Sub-Riemannian Geodesics on $SL(2, \mathbb{R})$

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus obtained in \cite{BoscaRossi}. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups \cite{AGBD}, \cite{Biggs}, \cite{HB2} to identify exactly automorphic structures on which our results apply.


Introduction
A sub-Riemannian structure is a triple (M, ∆, g), where M is a connected smooth manifold, ∆ is a constant rank distribution 1 and g a smooth metric defined on ∆. The distribution ∆ is usually described as the span of a set of vector fields {f 1 , ..., f m } (a frame), that is, ∆ = span{f 1 , ..., f m } and it is assumed to be nonintegrable, that is, [∆, ∆] ∆. The vector fields {f 1 , ..., f m } are also assumed to be orthonormal with respect to the (sub-Riemannian) metric g, defined on ∆, i.e., g p (f j (p), f k (p)) = δ j,k . If ∆ = T M nonintegrability is clearly not possible and one recovers the case of a Riemannian manifold. A horizontal curve γ : [a, b] ⊆ R → M is a Lipschitz continuous curve satisfying, a.e.,γ(t) ∈ ∆ γ(t) and minimizing geodesics are horizontal curves which minimize the sub-Riemannian distance between two points p and q in M (also called the Carnot-Caratheodory distance) where such a distance is defined by (with γ(0) = p, γ(T ) = q) d(p, q) = min γ T 0 g γ(t) (γ(t),γ(t))dt, with the minimum taken over all the horizontal curves γ joining p and q. Such a minimum exists, under the assumption that the frame {f 1 , ..., f m } is bracket generating, i.e., the Lie algebra generated by these vector fields has full dimension.This is the classical Chow-Raschevski theorem (see, e.g., [2]), and this will be the case in the problem we shall treat here. The problem of finding sub-Riemannian geodesics is equivalent (in that it has the same solution) to problems in control theory, where one considers the systeṁ x = q j=1 u j (t)f j , and wants to find control functions {u j } in order to drive the state x from x(0) = p to x(T ) = q, in minimum time subject to a L ∞ bound on the norm of the control or, equivalently, with fixed time T , and minimizing an energy type of functional T 0 j u 2 j (t)dt (see, e.g., [5]). In particular, sub-Riemannian geodesics γ parametrized by arclength ( γ = 1, a.e.) coincide with minimum time trajectories when the norm of the control is bounded by 1 [2] and, in fact, the minimum time can be taken as the sub-Riemannian distance. In view of this fact, in the following discussion, we shall use the words minimum time and distance interchangeably. Methods of geometric control theory [3], such as the Pontryaging Maximum Principle are used to solve sub-Riemannian problems.
Of special importance for the methods we shall use in this paper is the concept of isometry. A (global) isometry between two sub-Riemannian structures (M, ∆, g) and (M , ∆ , g ) is a diffeomorphism φ : M → M which preserves both the distributions ∆, ∆ and the metrics g, g , that is, φ * ∆ = ∆ and g = φ * g . It follows from the definition that isometries map minimizing geodesics to minimizing geodesics. An important special case is when the isometry φ maps a sub-Riemannian structure to itself. In this case, the set of all such isometries form a group Iso(M, ∆, g) Families of geodesics that can be mapped one to the other by elements of such group can be identified as representing a curve in M/Iso(M, ∆, g), the quotient space of M under the action of the transformation group Iso(M, ∆, g), and, in fact, the problem of finding the geodesics can be treated on the quotient space M/Iso(M, ∆, g). This is the essence of the method of symmetry reduction [5] which will be used in this paper. For a comprehensive introduction to sub-Riemannian geometry we refer to [2]. This paper deals with a sub-Riemannian structure on SL(2) = SL(2, R), the Lie group of area preserving transformations on R 2 . For this structure, we shall give a complete and explicit description of the optimal synthesis, i.e., the description of all the geodesics. For Lie groups, the frame of vector fields {f 1 , ..., f m } defining the distribution ∆ is usually taken to be made of right (or left) invariant vector fields, i.e., for every a ∈ M , R a * f j (x) = f j (R a x) (L a * f j (x) = f j (L a x)), for every x ∈ M and j = 1, ..., m, where R a (L a ) denotes the right (left) translation on the group M by a. This way the distribution ∆ is described by a set of elements in the tangent space at the identity 1, T 1 M , identified with the Lie algebra of the Lie group. We shall focus here on right invariant structures but similar things can be said for left invariant ones. The metric g is also assumed to be right invariant, that is, for X p , Y p ∈ T p M and a ∈ M , g Ra(p) (R a * X p , R a * Y p ) = g p (X p , Y p ). Therefore the metric can be defined by giving an inner product on the portion of the Lie algebra T 1 M corresponding to the distribution ∆ at 1 and it is then extended by right invariance at any point p ∈ M . For right invariant structures the right multiplication by a, R a , is an isometry R a * : (M, ∆, g) → (M, ∆, g) and, in particular, it maps minimizing geodesics to minimizing geodesics. Therefore, there is no loss of generality in studying only the geodesics starting from the identity 1, because, in general, the geodesic minimizing the length between p and q is obtained as R p • γ, where γ is the length minimizing geodesic from 1 to qp −1 . In view of this fact, in the sequel we shall only consider geodesics with the initial point given by the identity. In applying symmetry reduction, one considers therefore the subgroup of Iso(M, ∆, g), of isometries φ : (M, ∆, g) → (M, ∆, g) which leave the identity of the group M unchanged i.e., φ(1) = 1. This subgroup will be denoted by Iso 1 (M, ∆, g). If Iso 1 (M, ∆, g) is a continuous group, the quotient space M/Iso 1 (M, ∆, g) will have dimension strictly less than the dimension of M . In particular, if Iso 1 (M, ∆, g) is a compact connected Lie group, it follows from the theory of Lie transformation groups (see, e.g., [8]) that M/Iso 1 (M, ∆, g) is a stratified space with one particular stratum, corresponding to points with minimal isotropy type, which is a connected (open and dense) manifold. This is called the regular part of M/Iso 1 (M, ∆, g) and, if the minimal isotropy group is discrete, it has dimension equal to dim(M ) − dim (Iso 1 (M, ∆, g)). The remaining part of M/Iso 1 (M, ∆, g) is called the singular part. In applying symmetry reduction, the regular part of the quotient space M/Iso 1 (M, ∆, g) may be given a Riemannnian metric so that Riemannian geodesics in M/Iso 1 (M, ∆, g) are projections of sub-Riemannian geodesics in M with the same length [9] [16]. This is the approach we will follow in this paper to describe all the sub-Riemannian geodesics, i.e., the complete optimal synthesis on SL(2).
In physics, SL(2) is referred to as 2+1 dimensional anti-de Sitter space, since it is regarded as a three dimensional hypersurface sitting inside a four dimensional space, with a natural Lorentzian metric of signature (2,1). In general, it is a difficult problem to connect two arbitrary points by Lorentzian (semi-Riemannian) geodesics, (see, e.g., [17]). In sub-Riemannian geometry, the anti-de Sitter fibrations and the Hopf fibrations viewed as sub-Riemannian manifolds have been studied steadily. In particular, by applying the free action from the totally geodesic fiber of these two fibrations, the radial part of the sub-Laplacian can be calculated explicitly (e.g., [18,19]).
The paper is organized as follows. We start by describing the rank 2 sub-Riemannian structure (M, ∆, g) on SL(2) we want to study (section 2) and in section 3 we pick SO(2) as a Lie subgroup of Iso 1 (M, ∆, g) which we plan to use to carry out symmetry reduction. 2 The quotient space SL(2)/SO(2) is described in section 4. It is given by the part of the R 2 plane outside a disk of radius one and including the circle of radius one. The regular part corresponds to the part strictly outside the disk. Our goal is to arrive at a complete description of all the sub-Riemannian geodesics in terms of certain Riemannian geodesics in the regular part of SL(2)/SO (2). These are the images under the natural projection π : SL(2) → SL(2)/SO(2) of sub-Riemannian geodesics. In order to achieve this, we have to define a Riemannian metric on the regular part of SL(2)/SO(2) so that the natural projection π preserves the lengths. In section 5 we describe the tangent vector in T SL(2)/SO(2) obtained by projecting the tangent vectors defining the sub-Riemannian structure on SL(2) at any point. By imposing that these tangent vectors are orthonormal we obtain the appropriate Riemannian metric in section 6, which leads to writing down the geodesic equations in section 7. The geodesics in the quotient space are obtained in section 8 by projecting the sub-Riemannian geodesics which were obtained in [7]. These curves in fact satisfy the geodesic equations as verified in Appendix A. Visualizing the geodesics in the (two dimensional) plane allows us to easily select the geodesic joining the identity to any desired final point. This is the main contribution of this work which is described in sections 9 and 10. In these sections, we give the complete optimal synthesis, that is, the description of all the geodesics and show how to solve the length minimization problem between any two points in practice. As a byproduct, we obtain an alternative proof of the characterization of the cut-locus which was first obtained in [7]. The results of [7] are the most related to the ones presented in this paper. As compared to their characterization of the geodesics, the description given here is more explicit and constructive, giving a practical way to solve the length minimizing problem between any two points. Since the symmetry reduction technique was also used for the solution of a sub-Riemannian problem on SU (2) ( [4]), and the Lie algebras sl(2) and su(2) are both real forms of the same complex Lie algebra sl(2, C), it is natural to ask about the relation between the two problems. We do that in section 11. On any Lie group M we can define many sub-Riemannian structures even if we require right invariance. Structures (M, ∆, g) and (M, ∆ , g ) which are related by an automorphism φ of the Lie group M , i.e., φ * ∆ = ∆ g = φ * g , are seen as equivalent, and sub-Riemannian geodesics in one structure are mapped to sub-Riemannian geodesics in the other structure. With the help of the classification of right invariant sub-Riemannian structures on three dimensional Lie groups [1], [6], [13], we identify in section 12 the structures on SL(2) which are equivalent via automorphism to the one we have treated which leaves as an open problem to characterize the optimal synthesis on the remaining sub-Riemannian structures on the Lie group SL(2).
2 Sub-Riemannian structure on SL(2) On SL(2) we consider the three right invariant vector fields corresponding to the elements of the Lie algebra sl(2), {A 0 , A 1 , A 2 }, defined by and satisfying the commutation relations The basis {A 0 , A 1 , A 2 } gives a Cartan decomposition (see, e.g., [14]) of sl (2) in that by setting K = span{A 0 }, P = span{A 1 , A 2 }, we have from (2) We shall denote by, f 0 , f 1 , f 2 , the right invariant vector fields corresponding respectively to A 0 , A 1 , and A 2 , and by f 0,1,2 (X) their values at X ∈ SL(2). These vector fields are orthonormal at every point X ∈ SL(2) with the right invariant metric given by (the definition on the Lie algebra sl(2)) 3 g(B, C) := 2T r(BC T ).
The sub-Riemannian structure ∆ on SL(2) we shall study is given by the sub-bundle of the tangent bundle determined by {A 1 , A 2 } with the sub-Riemannian metric given by the restriction of the metric g in (4) to ∆(X) ⊂ T X SL (2). We want to describe sub-Riemannian minimizing geodesics connecting any two points in SL (2). Because of the right invariance of the distribution, it is enough to study geodesics starting from the identity.
In the next section we describe the quotient space SL(2)/SO (2). The space SL(2)/SO(2) is not a manifold but it is a stratified space which contains a subset (a stratum) which is connected and open and dense in SL(2)/SO(2), called the regular part of the quotient SL(2)/SO (2), and which can be given the structure of a manifold. It is on this regular part that we will define a Riemannian metric. According to standard results in Lie transformation groups [8], the regular part is the image under the natural projection π of the set of points in SL(2) which has minimal isotropy group. 4 The equivalence relation ∼ defining SL(2)/SO(2) is such that, given To see this notice that for a matrix the two quantities are invariant under the action of K ∈ SO(2), x because it is 1 2 the trace of X, and y because, we can write where the invariance is due to 1 2 k + y − k − y = y. Viceversa assume that two matrices X 1 and X 2 in SL(2) have the same values of x and y. Then we have to show that there exists K ∈ SL(2) such that X 2 = KX 1 K T . Write 4 In general, for a Lie transformation group G on a manifold M , one considers an equivalence relation between subgroups H 1 and H 2 of G saying that H 1 is equivalent to H 2 if there exists a k ∈ G such that kH 1 k −1 = H 2 . Denote the equivalence class containing H by (H). Some of these equivalence classes contain subgroups which are isotropy groups of some elements of M . These are called isotropy types. On isotropy types, one can consider a partial ordering by saying that (H 1 ) ≤ (H 2 ) if H 1 is conjugate (via an element of G) to a subgroup of H 2 . According to a basic result in the theory of Lie transformation groups, there exists a minimal isotropy type in this ordering which corresponds to a stratum in M/G which is a connected, open and dense manifold in M/G. This is called the regular part of the quotient space M/G the remaining part is the singular part. A more detailed discussion can be found in [5] and a complete treatment in [8]. and notice that, from the condition det(X 1 ) = det(X 2 ) = 1, we obtain Thus the points (m 1 , k 1 ) and (m 2 , k 2 ) are on the same circle with radius x 2 + y 2 − 1. Now choose θ such that With this choice, a direct calculation shows that using we have The above argument and equation (9) show that the points (x, y) of the x − y plane such that x 2 + y 2 ≥ 1 are in one to one correspondence with the points of the quotient space SL(2)/SO(2) which can be identified therefore with the portion of the plane outside the open disc of radius 1, together with the boundary of the unit disc itself. Points on the boundary of the disc correspond to points in SL(2) whose isotropy group (see, e.g. [8]) is the full SO(2) group. This is the singular part of the quotient space, and the corresponding (singular) part on SL (2) correspond to matrices such that x 2 + y 2 = 1, that is for (6), using so that, the matrices in the singular part have the form with x 2 + y 2 = 1 The rest of the plane outside the disk is the regular (open, dense and connected) part of the quotient space, which we denote by (SL(2)/SO(2)) reg . If we use the representation (8) of a matrix X ∈ SL(2), , matrices corresponding to points in this region have m 2 + k 2 = x 2 + y 2 − 1 > 0. (11) is in the isotropy group of this element, then using KXK T = X and specializing (10), we obtain Taking the inner product with m k gives cos(2θ)(m 2 + k 2 ) = (m 2 + k 2 ), that is, θ = kπ. Thus the isotropy group of this type of elements is made of only ±1. This is the (discrete) minimal isotropy type. This regular part of the orbit space will be given the structure of a Riemannian manifold in the next section. The orbit at each point (x, y) is a circle with square radius x 2 + y 2 − 1. 5 The x axis (y = 0) represents the matrices X in SL(2) which are in the same class as their inverse. These are Conversely, if X and X −1 are in the same class, they must have the same y.
However the y of X and X −1 are opposite to each other (while the x's are the same) which implies y = 0 and therefore the (1, 2) and the (2, 1) element of the matrix are the same, as in X = a b b d .

Projections of tangent vectors
Consider a point X belonging to SL (2) and the values at this point of the vector fields defining the given distribution ∆, i.e., f 1 (X) and f 2 (X). We now calculate, for every point π(X) in the regular part of SL(2)/SO (2), π * f 1 (X) and π * f 2 (X). We do this in the basis corresponding to the coordinates x and y defined in the previous section, i.e., ∂ ∂x , ∂ ∂y . We have Using (6), (7), and (1) we get Therefore and analogously, Therefore we have An analogous calculation leads to The expressions (13) and (14) show that when x 2 + y 2 > 1, that is, in the regular part π * , is an isomorphism π * : ∆(X) → T π(X) SL(2)/SO(2). 6 In the following section we define a Riemannian metric on the regular part of SL(2)/SO(2) to make π * an isometry between the sub-Riemannian manifold SL(2) and the Riemannian manifold SL(2)/SO(2).

Riemannian metric on SL(2)/SO(2)
Consider the (diagonal) metric g Q on the regular part of the quotient space given in the (x, y) coordinates by (at the point (x, y)) , A direct verification using (13) and (14) shows that This means that, since f 1 and f 2 are orthonormal in the original sub-Riemannian metric (4), π * is an isometry from the sub-Riemannian manifold (SL(2), ∆, g) to the Riemannian manifold (SL(2)/SO(2)) reg with the metric g Q . As an example of verifying (17), we have, using ad − bc = 1, using (7). 6 The map is non singular iff (a − d) 2 + (b + c) 2 > 0, which is true if and only if a 2 + d 2 + b 2 + c 2 > 0 which is true (adding 2(ad − bc) = 2 on both sides) if and only if (a + d) 2 + (b − c) 4 > 4, which is true, recalling the definitions of x and y (7).

Geodesic equations on SL(2)/SO(2)
To compute the geodesic equations associated with the given metric (15) (16) in (SL(2)/SO(2)) reg , we first compute the Christoffel symbols for the Levi-Civita connection associated to the given metric with the classical formulas given for example in [12] (formula (10) Chapter 2). The result is The geodesic equations are given in formula (1) of Chapter 3 of [12]. They are in our case:

Geodesics in SL(2)/SO(2)
The sub-Riemannian geodesics on SL(2) are such that, when projected onto the quotient space SL(2)/SO (2), (locally) minimize the distance with respect to the metric described in the previous section. Therefore the projections of the sub-Riemannian geodesics have to be geodesics for the metric we have defined in section 6. They have to satisfy equations (18). The sub-Riemannian geodesics in SL(2) were calculated in [7] (subsection 3.3.1) using general results on sub-Riemannian manifold having a K − P structure, that is, whose defining distribution corresponds to the P part of a Cartan decomposition as defined in (3). The sub-Riemannian geodesics (parametrized by arclength) are of the form (cf. (1)) where P is a linear combination of A 1 and A 2 with unit norm, and c a real parameter. Since we are interested in the projection of these trajectories onto the quotient space, we can, without loss of generality, consider, instead of X(t), KX(t)K T for an arbitrary K ∈ SO(2). In particular we can take P = A 1 . Furthermore assume Then From and (20) we see that the projection of the sub-Riemannian geodesic (19) π(X(t)) is equal to the one for X −1 (t), π X −1 (t) reflected about the x axis and π(X) and π(X −1 ) correspond to values of c which are opposite of each other. Therefore we can restrict ourselves to c ≥ 0 in (19) and consider only projections of sub-Riemannian geodesics in the upper half plane. Using (20) and the explicit expression of (19) calculated in [7], we obtain π(X(t)) = (x(t), y(t)) with, setting s : where for c ≥ 0, the functions k 1 = k 1 (s) and k 2 = k 2 (s) are defined as: In particular, 7 To simplify the notations, our definitions slightly differ from the ones in [7]. In particular our k 2 corresponds to their ck 2 .
For |c| < 1, For |c| = 1 Notice that for c = 0 the trajectory is along the x axis, in agreement with the above recalled symmetry. A direct but tedious calculation verifies that the trajectories in (21)-(26) satisfy the geodesic equations (18). For completeness, we present this calculation in Appendix A.

Geometric description of the optimal synthesis
The word optimal synthesis refers to a qualitative and graphical description of the optimal geodesics (from the identity) to any point. Because of symmetry reduction the projections of such geodesics can be plotted in the quotient space, SL(2)/SO (2). Such geodesics are plotted from the initial point (1, 0), which corresponds to the identity in SL (2), to the point where they cease to be optimal. The locus where sub-Riemannian geodesics cease to be optimal is the inverse image under the natural projection of the corresponding locus in the quotient space [5]. We shall call this locus the critical locus. 8 Another locus of importance is the cut locus, the locus of points that are reached simultaneously by two sub-Riemannian geodesics. Such a locus for SL(2) was described in [7]. In the following, we denote by R − CR (SR − CR) the 'Riemannian' ('sub-Riemannian') critical locus where the Riemannian geodesics (21), (22) (the sub-Riemannian geodesics (19)) lose their optimality. We have, with the natural projection π (cf. [5]) As a direct application of a result proved in [5] (cf. Corollary 3.6 in that paper), we have the following: Proposition 9.1. A sub-Riemannian geodesic on the manifold SL(2) which crosses the regular part touches the singular part in a point p and then returns to the regular part loses optimality at p.
We shall use this and the reduction to the quotient space to give a complete description of the optimal synthesis in this section. This is summarized in Figure 1. We shall discuss in the next section how to find the geodesics in practice. for any desired final condition. We remark that symmetry reduction is crucial in this treatment as it allows easy visualization of the geodesics in a plane.
As anticipated, the geodesics plotted in Figure 1 are symmetric with respect to the x-axis, that is, if (x c , y c ) = (x c (s), y c (s)) is the geodesic corresponding to c, then (x −c , y −c ) = (x c (s), −y c (s)). This follows from the fact that k 1 (k 2 ) is an even (odd) function of c. 9 Accordingly the geodesics corresponding to c = 0, starts from the point (1, 0) and moves rightwards along the x-axis. We have also that, for every c, 9.1 Geodesics for |c| < 1 For |c| < 1, formula (28) gives For any c, this is an increasing function of s. So the geodesics are spirals with increasing distance from the origin. In particular they never self intersect. The geodesic corresponding to c and the one corresponding to −c intersect when they cross the x-axis, with the same value of time s. In particular, notice that if y c (s) = k 1,c (s) sin(cs) − k 2,c cos(cs) = 0, that is, k 1,c (s) sin(cs) = k 2,c (s) cos(cs), then k 1,−c (s) sin(−cs) = −k 1,c (s) sin(cs) = −k 2,c (s) cos(cs) = k 2,−c (s) cos(−cs). Therefore y −c (s) = 0 and, by the above recalled symmetry about the x axis, x −c (s) = x c (s).
Furthermore the intersections at the x-axis of the geodesics corresponding to c and −c are the only ones where optimal geodesics with different c's intersect. To see this assume that the geodesic corresponding to c 1 and c 2 intersect. Because of (time) optimality they have to intersect at the same value of s since s has to be minimum. Using (29), we have However the function f (x) = sinh(x) x is strictly increasing with x ∈ (0, ∞). Therefore, we must have 1 − c 2 1 s = 1 − c 2 2 s which implies c 2 = ±c 1 . To find the point of crossing of the x axis and study how it varies with c, let us fix 0 < c < 1, and consider the function for y = y(s) given in (22) (24), whose derivative is This derivative shows that y grows until s = π c where it reaches its positive local maximum of y max = c √ 1−c 2 sinh √ 1 − c 2 π c and then decreases for π c < s < 2π c to reach at s = 2π c its local minimum at c . The first intersection with the x axis occurs for π c < s < 3π 2c since y( 3π 2c ) = − cosh √ 1 − c 2 3π 2c < 0. The function s int := s int (c), giving the time where the intersection occurs, is defined implicitly by y(s int ) = 0 for π c < s = s int < 3π 2c , that is, A detailed analysis based on the implicit function theorem given in Appendix B shows that s int = s int (c) is a differentiable decreasing function of c, for c ∈ (0, 1) with lim c→0 s int (c) = ∞ and lim c→1 s int (c) =s, wheres is the unique solution in (π, 3π 2 ) of tan s = s, which is,s ≈ 4.49341. The coordinate x of the (first) intersection, x int , is negative and |x int | can be obtained from (29), and it is This is, see Appendix B, a decreasing function of c, for c ∈ (0, 1) and, using the above limits for s int = s int (c), we have lim c→0 |x int (c)| = ∞ and lim c→1 |x int (c)| = √ 1 +s 2 , wheres ≈ 4.49341 was described above. This gives lim c→1 |x int (c)| ≈ 4.60333. Figure 1 shows in green two optimal geodesics corresponding to 0 < c < 1 and their corresponding mirror image corresponding to −c. The geodesics on the left correspond to c = ±0.9. The ones on the right correspond to c = ±0.95. The geodesics for |c| → 1 converge to the ones for c = ±1 which are given in (21), (22), (26) and plotted in black in the figure 1.

Geodesics for |c| > 1
For |c| > 1 , formula (28) gives For a fixed c, r 2 c is equal to 1 for the first time (after s = 0) at s = π √ c 2 −1 . After this point the geodesic cannot be optimal anymore according to Proposition 9.1, because it touches the singular part of the quotient space.
As it was done for the case |c| < 1, we study the intersection (if any) of the geodesics (that are still symmetric with respect to the x axis) with the x axis. Because of symmetry, we consider only the case c > 0. Similarly to what was done in the previous subsection, calculation of the derivative dy ds gives dy ds which shows that, as long as s < π c 2 −1 , y is an increasing function until s = π c and then decreasing until s = 2π c . The time s = π c always comes before the time s = π the geodesic in red) the geodesic corresponding to c = ±1.03 which is such that 1 < |c| < 3 2 √ 2 and (on the right of the geodesic in red) the geodesic corresponding to c = ±1.12 which is such that 3 The geodesics corresponding to |c| > 2 √ 3 are the ones that lose optimality by 'landing' on the unit circle according to Proposition 9.1. The time of landing, which is the length of the sub-Riemannian geodesic s int , is given by the condition, obtained from (31), √ c 2 − 1s int = π. Two such geodesics corresponding to c = ±1.2 and c = ±1.5 are depicted in purple in the figure. The more the geodesics correspond to higher values of |c| the shorter they are and the more their terminal point corresponds to a point on the circle with polar angle closer to zero. To see this, notice that for t = π

Cut locus and Critical Locus
The geodesics do not intersect before reaching the x-axis or the unit circle. More specifically, we have the following fact (cf. Appendix C).
Proposition 9.2. Denote by s int (c) the points where the geodesic corresponding to the value c intersect the x-axis or the unit circle. Consider two optimal geodesics γ c1 = γ c1 (s) defined for 0 ≤ s ≤ s int (c 1 ) and γ c2 = γ c2 (s) defined for 0 ≤ s ≤ s int (c 2 ). They can only intersect when c 2 = −c 1 and at the x axis with s int (c 1 ) = s int (c 2 ) = s int (−c 1 ).
Geodesics that reach the unit circle lose optimality on the unit circle according to Proposition 9.1. Geodesics also lose optimality when intersecting the x-axis in points (x, 0) with −∞ < x ≤ −1. In fact if the geodesic γ c did not lose optimality at the x-axis, the curve given by γ −c until the x-axis and γ c for an -interval would still be optimal, contradicting smoothness. Lifting back to sub-Riemannian geodesics according to (27), we have the following proposition. This is the set of the matrices in SL(2) of the form (12) together with the matrices of the form (cf. the last paragraph of section 4) For the sub-Riemannian problem a smoothness argument (see, e.g., [5]) shows that the sub-Riemannian cut locus (SR − CL) is included in the sub-Riemannian critical locus. Therefore from (27) we have Matrices of the form (32) definitely belong to the cut locus since they are such that for the minimum t min they satisfy, using (19), π e (cA0+P1)tmin e −cA0tmin = π e (−cA0+P2)tmin e cA0tmin , for matrices P 1 and P 2 in span{A 1 , A 2 }. Therefore, there exists K ∈ SO(2) such that e (−cA0+P2)tmin e cA0tmin = Ke (cA0+P1)tmin e −cA0tmin K T = e (cA0+KP1K T )tmin e −cA0tmin .
Therefore the two different sub-Riemannian geodesics {e (cA0+KP1K T )t e −cA0t | t ∈ [0, t min ]} and {e (−cA0+P2)t e cA0t | t ∈ [0, t min ]} are both optimal and lead to the same point which therefore belongs to the sub-Riemannian cut locus. 11 Matrices of the form (12) also belong to the sub-Riemannian cut locus because these matrices are obtained at the terminal time t int := 2s int = 2π √ c 2 −1 for the trajectories (19). In particular, a direct verification shows that the first factor in (19) at t int is equal to −1, independently of the matrix P as long as P = 1 (recall that P has unit norm in (19)). Therefore for 2 can all be obtained for an infinite number of geodesics each corresponding to a matrix P in the span of A 1 and A 2 and with P = 1.
In conclusion, we have Proposition 9.4. For the sub-Riemannian structure of SL(2) the critical locus and the cut locus coincide and are given by the inverse image under natural projection π of the union of the unit disc and the semi-infinite segment y ≡ 0, −∞ < x < −1. These are the matrices described in (12), (32).
This result gives an alternative description of the cut locus which is equivalent to the one given in [7] (Theorem 5), with a different proof. 12 10 Finding the sub-Riemannian geodesics; An example From a practical perspective, the main advantage of the characterization of geodesics described in the previous section is that it makes it easy to explicitly determine the sub-Riemannian geodesic between any two points in SL (2), that is, the parameter c the matrix P and the optimal final time t in (19). The fact that the problem was reduced to a two dimensional problem via symmetry reduction allows for a direct graphical solution. The procedure, which is adapted from what described in [5], can be summarized as follow: Given the desired initial and final conditions X i and X f in SL(2), because of right invariance, the sub-Riemannian geodesic from X i to X f is γX i where γ is the sub-Riemannian geodesic from the identity toX f := X f X −1 i . Therefore the problem is to determine γ. The algorithm is as follows: 1. Determine the equivalence class ofX f ad therefore the final point P f in the plane of figure 1.
2. Determine the rough range of the parameter |c| according to the desired final point, i.e., whether |c| = 1, 0 or 0 < |c| < 1, or 1 < |c| ≤ 2 √ 3 , or |c| > 2 √ 3 . In the various cases, the geodesic will be, respectively, from figure 1, plotted in black, following the x axis outwards starting from the point (1, 0), plotted in green, plotted in blue except for the geodesics corresponding to |c| = 3 2 √ 2 which is plotted in red, plotted in purple. 3. With a procedure of trial and error by considering geodesics corresponding to various values of c in the given range (possibly with a bisection algorithm, see example below) find the value of c which identifies the Riemannian geodesic crossing the point P f . 4. From the Riemannian geodesic, determine the final optimal value t f where the point P f is reached. This is the sub-Riemannian distance. (19) and the found values of c and t f using (for instance) P = A 2 find a given final condition, Y f . This is such that π(Y f ) = π(X f ).

Example
Assume we want to find the optimal sub-Riemannian geodesic from the point X i = 0 −1 1 0 ∈ SL(2) to the point . Using (19), the geodesic has the form 12 The matrices K loc Id in [7] correspond to the matrices (12) while the matrices K sym Id correspond to (32). The proof that symmetric matrices in SL(2) with trace < 0 (as in [7]) correspond to symmetric matrices with trace < −2 (as in (32)) boils down to showing that, for positive a and d, ad > 1 implies a + d > 2. In fact since (a − 1) 2 ≥ 0, a + 1 a ≥ 2 and since from ad > 1, d > 1 a we have a + d > a + 1 a ≥ 2. and the problem is to find the minimum final time t f , the optimal parameters c and P ∈ span{A 1 , A 2 }, with P = 1 such that e (cA0+P )t f e −cA0t f = X f X −1 i . In this case, we have which belongs to the class in SL(2)/SO (2) corresponding to the point (x, y) = (0, 3 2 ). By locating this point on the diagram of Figure 1, we find that the corresponding value of c has to be positive and such that 2 To find the correct value of c we plot different geodesics in the interval 2 √ 3 , ∞ looking for the one crossing the y axis at the point with y = 3 2 . In this case, we can in fact restrict the range of values of c by excluding the geodesics which reach the unit circle (where they lose optimality) before crossing the y axis. In particular using (31) we know that a geodesic (21), (22), (25), reaches the unit circle for the first time at s = π √ c 2 −1 . Imposing that this happens when x = 0 (using (21) (25)) gives that c must satisfy c = m  This gives for s the value s 0 ≈ 2.78115. With these values we calculate (19) with t = 2s 0 and c = c 0 , and P = A 2 (point 5 of the algorithm) which gives This matrix is in the same equivalence class as (33) but not exactly equal to it (although very close). According to step 6 of the algorithm, we need to find a matrix K ∈ SO (2) such that

This gives
Therefore the correct matrix P is

Relation with the sub-Riemannian problem on SU (2)
In [4] the sub-Riemannian K − P problem for the Lie group SU (2) was solved using the technique of symmetry reduction advocated here. It is interesting to compare the results. The Lie algebras su(2) and sl (2) are related in that they are both real forms of the complex Lie algebra sl(2, C). 13 The Lie algebra su(2) also has a K − P Cartan decomposition, su(2) = K ⊕ P with K = span{A 0 } and P := span{iA 1 , iA 2 } (cf. (1)). The metric for the sub-Riemannian problem is defined as in (4) with transposition T replaced by transpose conjugate † . In both cases the action of the Lie group SO(2) can be taken as the symmetry. The invariants (parametrizing the quotient space) under the (conjugation) action X → KXK T (K ∈ SO(2)) are defined in the same way as in (6) (7), x := T r(X) 2 , y = X12−X21 2 , 14 which are in this case constrained inside the unit disc (unit circle included) in the x − y plane [4]. The quotient map π has the same form as π in (5) but now it acts on SU (2). The sub-Riemannian geodesics have the form (19) but with P replaced by −iP . Therefore, the sub-Riemannian geodesics in the SU (2) case can be obtained from (19) by the substitution c → iω, t → −is and are therefore given by X U (s) := e (A0−iP )s e −ωA0s , which are parametrized by a parameter ω. Since the projection π takes the same form, Riemannian geodesics in the SU (2) case 15 can be obtained by doing the same substitution c → iω, t → −is in the Riemannian geodesic for the SL(2) case. To this purpose, we notice that formulas (21) (22) with the choices (24) (25) are both defined when c = iω independently of the absolute value of |c| = |ω| and give the same expressions for the Riemannian geodesics in the SU (2) case. So, for example, we can use the expression of the Riemannian geodesic for |c| > 1 in the SL(2) case (obtained from (21), (22), (25) replacing the variable s with t, with some abuse of notation) and we obtain the Riemannian geodesic in the SU (2) case, 13 Recall that for a complex Lie algebra L, we can consider L as a Lie algebra over the reals which has twice the (complex) dimension as L and we denote it by L R . The real Lie algebra L R has a complex structure J, that is, an endomorphism L R → L R such that J 2 = −Id, with Id denoting the identity, given by multiplication by the imaginary unit i. A real form of L is a real Lie algebra L 0 such that L R = L 0 ⊕ JL 0 . Both sl(2) and su(2) play this role for sl(2, C).
14 Notice that every matrix in SU (2) can be written as X = w z −z * w * , therefore both x and y are real.
for all values of ω. For a fixed value of ω the Riemannian geodesic in the SU (2) case starts at the (1, 0) point and ends, losing optimality, on the unit circle, which, similarly to the SL(2) case, is the singular part of the quotient space. A plot of the Riemannian geodesics in the SU (2) case in figure 3, reveals that there is a one to one correspondence between geodesics and values of ω and all values of ω are possible with a symmetry ω ↔ −ω about the x-axis. The geodesic corresponding to ω = 0 connects going leftwards the point (1, 0) with the point (−1, 0), while the others, as |ω| increases, land on a point of the unit circle whose polar coordinate angle is closer to zero, losing optimality. Figure 3: Riemannian geodesics in the SU (2) case which are in blue. In red are depicted the boundaries of the reachable sets for various values of the time s. These are obtained by keeping s fixed and varying ω in an appropriate range. We refer to [5], [16] for a discussion of the role of reachable set in optimal control problems in the context of symmetry reduction.
It is possible to obtain all the SU (2) Riemannian geodesics (all possible values of ω) as corresponding to SL(2) Riemannian geodesics with a range of values of c different from (−∞, ∞). Furthermore, we can establish a correspondence between SU (2) and SL(2) Riemannian geodesics so that the corresponding geodesics land on the same point on the unit circle. In order to do this, assume first ω ≥ 0. Then the corresponding value of c is chosen as This is a monotonically decreasing finction of ω from c = − 2 √ 3 for ω = 0 to −∞ for ω → ∞. Formula (36) after squaring both terms and direct manipulations gives The point where the SU (2) geodesic (35) reaches the unit circle is the point The point where the SL(2) geodesic (34) reaches the unit circle is the point Because of (37) the correspondence (36) guarantees that these two points coincide. An analogous calculation for ω ≤ 0 shows that the choice which is a monotonically decreasing function (as ω goes from 0 to −∞, c goes from 2 √ 3 to +∞) ensures that the two geodesics corresponding to ω and c end (and lose optimality) at the same point of the unit disc.

Automorphic structures on SL(2)
As we have mentioned, sub-Riemannian structures (M, ∆, g), (M, ∆ , g )), with M a Lie group, can be considered as equivalent if there exists an automorphism φ of M , with φ * ∆ = ∆ and φ * g = g. In this case, the structures can be identified and geodesics of one structure can be mapped to geodesics in the other structure via the automorphism φ. It is therefore of interest to identify the structures which are related by an automorphism to the one we have studied in the previous sections, since for these structures as well we will have found the complete optimal synthesis.
The study of the automorphisms of Lie groups is an important chapter of the theory of Lie groups (see, e.g., [14], and [10] for a recent expository paper). In the following, we always assume that M is compact.  (2) is not simply connected. The automorphisms of all classical Lie groups have been described in detail in standard references [11]. Here we present, for the SL(2) case, an elementary and explicit derivation based on the above map φ → φ * . In particular, we first describe the group aut (sl(2)) using the argument in [13] and then we find Aut (SL(2)) by displaying for every α ∈ aut (sl(2)) a φ ∈ Aut (SL(2)) such that φ * = α.
In describing aut (sl(2)), we consider the coordinates of sl(2) in the basis {A 0 , A 1 , A 2 }. With this basis, the sub-Riemannian structure we have treated above is identified by the pair of vectors { e 2 , e 3 } in R 3 . The automorphism group aut (sl(2)) is identified with a subgroup of the linear group on R 3 . We have the following.  3)) is also known as the Lorentz group and has applications in relativity. SO(1, 2) has two connected components (Lemma X.2.4 in [14]) given by matrices with determinant equal to +1 or −1, respectively. The component SO 0 (1, 2) corresponds the matrices with determinant equal to 1. This is the component which contains the identity. Lemma 12.1 was proven in [13] and we will give the proof with some extra details in Appendix D.
Lemma 12.2. Every matrix X in SO 0 (1, 2) can be written as for three real parameters θ 1 , θ 2 , z and one of three choices 0, 1, 2, where, and I 0 is the 3 × 3 identity, I 1 := diag(−1, −1, 1), Notice that the definition of I 0,1,2 is somehow redundant, since I 2 = O π 2 I 1 O − π 2 , but it was set this way to simplify the exposition in the proof below.
If h 13 = 0, setting h 13 = sinh(z), for some real z, we consider two possibilities h 11 > 0 and h 11 < 0. If h 11 > 0, from the first one of (46) and h 2 31 = h 2 13 we can write h 11 = cosh(z) and h 33 = ± cosh(z), h 31 = ± sinh(z). From the second one of (46) it follows that the same sign has to be used in h 31 and h 33 . The condition det(Ĥ) = 1 imposes that if the chosen sign is + then h 22 = 1 andĤ has exactly the form of H in (39). If we choose the sign −, then, the condition det(Ĥ) = 1 imposes that h 22 = −1 and againĤ = O(π)H where O and H were defined in (39). In the case, h 11 < 0, we write h 11 = − cosh(z), and again h 33 = ± cosh(z), h 31 = ± sinh(z). With these definitions, the second formula of (46) gives that the opposite signs have to be chosen. By choosing − for the cosh and + for the sinh, the condition on the determinant gives that h 22 = +1 and thereforeĤ = I 2 H(−z). By choosing + for the cosh and − for the sinh we have that h 22 = −1 andĤ = I 1 H(−z). In all cases considered above forĤ, from formula (40) we have that A has the factorization (38) (withθ 1,2 = −θ 1,2 ).
Notice that Aut (SL(2)) is different from Inn (SL(2)) and the group of outer automorphisms, that is, Aut (SL(2))/Inn (SL(2)), is a cyclic group of order two with two classes given by the class containing the identity (and corresponding to the inner automorphisms) and the class containing the transformation φ − (X) Proof. The map φ → φ * from Aut(SL (2)) to aut(sl (2)) is an injective homomorphism and thus one can naturally consider Aut(SL (2)) as a subgroup of aut(sl (2)) (see, e.g., [10]). The group G defined in the statement of the theorem is a subgroup of Aut(SL (2)). Thus the proposition is proved if we prove that for every α ∈ aut(sl (2)) there exists a φ ∈ G such that φ * = α. We notice that the automorphisms φ in G on SL(2) have formally the same (conjugacy) expression if we consider the corresponding a φ * on sl (2), that is A → KAK −1 . Therefore, the proposition is proved, using Lemma 12.1, if we show that for every element in SO 0 (1, 2) aut (sl (2)), there exists a K ∈ SL ± (2) which realizes the same transformation via A → KAK −1 A ∈ sl(2). In fact, it is enough to show this for the three types of factors which appear in (38) 2). The automorphisms in aut(sl (2)) of the type I 1 are obtained with K = 0 1 1 0 since we have, with this choice, KA 0 K −1 = −A 0 , as a similar direct verification shows.
The structures on which our results in the previous sections apply can be identified by two elements in sl(2) {B 1 , B 2 } and their associated right invariant vector fields. The metric is implicitly defined by imposing that these two matrices are orthonormal. Our results apply to all structures {B 1 , B 2 } automorphic to the structure {A 1 , A 2 }, that is, such that there exists a K ∈ SL ± (2) with KA 1 K −1 = B 1 and KA 2 K −1 = B 2 . Equivalently, these are structures whose coordinates in the {A 0 , A 1 , A 2 } basis are given by the last two columns of a matrix in SO 0 (1, 2). For example, choosing a matrix of the form H(z) in (39), we might have We have a T j I 1,2 a k = δ j,k and since b j,k = X a j,k with X ∈ SO 0 (1, 2) b T j I 1,2 b k = δ j,k . In particular the inner product v, w = v T I 1,2 w is positive definite on any pair of vectors representing a sub-Riemannian structure automorphic to the one we have treated. Structures where such inner product is positive definite are called elliptic, and therefore the structures considered in this paper are a special case of elliptic sub-Riemannian structures. In fact, the inner product v, w = v T I 1,2 w is the same as the Killing inner product defined by A, B = T r(ad A ad B ) (cf. (69)) which is equal, up to a scaling factor, to the inner product defined in (4). Structures on SL(2) such that the Killing inner product is indefinite are called hyperbolic [1]. Sub-Riemannian structures that have the same geodesics as described here (or mapped to them by an automorphism) are also the ones corresponding to pairs of coordinate vectors {λX a 1 , λX a 2 }, for some scaling factor λ > 0, since the scaling factor can be taken into account be rescaling the metric which corresponds to a uniform scaling of the sub-Riemannian length on the whole manifold. The explicit description of geodesics for sub-Riemannian structures (elliptic or hyperbolic) non automorphic to the one considered here is an open problem, although important general properties have been proved [15].
For simplicity we write s(c) instead of s int (c). We have The function x = x(k) is defined implicitly by for π < x < 3 2 π, which is obtained from (30). From the implicit function theorem, we have . (55) Replacing this into (53), multiplying it by c and expressing everything in terms of k we obtain (after multiplication by k 2 ) From standard trigonometric and hyperbolic identities we obtain that the denominator that appears in (56) is Therefore the sign of ds dc coincides with the sign of (1 + k 2 ) tan(x) − x cosh 2 (kx) − k 2 x tan 2 (x) + tanh 2 (kx) .
Therefore the function s = s(c) is monotone and it has a limit as c → 0 + and c → 1 − . In particular since s(c) > π c , we have lim c→0 + s(c) = ∞. Furthermore, denoting bys the lim c→1 − s(c) we have that π ≤s ≤ 3π 2 and from (30)s satisfies tan(s) = lim We want to show that |x int (c)| is e decreasing function of c. Equivalently we can consider |x int | as a function of k and show, using (52), that it is an increasing function of k. Therefore we consider the function sinh( written in terms of x = cs = x(k) and k, that is, the function, and prove that dg dk > 0. Using the chain rule, we have, The derivative dx dk was already computed in (55) and using (57) and (54) and replacing into (60), we get (1 + k 2 ) tan 2 (x)   , which, using 1 cosh 2 (kx) = 1 − tanh 2 (kx) and (54), becomes after simplifications, dg dk = 1 k 2 √ 1 + k 2 cosh(kx) − tanh(kx) + kx(1 + tan 2 (x)) − k tan(x) tan 2 (x) .
Function s int = s int (c) As before, we define x = cs, so that from π c < s < 2π c , we have π < x < 2π. Furthermore, we define k := (second order) equations. The computation (48) (49) shows that the slope of the geodesic γ c (near s = 0) iṡ ẏ x (s) = tan(cs). Furthermore the slope varies as d dsẏ x (s) = d ds tan(cs) = c cos 2 (cs) , which gives c at s = 0. Therefore the geodesic corresponding to c 1 is more 'flat' than the one corresponding to c 2 and ends at a value of x int which is less than the value for c 2 . If there is a point of (transversal) intersection, there must be at least two because otherwise we would have x int (c 1 ) ≥ x int (c 2 ).
Consider then two such points P 1 and P 2 with P 1 occurring before P 2 . We know that γ c2 is a length minimizing geodesics between P 1 and x int (c 2 ) (it is, in fact, length minimizing between each point γ c2 (t 0 ) and x int (c 2 ), for 0 < t 0 < s int (c 2 )) and γ c1 is length minimizing between P 1 and x int (c 1 ). However, this implies that both γ c1 and γ c2 are length minimizing between P 1 and P 2 . This implies that the curve that goes from P 1 to P 2 following, say, γ c1 and then switches to, say, γ c2 is length minimizing from P 1 to, say, x int (c 2 ). The only points where two geodesics can and do intersect is on the x axis where the two geodesics corresponding to c and −c for 0 < |c| ≤ 2