INVARIANT NONHOLONOMIC RIEMANNIAN STRUCTURES ON THREE-DIMENSIONAL LIE GROUPS

. We consider Riemannian manifolds endowed with a nonholonomic distribution. These structures model mechanical systems with a (positive def- inite) quadratic Lagrangian and nonholonomic constraints linear in velocities. We classify the left-invariant nonholonomic Riemannian structures on three- dimensional simply connected Lie groups, and describe the equivalence classes in terms of some basic isometric invariants. The classiﬁcation naturally splits into two cases. In the ﬁrst case, it reduces to a classiﬁcation of left-invariant sub-Riemannian structures. In the second case, we ﬁnd a canonical frame with which to directly compare equivalence classes.

1. Introduction. A nonholonomic Riemannian manifold is a Riemannian manifold (M, g) together with a nonholonomic (i.e., nonintegrable) distribution D. There are two (inequivalent) geometries that may be defined on a nonholonomic Riemannian manifold. In the first, the dynamics are specified by means of the Chetaev equations; this approach may be viewed as a special case of nonholonomic mechanics. In the second (a special case of vakonomic mechanics), the dynamics follow a classical variational principle: geodesics are length minimizers of the Carnot-Carathéodory distance. While the term nonholonomic Riemannian geometry has, in the past, been used to refer to both of these geometries, we shall use it to mean only the first. The second we shall refer to, by the usual modern terminology, as sub-Riemannian geometry.
From the point of view of physics, nonholonomic Riemannian geometry describes a free particle moving in M with kinetic energy Lagrangian L(x,ẋ) = 1 2 g(ẋ,ẋ), and subject to (time-independent) nonholonomic constraints linear in velocities (given by D). The study of this area has attracted numerous geometers. Among the pioneers we mention, in particular, E. Cartan and Synge (who first introduced the nonholonomic connection), Vrȃnceanu, as well as Schouten and Wagner (who introduced the curvature tensors of the nonholonomic connection). More recent contributions, using the language of modern differential geometry, have been made by Vershik and collaborators [34,33,35,36], Lewis [20,21] and Bloch, Crouch and collaborators [2] (and references therein), to mention but a few. (The last two, in particular, have studied control systems on nonholonomic Riemannian manifolds.) Some standard references for nonholonomic mechanics are [2,6,7].
Left-invariant nonholonomic Riemannian structures on Lie groups (often referred to as LL systems in the context of nonholonomic mechanics) are basic prototypes of nonholonomic Riemannian manifolds: their study is a first step towards an understanding of more general structures. In contrast to invariant sub-Riemannian structures, these structures have not been extensively studied. To the best of our knowledge, the work that has been done has been mainly devoted to reduction (see, e.g., [2,7] and references therein), integrability (including the existence of invariant measures/volumes; see, e.g., [16,14]) and problems from mechanics (particularly, involving symmetries and conservation laws), e.g., the Chaplygin problem [5,24,11,4], the Suslov problem [29,13] and problems with nonlinear constraints [31,30,27]. (Also of interest for integrability are the so-called LR systems, where the metric is left invariant and the distribution is right invariant; see, e.g., [37,12].) In particular, little effort has been devoted to tackling the questions of equivalence and classification of nonholonomic Riemannian structures. Cartan [3] was the first to apply his method of equivalence in nonholonomic mechanics (specifically, nonholonomic Riemannian manifolds), though only to the case of a strongly nonholonomic distribution. (The papers [15,32] are a modern exposition of Cartan's ideas; the former paper in particular discusses some generalizations of Cartan's work to more general nonholonomic distributions.) However, regarding specific problems of equivalence and classification, we are aware only of the papers [9,10]. In the first, the author makes use of the method of equivalence in order to determine differential invariants for three-dimensional nonholonomic Riemannian manifolds with contact distributions. (The paper does not, however, give a classification of these structures.) In the second paper, the authors again make use of the method of equivalence to derive the differential invariants, this time studying (four-dimensional) Engel manifolds.
In this paper we consider the equivalence and classification of nonholonomic Riemannian manifolds. In particular, we classify the left-invariant nonholonomic Riemannian structures on simply connected three-dimensional Lie groups. In section 2 we recall and develop some basic elements of nonholonomic Riemannian manifolds. In sections 2.1 and 2.2 we recall the definition of the Schouten curvature tensor, introduce suitable notions of sectional curvature, Ricci curvature, etc., and introduce some natural equivalence relations between nonholonomic Riemannian manifolds (geodesic equivalence, affine equivalence, and equivalence up to "nonholonomic isometry" or N H-isometry). While many of the definitions and results of these sections are new, they are a natural and easy consequence of working by analogy with the Riemannian case. Lastly, in section 2.3 we discuss some specifics of left-invariant nonholonomic Riemannian structures on Lie groups. In particular, we discuss the case of left-invariant nonholonomic Riemannian structures on Lie groups whose nonholonomic geodesics are left cosets of one-parameter subgroups.
Section 3 comprises the main contribution of this paper. In this section we classify the (left-invariant) nonholonomic Riemannian structures on three-dimensional simply connected Lie groups, up to N H-isometry and rescaling. We also identify (in section 3.1) some basic isometric invariants for structures in three dimensions.
The classification of structures naturally splits into two classes, depending on an invariant ϑ ≥ 0. In the first case (ϑ = 0), the classification reduces to a classification of (left-invariant) sub-Riemannian structures in three dimensions; these structures were recently classified by Agrachev and Barilari [1]. The second case (ϑ > 0) is of the most interest. For these structures, we construct a canonical frame by using an intrinsic contact form. The commutator relations of the canonical frame uniquely determine the nonholonomic Riemannian structure. For the invariant structures, the canonical frame is left invariant; consequently, any N H-isometry between these structures must be the composition of a left translation with a Lie group isomorphism. This leads to the introduction of three further invariants 0 , 1 and 2 , which, together with ϑ, form a complete set of differential invariants for structures on the unimodular groups. For structures on (most) non-unimodular groups, we show that there are at most two non-N H-isometric structures with the same invariants ϑ, 0 , 1 and 2 . In section 3.4 we look more closely at those structures whose nonholonomic geodesics are the left cosets of one-parameter subgroups. In particular, we characterize these structures in terms of the invariants.
In section 4 we discuss two classical problems, viz. the Chaplygin and Suslov problems, and their relation to our classification.
Finally, in appendix A, we briefly recall the Bianchi-Behr classification of the real three-dimensional Lie algebras, and discuss some algebraic means of identifying each algebra (used in section 3).
2. Nonholonomic Riemannian geometry. Let g be a Riemannian metric on an n-dimensional smooth manifold M . The geodesics of (M, g) are exactly the geodesics of the corresponding Levi-Civita connection ∇; these geodesics coincide with the extremal curves of the mechanical system given by the Lagrangian L(x,ẋ) = 1 2 g(ẋ,ẋ). Indeed, the Euler-Lagrange equations for L, d dt ∂L ∂ẋ i − ∂L ∂x i = 0, are the second-order ODEs g ij (ẍ j + Γ j k ẋ kẋ ) = 0, where g ij are the components of g and Γ j k are the Christoffel symbols of ∇. These equations have the obvious intrinsic form g (∇ċċ) = 0. (Here g : Let D be a nonintegrable distribution of rank m < n on M ; D represents a nonholonomic constraint linear in velocities. The distribution D is said to be completely nonholonomic if the filtration D (1) ⊆ D (2) ⊆ · · · defined by D (1) = D, T M and D (N ) = T M . The simplest case N = 2 is of particular interest; in this instance D is called strongly nonholonomic. (A rank-two nonintegrable distribution on a three-dimensional manifold M is always strongly nonholonomic.) If D is completely nonholonomic, then the Chow-Rashevskii theorem guarantees that every two points in M can be joined by an integral curve of D.  [36]). A nonholonomic Riemannian manifold is a triple (M, g, D), where M is a smooth manifold of dimension n, g is a Riemannian metric on M , and D is a smooth nonintegrable distribution of rank m (1 < m < n) on M .
Let D ⊥ be the distribution orthogonal to D (so that T M = D ⊕ D ⊥ ) and let P : T M → D and Q : T M → D ⊥ be the associated projection operators.
for every t.
The nonholonomic geodesics of (M, g, D) coincide with the nonholonomic extremals of L (see, e.g., [35,20]). Indeed, the Chetaev equations for L are d dt where ϕ a = B a i dx i span the annihilator D • = g (D ⊥ ) of D and λ a are Lagrange multipliers, determined by the constraints. Specifically for the kinetic energy Lagrangian, the equations 1 are given by The nonholonomic geodesics of (M, g, D) can be characterized as the geodesics of an affine connection on M . The condition ∇ċ (t)ċ (t) ∈ D ⊥ c(t) is equivalently written as P(∇ċ (t)ċ (t)) = 0. Accordingly, the nonholonomic connection ∇ is defined as It is not difficult to show that ∇ is an affine connection, i.e., it is tensorial in its first argument and a derivation in the second. Furthermore, the nonholonomic geodesics are exactly the geodesics of ∇ (see, e.g., [35,20]), i.e., integral curves c of D such that ∇ċ (t)ċ (t) = 0 for every t. (In fact, ∇ċ (t)ċ (t) = 0 is exactly the geometric expression of the reduced Chetaev equations; see, e.g., [18,28].) The nonholonomic connection ∇ can be characterized (like the Levi-Civita connection) as the unique connection that is "metric" and "torsion-free" (see, e.g., [19]). Specifically, the nonholonomic connection is the unique affine connection Notice that the connection ∇ depends only on (D, g| D ) and the complement D ⊥ . Accordingly, one could equivalently define a nonholonomic Riemannian structure as (M, D, D ,ḡ), where D is a distribution complementary to D andḡ is a metric defined only on D.
2.1. Curvature. Let (M, g, D) be a nonholonomic Riemannian manifold with associated nonholonomic connection ∇. As D is nonintegrable, the Riemannian curvature tensor is not defined. The Schouten (1, 3) curvature tensor [8] where X, Y, Z ∈ Γ(D). The associated (0, 4) curvature tensor K, given by satisfies the following symmetries: In contrast to the (0, 4) Riemannian curvature tensor, K is in general not skewsymmetric in the final two arguments, nor is it symmetric if one swaps the first two arguments with the last two. We decompose K into two tensors R and C, where R is the component of K that is skew-symmetric in the last two arguments and C is the component that is symmetric in the last two arguments. Specifically, we have R satisfies, in addition to (S1 ) and (S2 ), the following symmetries: . (This follows from (S1 ), (S2 ) and (S3 ).) (On the other hand, C(W, X, Y, Z) = C(W, X, Z, Y ).) That is, R satisfies all of the symmetries of the Riemannian (0, 4) curvature tensor, and hence behaves analogously to the Riemannian tensor. Thus we may define a sectional curvature, a Ricci tensor and a (Ricci) scalar curvature in terms of R.
Let Π x be a two-dimensional subspace of D x and let (U x , V x ) be a basis for Π x . We define the sectional curvature of Π x as As in the Riemannian case, one can show that R is well defined (i.e., does not depend on the choice of U x and V x ) and determines R. We shall also write R(Π x ) as R(U x , V x ). The Ricci tensor Ric : Γ(D) × Γ(D) → C ∞ (M ) is the (0, 2)-tensor given by where (X p ) m p=1 is an orthonormal frame for D. (The definition of Ric does not depend on the choice of X 1 , . . . , X m .) It follows from the symmetries of R that Ric is symmetric. We call the trace of the endomorphism g| D • Ric the scalar curvature, denoted S. (Here g| D : Γ(D) → Γ(D * ), X → g(X, · ), g| D = ( g| D ) −1 and Ric : Γ(D) → Γ(D * ), X → Ric(X, · ).) In terms of (X p ) m p=1 , we have In a similar fashion to the Ricci tensor, let A be the (0, 2)-tensor given by In general, A is not symmetric. Thus we define two tensors A sym and A skew to be the symmetric and skew-symmetric parts of A, respectively. That is, In terms of an orthonormal frame (X p ) m p=1 for D, we then have Both A sym and A skew are trace-free, so there is no analogue of the scalar curvature in this case.

2.2.
Equivalence. There are three natural equivalence relations (of increasing strength) between nonholonomic Riemannian manifolds. We shall describe each of these in turn. (However, we shall be chiefly concerned with the strongest of the three.) Geodesic equivalence. We shall say that two nonholonomic Riemannian manifolds (M, g, D) and (M , g , D ) are geodesically equivalent if there exists a diffeomorphism φ : M → M establishing a one-to-one correspondence between the geodesics of (the respective nonholonomic connections of the two structures) ∇ and ∇ , i.e., c is an nonholonomic geodesic of (M, g, D) if and only if φ • c is an nonholonomic geodesic of (M , g , D ).
; it is easy to show that B is tensorial in both arguments. Let V x ∈ D x and let c be the geodesic of ∇ such that c(0) = x, c(0) = V x . Let X ∈ Γ(D) be an extension of V x to a neighbourhood of x such that X coincides withċ along c in the neighbourhood. Then (We have used the fact that ∇ X Y (x) depends only on the values of Y along any curve tangent to X(x); see, e.g., [15].) That is, B(X, X) = 0 for every X ∈ Γ(D). By lemma 2.3, it follows that ∇ and φ * ∇ have the same geodesics.
The proof of the converse is straightforward.
Affinities. We say that two nonholonomic Riemannian manifolds (M, g, D) and (M , g , D ) are affinely equivalent if there exists a diffeomorphism φ : M → M such that φ * D = D and ∇ = φ * ∇ .
Any map φ satisfying the above properties is called an affinity.
Any map satisfying the above three properties is termed an N H-isometry. Clearly, every N H-isometry is an affinity. In addition, every N H-isometry φ preserves the decomposition T M = D ⊕ D ⊥ , i.e., we have φ * P(X) = P (φ * X) and φ * Q(X) = Q (φ * X) for every X ∈ Γ(T M ). The equivalence (and classification) of nonholonomic Riemannian manifolds under N H-isometries is the primary concern of this paper.
Since N H-isometries preserve the metric along D, the nonholonomic connection and the projection operators, it is easy to see that the curvature tensors K, K, R, C, the sectional curvature K, the Ricci tensor Ric, the tensors A sym and A skew and the scalar curvature S are all preserved under N H-isometries.

Invariant nonholonomic Riemannian structures.
In this section we consider a nonholonomic Riemannian manifold (M, g, D), where M = G is a (connected) Lie group with Lie algebra g = T 1 G. (We denote the identity element of G by 1.) The distribution D and metric g are both assumed to be left invariant, i.e., (L x ) * D = D and g = (L x ) * g for every x ∈ G, where L x : y → xy is the left translation.
Since D and g are left invariant, it follows that ∇ is also left invariant, i.e., ∇ = (L x ) * ∇ for every x ∈ G. (The associated tensors K, K, etc. are left invariant as well.) In particular, ∇ induces a bilinear map ∇ : Furthermore, by left invariance, the equation ∇ċ (t)ċ (t) = 0 for a curve c in G may be written aṡ where U (·) is a curve in D 1 . In terms of a left-invariant orthonormal frame (X p ) m p=1 for D, the second equation in 2 takes the forṁ where Ω p qr ∈ R are the connection coefficients of ∇ with respect to the frame and U = u p X p (1).
Cartan-Schouten connections. A left-invariant connection ∇ on G is called a Cartan-Schouten connection (cf. [25,26]) if the geodesics of ∇ are the left cosets of oneparameter subgroups t → x 0 exp(tV 0 ), V 0 ∈ g, x 0 ∈ G. Following this terminology, we shall call ∇ a (nonholonomic) Cartan-Schouten connection if the geodesics of ∇ are the left cosets of one- Proposition 2.6. The following statements are equivalent.
Any two left-invariant nonholonomic Riemannian structures whose nonholonomic connections are Cartan-Schouten connections and whose distributions are related by a Lie group isomorphism, are geodesically equivalent.
Proposition 2.7. Let (G, g, D) and (G , g , D ) be two left-invariant nonholonomic Riemannian structures whose nonholonomic connections are Cartan-Schouten connections. If φ : G → G is a Lie group isomorphism such that φ * D = D , then (G, g, D) and (G , g , D ) are geodesically equivalent (with respect to φ).
Proof. The result follows easily from the fact that, for a Lie group isomorphism Accordingly, there exists (up to geodesic equivalence) at most one structure whose connection is a Cartan-Schouten connection, for each left-invariant completely nonholonomic distribution on a Lie group. Moreover, it turns out that for (left-invariant) structures with Cartan-Schouten connections, geodesic equivalence coincides with affine equivalence. Proof. Suppose ∇ and ∇ are both Cartan-Schouten connections, and φ : G → G is a diffeomorphism such that ∇ X X = (φ * ∇ ) X X for every X ∈ Γ(D). It is clear that φ * ∇ is also a Cartan-Schouten connection. Hence, by proposition 2.6, we Since D is spanned by left-invariant vector fields, the result follows by tensoriality of T and proposition 2.5.
3. Classification of structures in three dimensions. In this section we discuss the equivalence, up to N H-isometry and rescaling, of three-dimensional nonholonomic Riemannian manifolds. In particular, we obtain a classification of all leftinvariant nonholonomic Riemannian structures on three-dimensional simply connected Lie groups. Structures on three-dimensional manifolds with a completely nonholonomic distribution D are notable in that D may be described as the kernel of an intrinsic contact form. This contact form, together with the associated Reeb vector field, is central to our discussion. In section 3.1, we identify some fundamental isometric invariants of nonholonomic Riemannian structures in three dimensions. Following this section, we consider the equivalence of such structures. There arise two natural cases (based on one of the invariants). In the first case (section 3.2), we show that two structures are N H-isometric if and only if their associated sub-Riemannian structures are "SR-isometric." Accordingly, the classification of invariant structures reduces to a classification of sub-Riemannian structures on three-dimensional Lie groups; this result is known [1]. The second case (section 3.3) is the generic case. Here we make use of the contact structure to define a canonical orthonormal frame. It turns out that any N H-isometry between left-invariant structures must preserve leftinvariant vector fields; this leads to the introduction of three further invariants. The commutator relations of the canonical frame uniquely determine the nonholonomic structure, and facilitate the classification of the invariant structures. Lastly, in section 3.4, we characterize those structures whose nonholonomic connection is a Cartan-Schouten connection in terms of the invariants.
3.1. Isometric invariants. Let (M, g, D) be a nonholonomic Riemannian manifold, where dim M = 3, and let (Y 1 , Y 2 ) be a (local) orthonormal frame for D. We shall assume that D is completely nonholonomic. This implies that the onedimensional annihilator D • is (locally) spanned by a contact form ω, i.e., a 1-form such that ω ∧ dω = 0. By imposing the condition dω(Y 1 , Y 2 ) = ±1, we fix ω up to sign. (The value of dω(Y 1 , Y 2 ) is-up to sign-independent of the choice of orthonormal frame for D.) Specified in this fashion, ω depends only on (D, g| D ); hence it is intrinsic to three-dimensional nonholonomic Riemannian manifolds (and is preserved, up to sign, by N H-isometries).
Let Y 0 ∈ Γ(T M ) denote the Reeb vector field of ω. That is, Y 0 is the unique vector field such that i Y0 ω = 1 and i Y0 dω = 0. As ω is unique up to sign, the same holds for Y 0 . Likewise, Y 0 depends only on (D, g| D ), and so any N H-isometry preserves Y 0 (up to sign).
There are two cases to consider, viz. Y 0 ∈ D ⊥ and Y 0 / ∈ D ⊥ . Clearly, these conditions are invariant under N H-isometry. Morever, in the former case, the nonholonomic Riemannian structure reduces to a sub-Riemannian structure, i.e., D ⊥ is determined by (D, g| D ). It is thus natural to consider the angle between Y 0 and D ⊥ in terms of an appropriate Riemannian metric on M (i.e., one preserved by N H-isometries). Letg be the unique extension of g| D to a Riemannian metric on M such that (Y 0 , Y 1 , Y 2 ) is an orthonormal frame. This extension is independent of the choice of orthonormal frame for D; furthermore, it depends only on (D, g| D ). The angle θ between Y 0 and D ⊥ is given by Having said that, we shall find it more convenient to use the invariant ϑ = tan 2 θ. Note that Y 0 ∈ D ⊥ exactly when ϑ = 0. Thus we have the two cases ϑ = 0 and ϑ > 0 to consider. We introduce a further three isometric (curvature) invariants. The first invariant, denoted κ, is defined to be the sectional curvature of D, i.e., κ(x) = R(D x ). (Alternatively, we have κ = 1 2 S, where S is the scalar curvature. Furthermore, .) The second two invariants, denoted χ 1 and χ 2 , are defined as (Both g| D • A sym and g| D • A skew are trace free, and their determinants can be shown to always be nonpositive and nonnegative, respectively.) By the symmetries (S1 )-(S4 ), we have that R is fully determined by the value of R(X 1 , X 2 , X 1 , X 2 ). Hence R ≡ 0 exactly when κ = 0. Similarly, C is fully determined by 2 20 , and we have Thus C ≡ 0 if and only if χ 1 = χ 2 = 0.
For a left-invariant structure on a Lie group, it is possible to find a global leftinvariant orthonormal frame (Y 1 , Y 2 ) for D. Furthermore, the contact form ω and Reeb vector field Y 0 are left invariant, as are the invariants ϑ, κ, χ 1 and χ 2 (hence they are constant).
Remark 3.2. Ehlers [9] has used Cartan's method of equivalence to determine a generating set for differential invariants of three-dimensional nonholonomic Riemannian manifolds (under N H-isometry), which he denotes K, p, q, r, s and t. Ehlers makes the claim that K, s 2 + t 2 and p 2 + qr form a complete set of differential invariants for nonholonomic Riemannian manifolds. However, this set is not complete. Indeed, looking ahead to our classification, consider the equivalence class 10 on G 3.2 with β = 7 2 . For this structure, we have K = − 191 32 , s 2 + t 2 = 1 and p 2 + qr = 81 1024 . Likewise, the structure 11 (on the same group) has the same invariants K, s 2 + t 2 and p 2 + qr for δ = 0, β = 5 2 . However, these two structures are not N H-isometric: for the first structure we have χ 2 = 0, whereas χ 2 = 5 4 for the second. The invariants ϑ, κ, χ 1 and χ 2 we have introduced in this paper may be expressed in terms of Ehlers' generating set of functions as ϑ = s 2 + t 2 , 4 was given in [9], as a sectional curvature of D. In this paper we have interpreted κ as a different sectional curvature of D; we have also given a geometric interpretation of ϑ.    Accordingly, in this case, the classification of the three-dimensional left-invariant nonholonomic Riemannian manifolds (G, g, D) reduces to the classification (under SR-isometries) of three-dimensional left-invariant sub-Riemannian manifolds (G, D, g| D ). These structures have already been (locally) classified on three-dimensional Lie groups (see [1]). (It turns out that the local classification may be globalized on the simply connected Lie groups.) We restate this result in terms of the Bianchi-Behr enumeration.
We now describe the commutator relations of the canonical frame and give explicit expressions for the invariants in terms of the structure constants of this frame.
Remark 3.10. When ϑ > 0, the N H-isotropy subgroup of identity Iso 1 (G, g, D) is a subgroup of the automorphism group of G, and hence may easily be calculated. Moreover, the N H-isometry group Iso(G, g, D) decomposes as the semidirect product L G Iso 1 (G, g, D), where L G = {L g : g ∈ G}.
The equivalence classes of left-invariant nonholonomic Riemannian structures on Lie groups with positive ϑ are better described by the invariants 0 , 1 and 2 than the curvature invariants κ, χ 1 and χ 2 . Accordingly, we shall prefer the i 's to the curvature invariants for these structures. In fact, for the structures on unimodular groups, ϑ, 0 , 1 and 2 form a complete set of differential invariants. (This is not the case for the set of invariants ϑ, κ, χ 1 and χ 2 .) For the structures on nonunimodular groups (with the exception of those on G h 3.5 , h = 1), we shall find that there are at most two structures with the same invariants ϑ, 0 , 1 and 2 . (For structures on G h 3.5 , h = 1 there are infinitely many structures with the same values for these invariants, but at most two with the same invariants ϑ, κ and χ 2 .) We shall distinguish between the case when G is unimodular and the case when G is non-unimodular. ( 1 + 2 + ϑ) 2 − 4 0 , χ 2 = 0.
Theorem 3.11. We have the following classification of left-invariant nonholonomic Riemannian structures on unimodular simply connected Lie groups, rescaled such that ϑ = 1.
Next, we suppose that G is not unimodular, i.e., at least one of c 1 10 + c 2 20 , c 0 20 + c 1 21 or c 2 21 is nonzero. Theorem 3.14. We have the following classification of left-invariant nonholonomic Riemannian structures on non-unimodular simply connected Lie groups, rescaled such that ϑ = 1.
If G is unimodular, then the classification under N H-isometry of (left-invariant) nonholonomic Riemannian structures for which ∇ is a Cartan-Schouten connection coincides with the classification of (left-invariant) sub-Riemannian structures (proposition 3.16). On the other hand, it is easy to see that, on the non-unimodular three-dimensional Lie groups, the following equivalence classes of nonholonomic Riemannian structures are those whose nonholonomic connection is a Cartan-Schouten connection: • On Aff(R) 0 × R, the equivalence class 9 with α = 0 and γ = 1.
• On G h 3.4 , the equivalence class 12 with β = 1.  4. Two classical problems. We consider two classical problems (and their generalizations) from nonholonomic mechanics, viz. the Chaplygin problem and the Suslov problem. These problems are typically modelled as left-invariant nonholonomic Riemannian structures on the Lie groups SE(2) and SO (3), respectively. However, we shall consider them on the universal covering groups SE(2) and SU (2). We discuss how these structures relate to our classification, and how some of the invariants may be used in distinguishing between different qualitative cases.
4.1. The Chaplygin problem. The Chaplygin problem models the motion of a planar rigid body equipped with a blade along which the body slides, that prohibits motion in directions orthogonal to the blade. The problem may be described as a left-invariant nonholonomic Riemannian structure on the Euclidean group SE(2). It is known that the dynamics exhibit three qualitatively different cases (of increasing analytical complexity) [11]: the Chaplygin "skate" (when the centre of mass is at the point of contact); the Chaplygin sleigh (the classical statement of the problem); and a generalization of the Chaplygin sleigh, called the hydrodynamic Chaplygin sleigh (first introduced in [11]). For further details on the Chaplygin problem, see, e.g., [5,24,2,11].
We consider the problem on the universal covering group which has Lie algebra The nonzero commutator relations are [E 2 , For compatibility with the notation used for the body's inertia tensor in [11], we shall work in the (ordered) basis (E 3 , E 1 , E 2 ). We assume, without loss of generality, that the constraint distribution D is specified by D 1 = span{E 1 , E 3 }. Let g be the left-invariant Riemannian metric on SE(2) specified with respect to (E 3 , E 1 , E 2 ) by The reduced equations of motionU The first invariant ϑ is given by We have the following cases: (i ) ϑ = 0. This occurs exactly when L 1 = Z = 0, and corresponds to the case with the simplest qualitative behaviour (the Chaplygin skate). Indeed, the reduced equations of motion are trivial, hence the nonholonomic connection is a Cartan-Schouten connection; furthermore, we have κ = (ii ) ϑ > 0. At least one of L 1 , Z is nonzero. The canonical frame (X 0 , X 1 , X 2 ) (rescaled so that ϑ = 1) is readily calculated: . In terms of the basis (X 0 (1), X 1 (1), X 2 (1)), the reduced equations of motionV + ∇ V V = 0 take the (simplified) form (Here V = v 1 X 1 (1) + v 2 X 2 (1).) The values of the parameters α 1 , α 2 in the equivalence class representative 4 may likewise be calculated: The structure ( SE(2), g, D) describes the Chaplygin sleigh exactly when L 1 = 0, Z = 0 (cf. [11]). This occurs exactly when α 2 = 0 (or equivalently, when κ 2 = χ 2 1 ; see remark 3.13). For α 2 > 0 (κ 2 = χ 2 1 ), the structure describes the hydrodynamic Chaplygin sleigh. 4.2. The Suslov problem. The Suslov problem describes the motion of a rigid body in R 3 about a fixed point subject to the (nonholonomic) constraint W • Ω = 0. (Here Ω is the angular velocity of the body, W is a fixed vector in the body frame and • denotes the standard dot product of R 3 .) For more details on the Suslov problem, see, e.g., [29,13].
We have the following cases: (i ) ϑ = 0. This occurs exactly when I 13 = I 23 = 0, and corresponds to the simplest qualitative case (when the nonholonomic connection is a Cartan-Schouten connection). The invariants are given by In particular, g 1 is (a rescaling of) the Killing form exactly when I 11 = I 22 . (ii ) ϑ > 0. At least one of I 13 , I 23 is nonzero. The canonical frame (rescaled so that ϑ = 1) is given by . In terms of (X 0 (1), X 1 (1), X 2 (1)), the reduced equations of motion again take the simplified form 17. (In fact, the same holds true for any structure on a unimodular Lie group with ϑ = 1.) The parameters α 1 , α 2 , δ in the equivalence class representative 6 are: We have δ = 0 (equivalently, 0 = 1 (1 + 2 )) exactly when I 11 = I 22 , I 13 = 0 or I 23 = 0. Thus δ may be used to distinguish between the different qualitative cases discussed in [13].
On the other hand, the eigenvalues of ad U can be used to determine the parameter h in the infinite families g h 3.4 and g h 3.5 . Indeed, in terms of (E 1 , E 2 , E 3 ), the eigenvalues of ad U , U = u i E i are {0, u 3 (h − 1), u 3 (h + 1)} and {0, u 3 (h − √ −1), u 3 (h + √ −1)} Type Bianchi a n 1 n 2 n 3 Simply connected Lie group Unimodular Nilpotent Compl. solvable Solvable Semisimple Table 1. Bianchi-Behr classification of real three-dimensional Lie algebras on g h 3.4 and g h 3.5 , respectively. The scalars u 3 (h − 1) + u 3 (h + 1) (i.e., the ratio between the sum and difference of the nonzero eigenvalues, ignoring sign) are invariant under automorphism, and determine h.
Remark A.3. The two semisimple algebras g 3.6 and g 3.7 may be distinguished by inspecting the Killing form: for g 3.7 it is definite, whereas for g 3.6 it is indefinite.
Remark A.4. Apart from SL(2, R) (the universal cover of SL(2, R)) there exists, up to Lie group automorphism, at most one completely nonholonomic left-invariant distribution on each three-dimensional simply connected Lie group. On SL(2, R) there exist exactly two such distributions up to automorphism, according as whether the Killing form restricted to the distribution (at identity) is definite or indefinite.
Following [1], if the Killing form is definite on a given distribution, we shall say that the distribution is of elliptic type, and denote the group as SL(2, R) ell . On the other hand, when the Killing form is indefinite on the distribution, we shall say that it is of hyperbolic type, and write SL(2, R) hyp for the group.