ideal is equivalent to the distribution being completely nonintegrable [1]. I(1) annihilates a distribution of rank r ≥ m. We will assume that coframes are arranged so that {ηr+1, ..., ηn} generates I(1). I Let e = (e1, ..., en) be a frame for TM for which η (eJ ) = δIJ (the Kronecker delta). We can then express the Levi-Civita connection ∇ associated with ds2 in terms of local one-forms ωIJ = −ωJI where X ∇X (eJ ) = ωIJ (X)eI . I D’Alembert’s Principle states that if a free particle whose path is c(t) with kinetic 1 energy T = 2 hc,˙ c˙i is subject to the constraintc ˙ ∈ Ec(t), then the constraining force 2 is ds perpendicular to Ec(t). This suggests making the following definition:

Definition 1. The nonholonomic connection D on (M n, ds2, E) is defined by X DX ej = ωij(X)ei. i 248 KURT EHLERS

The path c(t) is a nonholonomic geodesic if it satisfies the nonholonomic geodesic equations d ( c˙ +c ˙ ω (c ˙))e = 0 (1) dt i j ij i k wherec ˙k = η (c ˙). For a history of the nonholonomic connection see [11]. See [9] and references therein for current research on affine connections in mechanics and control theory. Suppose that we have another nonholonomic connection D¯ on E associated with the Riemannian metric ds¯ 2. When are the geodesics for these connections the same? Cartan answered this question in terms of the connection forms and coframes. The most general change of coframes that preserves the distribution E is of the form µ ¶ µ ¶ µ ¶ η¯i A b ηi = . (2) η¯α 0 a ηα where A ∈ GL(m), a ∈ GL(n − m), and b ∈ M(m, n − m). To preserve the subRiemannian metric ds2 = η1 ⊗η1 +···+ηm ⊗ηm we must insist that A ∈ O(m). For subRiemannian geometry there are no further restrictions (see ([10]). Cartan observed that in the nonholonomic case there is an additional restriction on the matrix b. Consider the modified metric ds¯ 2 =η ¯1 ⊗ η¯1 + ···η¯n ⊗ η¯n and associated nonholonomic connection D¯. The corresponding geodesic equations are

(s ¨i +s ˙jω¯ij(s ˙))¯ei = 0 (3) k wheres ˙k =η ¯ (s ˙). In comparing (1) and (3) there is no loss in generality in assuming A = id and we observe Proposition 1. (Cartan 1928 [3]) The geodesics of D and D¯ are the same iff ωij(T ) =ω ¯ij (T ) for all T ∈ E. The most general change of coframes preserving the nonholonomic structure is given in the following theorem. Since this theorem provides the starting point for our analysis we repeat its proof following [8]. Theorem 1. (Cartan 1928 [3]) Let (M n, ds2 = h · , · i, E) be a nonholonomic struc- ture with rank(E) = m and adapted coframe η = (ηi, ηα)tr. The most general change of coframes preserving the nonholonomic geometry are of the form η¯ = gη where g ∈ G0 and G0 is the group of all matrices of the form µ ¶ A b (4) 0 a

λ (1) for which b = [biα] with biλη ∈ I , A ∈ O(m), and a ∈ H < GL(n − m). Proof. For simplicity, assume that A = id , then ηj ≡ η¯j (mod ηα). By the the α preceding proposition ωij ≡ ω¯ij (mod η ). Subtracting the structure equations for dηi and dη¯i we get i i j α j α α dη − dη¯ = −ωij ∧ η − ωiα ∧ η +ω ¯ij ∧ η¯ +ω ¯iα ∧ η¯ ≡ 0(mod η ). i i λ By (4) with A = id we see thatη ¯ = η + biλη so i i λ λ dη¯ = dη + dbiλη + biλdη . λ α λ (1) Therefore biλdη ≡ 0 (mod η ) or equivalently biλη ∈ I . (Note that H may −1 be a subgroup of GL(n − m) to ensure that g ∈ G0 for all g ∈ G0. One can verify that this condtion also ensures that {η¯r+1, ..., η¯n} generates I(1).) GEOMETRIC EQUIVALENCE ON NONHOLONOMIC THREE-MANIFOLDS 249

∗ Definition 2. The subbundle B0 ⊂ F M of with structure group G0 ⊂ GL(n) is the initial G-structure for nonholonomic geometry. Sections of B0 are called 0-adapted coframes. For a nonholonomic structure on an three-dimensional manifold, the nonintegra- bility condition dη3 ∧ η3 6= 0 implies that I(1) = 0 so the block b in (4) must be the zero matrix. The initial G-structure for nonholonomic geometry on 3-manifolds has structure group ½µ ¶ ¾ A 0 G = |A ∈ O(2), a 6= 0 . (5) 0 0 a The initial G-structure completely characterizes the nonholonomic geometry in the sense that a given a G-structure, we can reconstruct the geometric structure (see [5]). Two G-structures, τM : B(M) → M and τN : B(N) → N, are said to be equivalent if there is a diffeomorphism f : M → N for which f1(B(M)) = B(N) where f1 is the induced bundle map. If we think of β ∈ B(M) as a linear isomor- 3 −1 phism β : TτM (β)M → R , then f1(β) = β ◦ (f∗) where f∗ is the differential of f. Our original question of whether there is a diffeomorphism between manifolds that preserves the nonholonomic geometry can be answered by determining whether two G-structures are equivalent. This question is answered in terms of the differential invariants for the G-structures found by applying the method of equivalence.

3. Application of the method of equivalence. In this section we apply the method of equivalence to obtain a subbundle of B0 that has a canonical coframing. Once this coframe is obtained, all local invariants and symmetries of the geometric structure can be obtained using the general theory of e-structures [5]. 3 1 2 3 tr The R -valued one-form Ω = (Ω , Ω , Ω ) on B0 defined by Ω(v) = b(τ∗v) where b ∈ B0 and v is a vector field on B0, is called the tautological one-form. (Ω is also known as the soldering form in the physics literature.) If η is a local section of −1 B0, then Ω has local expression Ω = g η. The tautological one-form is semi-basic and provides a partial coframe for B0. Its most important property is contained in the following theorem [5]: Theorem 2. Suppose G is a connected group and B(M) and B(N) are G-structures over manifolds M and N with tautological one-forms ΩM and ΩN . If there is a ∗ smooth map F : B(M) → B(N) such that F ΩN = ΩM , then these G-structures are equivalent. To find the map F , we use a generalization of Cartan’s method of the graph (see [12]). Roughly speaking, we look for an integral manifold Σ ⊂ B(M) × B(N) of the one form ΩN − ΩM that projects diffeomorphically onto each factor. Σ is then ∗ the graph of a smooth function g : B(M) → B(N) for which g ΩN = ΩM . By the preceding theorem, B(M) and B(N) are then equivalent. Unfortunately ΩM and ΩN do not provide full coframes for M and N as is required for the method of the graph. To apply the method in a meaningful way, we need to have a full coframe uniquely associated to the G-structure. In the following section we apply Cartan’s method of equivalence to obtain a sub-bundle of the initial G-structure that is endowed with a canonical coframe.

3.1. Reduction to a G1 structure. The structure equations for the tautological one-form on B0 are dΩ = −Γ ∧ Ω + T where T is a semi-basic two-form and Γ is a pseudo-connection on B0 [2]. A pseudo-connection is a TeG0-valued one-form satisfying Γ(ξB0 ) = ξ where ξB0 is the vector field on B0 generated by ξ ∈ TeG0 using the action of G0 on B0. Γ is referred to as a pseudo-connection because it is 250 KURT EHLERS not assumed to be equivariant with respect to the group action. T is the torsion associated with Γ. Together with the tautological one-form, the pseudo-connection provides a coframing for B0. We have           1 1 1 1 1 2 3 Ω 0 γ 0 Ω T23 T31 T12 Ω ∧ Ω  2     2   2 2 2   3 1  d Ω = − −γ 0 0 ∧ Ω + T23 T31 T12 Ω ∧ Ω 3 3 3 3 3 1 2 Ω 0 0 α Ω T23 T31 T12 Ω ∧ Ω i where Tjk : B0 → R. 1 We can use Lie algebra compatible absorption ([5]) to choose γ so that T12 and 2 3 3 T12 are simultaneously zero. We can also choose α so that T23 = T31 = 0. Note that 3 3 3 T12 cannot be made equal to zero because of the contact condition dΩ ∧ Ω 6= 0. The structure equations are then           1 1 1 1 2 3 Ω 0 γ 0 Ω T23 T31 0 Ω ∧ Ω  2     2   2 2   3 1  d Ω = − −γ 0 0 ∧ Ω + T23 T31 0 Ω ∧ Ω 3 3 3 1 2 Ω 0 0 α Ω 0 0 T12 Ω ∧ Ω 1 2 3 Now {γ, α, Ω , Ω , Ω } provides a coframe for B0, but it is not unique, we can still add arbitrary multiples of Ω3 to α. Following Cartan’s prescription, we check the ∗ 3 −1 3 action of G0 on the torsion space. Let g ∈ G0, then RgΩ = a Ω . Differentiating both sides of this identity, we get ∗ 3 ∗ 3 ∗ 3 ∗ 3 ∗ 1 2 d(RgΩ ) = RgdΩ = Rg(α ∧ Ω ) + (RgT12)Rg(Ω ∧ Ω ) ∗ 3 1 2 3 = detA(RgT12)Ω ∧ Ω (mod Ω ) and −1 3 −1 3 1 2 3 d(a Ω ) = a T12Ω ∧ Ω (mod Ω ) ∗ 3 −1 3 3 We deduce that RgT12 = detAa T12 and we can use G0 to force T12 ≡ 1. The Stabilizer subgroup, G1 ⊂ G0 for this choice of torsion consists of matrices of the form µ ¶ A 0 (6) 0 det(A) with A ∈ O(2). We now consider the reduced G1 structure B1. The structure equations are           1 1 1 1 2 3 Ω 0 γ 0 Ω T23 T31 0 Ω ∧ Ω  2     2   2 2   3 1  d Ω = − −γ 0 0 ∧ Ω + T23 T31 0 Ω ∧ Ω 3 3 3 3 1 2 Ω 0 0 0 Ω T23 T31 1 Ω ∧ Ω 1 2 1 3 If we replace γ with γ + 2 (T31 + T23)Ω a straight forward computation shows that 1 2 1 2 3 we can take T23 = −T31 ≡ p. To simplify notation we set q = T31, r = T23, s = T23, 3 and t = T31. Now the choice of pseudo-connection is unique, the only solution to A ∧ Ω = 0 is A = 0, and we have a canonical coframe {γ, Ω1, Ω2, Ω3} that is preserved under all automorphisms of B1.A G-structure with a canonical coframing is referred to as an e-structure (see [5]). We summarize the preceding results in the following: Theorem 3. Let {M, E, h·, ·i} be a three-dimensional manifold endowed with a non- ∼ holonomic contact structure. There is a G1 = O(2) structure B1 → M endowed with a canonical coframe {γ, Ω1, Ω2, Ω3} with structure equations           Ω1 0 γ 0 Ω1 p q 0 Ω2 ∧ Ω3 d  Ω2  = −  −γ 0 0  ∧  Ω2  +  r −p 0   Ω3 ∧ Ω1  (7) Ω3 0 0 0 Ω3 s t 1 Ω1 ∧ Ω2 GEOMETRIC EQUIVALENCE ON NONHOLONOMIC THREE-MANIFOLDS 251 where p, q, r, s, and t are the torsion functions associated with the pseudo-connection.

3.2. Bianchi Identities and the Differential Invariants on B1. In this sec- tion we investigate the quadratic relations between the torsion functions and define three differential invariants on M. These three functions are preserved under all automorphisms of B1 and by the general theory of e-structures form a complete set of differential invariants for nonholonomic contact structures on three-dimensional manifolds (see [5]). Bianchi identities are relations between the torsion functions that arise from the fact that d2 = 0. Computing d2ΩI = 0 for I = 1, 2, 3 we get, respectively 0 = −dγ ∧ Ω2 + (dp + (q + r)γ) ∧ Ω2 ∧ Ω3 + (dq − 2pγ) ∧ Ω3 ∧ Ω1 (8) +(qs − pt)Ω1 ∧ Ω2 ∧ Ω3 0 = dγ ∧ Ω1 + (dr − 2pγ) ∧ Ω2 ∧ Ω3 − (dp + (q + r)γ) ∧ Ω3 ∧ Ω1 (9) −(rt + ps)Ω1 ∧ Ω2 ∧ Ω3, 0 = (ds − tγ) ∧ Ω2 ∧ Ω3 + (dt + sγ) ∧ Ω3 ∧ Ω1 + (q − r)Ω1 ∧ Ω2 ∧ Ω3 (10) We will also need the the expansions of the derivatives of the torsion functions in terms of the canonical coframe on B1. They are X I ds = tγ + SI Ω (11) X I dt = −sγ + TI Ω (12) X I dp = −(q + r)γ + PI Ω (13) X I dr = 2pγ + RI Ω (14) X I dq = 2pγ + QI Ω (15)

where the SI ,TI ,PI ,RI and QI are the covariant derivatives of the respective torsion functions in the Ωi directions. Proposition 2. The two-form dγ is semi-basic. In other words we can write 2 3 3 1 1 2 dγ = H1Ω ∧ Ω + H2Ω ∧ Ω + KΩ ∧ Ω (16) Proof. Wedging (9) with Ω3 gives dγ ∧ Ω3 ∧ Ω1 = 0 and wedging (10) with Ω3 gives dγ ∧ Ω2 ∧ Ω3 = 0. Wedging (8) with Ω1 and (9) with Ω2 and adding gives dγ ∧ Ω1 ∧ Ω2 = 0. Together, these imply the assertion. We define a function K : M → R by the relation dγ = KΩ1 ∧ Ω2 (mod Ω3). Proposition 3. K, s2 + t2, and p2 + qr are well defined functions on M. Proof. Differentiating (16) and wedging with Ω3 gives 0 = d2γ = dK ∧ Ω1 ∧ Ω2 ∧ Ω3. The differential of K is therefore semi-basic and K must be constant in the fiber direction. Similarly, using (11) and (12) we compute d(s2 + t2) ≡ 0 (mod Ω1,Ω2,Ω3), and using (14), (13), and (15) we compute d(p2 + qr) ≡ 0 (mod Ω1,Ω2 ,Ω3). The functions s2 + t2 and p2 + qr must therefore be constant in the fiber direction. By the general theory of e-structures ([5], [2]), the functions K, s2 + t2, and p2 + qr form a complete set of differential invariants for the nonholonomic manifold . Other relations based on the Bianchi identities that will be useful in the following sections are

r − q = S1 + T2, rt + ps = R1 − P2, and pt − qs = P1 + Q2. (17) 252 KURT EHLERS

The induced action of the group G1 ≡ O(2) on the torsion space can also be deduced from the Bianchi identities. In particular we have (mod Ω1, Ω2, Ω3) µ ¶ µ ¶ µ ¶ s 0 γ s d = , (18) t −γ 0 t and µ ¶ µ ¶ µ ¶ 1 (q + r) 0 2γ 1 (q + r) d 2 = 2 . (19) p −2γ 0 p 1 1 G acts on the torsion plane (s, t) by the weight one representation and on ( 2 (q + r), p) by the weight two representation. Both torsion planes rotate as we move in the fiber direction.

4. Interpretation of the invariants. 4.1. Nonholonomic manifolds with maximal symmetry. The four-dimensional B1-structure has a canonical coframing. It follows from the framing lemma ([10], [5]) that the maximal dimension of the (local) Lie group G of diffeomorphisms of B1 that preserves the coframing {γ, Ω1, Ω2, Ω3} is of dimension 4 (the dimension of B) and that this bound is achieved if and only if K and the torsion functions p, q, r, s are constant. In this case G acts freely and transitively on B and the covectors comprising the coframe can be identified with the Maurer-Cartan forms on G.

Proposition 4. The dimension of the (local) symmetry group of B1 is four if and only if p = q = r = s = t = 0 and K is constant. K is therefore the only local invariant for nonholonomic contact manifolds with maximal symmetry.

Proof. By the framing lemma the torsion functions must all be constant. G1 acts on the torsion plane (s, t) by rotations (18), so if s and t are constant, they must be zero. Since s and t are zero, S1 and T2 are both zero and (17) implies that q = r. Equation (19) then implies that the torsion planes (p, q) and (p, r) rotate as we move in the fiber direction. As a result p, q, and r must all be zero. The Bianchi identities are now 0 = d2Ωi = dγ ∧ Ωi, i = 1, 2. Wedging (16) with Ωi 1 2 1 2 3 we see that Hi = 0 so dγ = KΩ ∧ Ω . We can write dγ = KΩ ∧ Ω = KdΩ so we must have γ = KΩ3. The structure equations are therefore dΩ1 = KΩ2 ∧ Ω3, dΩ2 = KΩ3 ∧ Ω1, and dΩ3 = Ω1 ∧ Ω2. If we identify {Ω1, Ω2, Ω3} with the Maurer-Cartan forms on M, then we can identify the torsion constants with the structure constants of M. Locally, M is equivalent to SO(3) if K > 0, the Heisenberg group if K = 0, or Sl(2, R) if K < 0 (see [7]). In these cases the nonholonomic contact structure is given by D = (Ω3)⊥ and we can 3 interpret Ω as a connection form on the principal bundle π : M → Σ where π∗ is a Riemannian submersion and Σ is a surface with constant Gaussian curvature K. The four dimensional symmetry groups of these structures are the base manifolds M centrally extended by S1. Note that while the classifications in the maximal symmetry case for nonholonomic and subRiemannian cases are analogous (see [7]), the geometries are not the same. A standard example of a nonholonomic system with four dimensional³ symmetry´ is 2 the nonholonomic system on the Heisenberg group with metric ds2 = 1 + y dx⊗ ³ ´ 4 x2 xy dx + 1 + 4 dy ⊗ dy + dz ⊗ dz + xdy ⊗ dz − ydx ⊗ dz − 2 dx ⊗ dy and contact distribution annihilated by η3 where 1 (η1, η2, η3)tr = (dx, dy, dz + (xdy − ydx))tr 2 GEOMETRIC EQUIVALENCE ON NONHOLONOMIC THREE-MANIFOLDS 253 is the standard left invariant coframing. We have s2 + t2 = p2 + qr = K = 0. In 1 2 3 tr 2 this case (η , η , η ) is a B1-adapted coframe and ds can be identified with the canonical metric.

4.2. A geometric interpretation of the invariants. We have observed that K can be interpreted as the Gaussian curvature in certain circumstances. Hughen ([7]) observed that the invariants for subRiemannian geometry can be interpreted in terms of the sectional curvature, in the plane of E, of a canonically defined Riemannian metric. We can make a similar interpretation in the nonholonomic 2 case. We consider the canonical metric introduced by Cartan given by ds = (Ω1)2+ 2 2 3 2 1 2 3 tr (Ω ) +(Ω ) where (Ω , Ω , Ω ) is the tautological one-form on B1 (see [3], section 9, see also [8]). Since any section of B1 is an orthonormal coframe in this metric, we can compute the Levi-Civita connection form A relative to any B1 adapted coframe using (7) and the structure equation dΩ = −A ∧ Ω. We have   0 α3 −α2 A =  −α3 0 α1  α2 −α1 0 1 1 2 3 1 1 2 3 1 3 with α1 = ( 2 − p)Ω − rΩ − sΩ , α2 = −qΩ + ( 2 + p)Ω − tΩ , and α3 = γ − 2 Ω The curvature matrix is Θ = [Θij ] = dA − A ∧ A. We compute 3 θ = dα − α1 ∧ α = (K + p2 + rq − )Ω1 ∧ Ω2 (mod Ω3) 12 3 2 4 The sectional curvature in the direction of the distribution is therefore K − p2 − 3 rq − 4 . If s = t = 0 then r = q and we recover the result obtained by Hughen for subRiemannian geometry.

4.3. Examples with three-dimensional symmetry. The group G1 acts on the 1 torsion planes (s, t) and ( 2 (r + q), p) by rotations. If s, t, p, or r + q 6= 0 we can use G1 to simplify the torsion. The stabilizer subgroup G2 for the simplified choice of torsion will then be discrete, consisting of the diagonal matrices in SO(3). The resulting G2-structure B2 is three-dimensional. The framing lemma implies that the dimension of such a structure is three if and only if the torsion is constant. In these cases the torsion can be identified with the structure constants of the Lie algebra of the (local) Lie group of symmetries of the nonholonomic structure. Ac- cording to Lie’s third theorem (see [5] or [4]), we can determine the group from these structure constants and, in principle, make a systematic classification of non- holonomic structures with three-dimensional symmetry. We will not pursue this study in the present work1. There are two situations to investigate: s or t 6= 0, and 1 p or 2 (r + q) 6= 0.

4.3.1. Case 1: s or t 6= 0. In this case we can use the action of G1 to force t = 0 and s > 0. The pseudo-connection is then basic and we can express it as γ = 1 2 3 λ1Ω + λ2Ω + λ3Ω . The structure equations in this case are       1 2 3 Ω (λ3 + p) q −λ1 Ω ∧ Ω 2 3 1 d  Ω  =  r (λ3 − p) −λ2   Ω ∧ Ω  (20) Ω3 σ 0 1 Ω1 ∧ Ω2

1See [7] for this study in the case of subRiemannian geometry where there are nine cases to analyze. The classification in the nonholonomic case is more complicated because of the additional structure constants. 254 KURT EHLERS √ where σ = s2 + t2. The Bianchi identities 0 = d2Ωi for i = 1, 2, 3 give respectively

0 = qσ + rλ1 − pλ2 − λ2λ3 = (λ3 − p)(σ + λ1) − qλ2 = q − r − σλ2

Example. Consider the nonholonomic structure induced by the coframing (η1 = dx, η2 = dy, η3 = eσydz+xdy)tr on R3. The structure equations are dΩ1 = dΩ2 = 0, and dΩ3 = σ Ω3 ∧ Ω1 + Ω1 ∧ Ω2. The corresponding three-dimensional Lie algebra is solvable but not nilpotent. The factor σ will show up explicitly in the geodesic equations. In subRiemannian geometry this case does not occur, the group action can be used to force the torsion function σ to be zero.

1 4.3.2. Case 2: p or q + r 6= 0. The group G1 acts on the torsion plane (p, 2 (r + q)) by rotations. We can use this action to force r = q. The stabilizer group again consists of the diagonal matrices in SO(3). The structure equations are       1 2 3 Ω (λ3 + P ) q −λ1 Ω ∧ Ω 2 3 1 d  Ω  =  −q (λ3 − P ) −λ2   Ω ∧ Ω  (21) Ω3 s t 1 Ω1 ∧ Ω2

1 where P = ||(p, 2 (r + q))||. The Bianchi identities are

0 = q(s − λ1) − (λ3 + P )(λ2 + t)

0 = (λ3 − P )(s + λ1) + q(t − λ2)

0 = 2q − sλ2 + λ1t.

If we consider the case s = t = 0, then q = 0 and the Bianchi identities reduce to (λ3 − p)λ1 = (λ3 + p)λ2 = 0. A complete classification of the resulting Lie algebras for this case can be found in [7]. The cases for which λ1 and λ2 are also zero include several standard examples of nonholonomic systems. For these cases the Lie algebras are so(3) for λ3 > p, e(2) for λ3 = p, sl(2, R) for −p < λ3 < p, e(1, 1) for λ3 = −p, and sl(2, R) for λ3 < −p. Example: a skate on SE(2). Consider a vertical ice skate moving on the xy- plane. The configuration space is SE(2). We take the Cartesian coordinates (x, y) to represent the point of contact of the skate on the ice and the coordinate θ to represent the orientation of the skate with respect to the positive x-axis. The skate is allowed to spin in the θ direction but can only translate in the direction in which it is oriented. The kinetic energy of the skate is given by 2T = m((dx)2 + (dy)2) + I(dθ2) where m is the mass of the skate and I is its moment of inertia about the point³ of´ contact.³ We´ start with an orthonormal³ frame´ {X1,X³ 2,X´3} where cos√ θ sin√ θ √∂θ sin√ θ cos√ θ X1 = m ∂x + m ∂y,X2 = , and X3 = − m ∂x + m ∂y, and √ √ I √ √ √ dual coframe ( m cos θdx + m sin θdy, Idθ, − m sin θdx + m cos θdy)tr. We compute the connection form and torsion functions for the reduced B1-structure 1 3 1 1 defined in (7) to be γ = −dφ + 2I Ω , p = 2I cos 2φ, and q = r = − 2I sin 2φ, 1 2 2 1 s = t = 0. The invariants for this structure are therefore K = 2I , p + q = 4I2 , 2 2 and s + t = 0. The structure equations for the reduced G2-structure B2 are dΩ1 = I−1Ω2 ∧ Ω3, dΩ2 = 0, and dΩ3 = Ω1 ∧ Ω2. We recognize the to be the structure constants for the Lie algebra se(2) and we can identify the symmetry group SE(2) for this structure with the configuration manifold. GEOMETRIC EQUIVALENCE ON NONHOLONOMIC THREE-MANIFOLDS 255

5. Conclusions and future work. We have applied the method of equivalence to obtain the differential invariants for nonholonomic geometry on a contact 3- manifold. We have discussed some of the conclusions that can be drawn from these invariants. There is much more that can be said even in this simplest case. It is possible to give a complete classification of nonholonomic structures on Lie groups in terms of the structure constants using the classification for 3-dimensional Lie algebras [11]. It would be interesting to compare our invariants to those found by others using different means (see [11] and references therein). We (Jair Koiller and the author) have applied the method of equivalence and obtained preliminary results for nonholonomic geometry on Engel manifolds. The initial G-structure in this case has a six dimensional structure group. The results of this study will be reported elsewhere.

Acknowledgments. I would like to thank Jair Koiller for introducing me to this prob- lem during a visit to Rio de Janeiro funded by FAPERJ. I would also like to thank Richard Montgomery and the referees for their insightful and helpful suggestions.

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