Homogeneous geodesics in sub-Riemannian geometry

We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step $1$ and $2$. Finally, we get a broad condition for existence of at least one homogeneous geodesic.


Introduction
In this paper by a sub-Riemannian manifold we mean a triple (M, ∆, B), where M is a smooth manifold, ∆ is a smooth distribution of constant rank on M , i.e., a sub-bundle of the tangent bundle T M , and B is a scalar product on ∆ smoothly depending on a point of the manifold M . We assume that the distribution ∆ is bracket generating, i.e., a finite number of Lie brackets of the vector fields that are tangent to the distribution span the whole tangent bundle T M .  The sub-Riemannian distance (or the Carnot-Caratheodory distance) between points p, q ∈ M is the infimum of sub-Riemannian length of admissible curves connecting the points p and q. A curve with natural parametrization is called a geodesic if its sufficiently small arcs are shortest arcs.
Thus, the sub-Riemannian structure on the manifold M provides a structure of metric space on M (for details and for sub-Riemannian geometry in general we refer to the book [2]). One can define an isometry more directly as a distance preserving homeomorphism. This definition is equivalent to Definition 1.2 for an equiregular sub-Riemannian structure, in particular, for a left-invariant sub-Riemannian structure. Moreover, in this case the group of isometries is a finite-dimensional Lie group [19].
A sub-Riemannian geodesic is called homogeneous if it is an orbit of a one-parametric group of isometries. A sub-Riemannian manifold is called geodesic orbit if any normal geodesic is homogeneous. Firstly this notion appears in paper [7] by W. Ambrose and I. M. Singer in Riemannian case. Later Riemannian geodesic orbit manifolds were studied by A. Kaplan [24] and O. Kowalski, L. Vanhecke [28]. We refer to the book by V. N. Berestovskii and Yu. G. Nikonorov [11] for a historical overview of the results obtained for homogeneous geodesics and geodesic orbit Riemannian manifolds.
The objective of this paper is a generalization to sub-Riemannian case of some of the results on homogeneous geodesics and the geodesic orbit property known for Riemannian manifolds. We use an approach from geometric control theory [3] and characterize normal geodesics as projections of phase curves of a Hamiltonian vector field on the cotangent bundle. Any geodesic starting at a fixed point is determined by an initial covector (momentum) instead of an initial vector in Riemannian geometry. Therefore, we study homogeneous properties of geodesics in terms of initial momenta.
The paper has the following structure. In Section 2 we consider a sub-Riemannian problem as an optimal control problem, then we define corresponding extremal curves in terms of a Hamiltonian vector field on the cotangent bundle. Then in Section 3 we obtain a criterion for homogeneous geodesics using vertical part of this Hamiltonian system and consider some examples. Especially we discuss geodesics of left-invariant sub-Riemannian structures on Carnot groups in Section 4. We prove there that only 2-step Carnot groups are geodesic orbit. Section 5 is devoted to several general properties of sub-Riemannian geodesic orbit manifolds. In particular, we get a criterion for a sub-Riemannian manifold to be geodesic orbit, we prove that the geodesic flow of a geodesic orbit sub-Riemannian manifold is integrable in non-commutative sense. Also we obtain that any weakly commutative (in particular, weakly symmetric) sub-Riemannian homogeneous space is geodesic orbit. Finally, in Section 6 we obtain some broad conditions for existence of at least one homogeneous geodesic on a sub-Riemannian manifold. Let us introduce some notation that we will use throughout the paper. We denote Lie groups by uppercase Latin letters and corresponding Lie algebras by same lowercase Gothic letters. The Killing form of a Lie algebra g is K g . If it is clear which Lie group we are speaking about, then we omit a subscript of this symbol. The left and right shifts by an element g of a Lie group are denoted by L g and R g respectively. We use the same letters for their differentials. The identity element of a group is denoted by symbol e. If U ⊂ V is a vector subspace of a vector space V , then U • ⊂ V * is its annihilator in the dual space. For a smooth function F on a cotangent bundle T * M we denote by F its symplectic gradient with respect to the canonical symplectic structure on T * M , i.e., the Hamiltonian vector field corresponding to the Hamiltonian F .
In this paper we will consider a connected Lie group of isometries G ⊂ Isom M acting on the manifold M transitively. So, we study homogeneous geodesics that are orbits of one-parametric subgroups of the group G. We denote by K = {g ∈ G | go = o} the isotropy subgroup of a point o ∈ M . Thus, we can consider M = G/K as a homogeneous space. It is well known (see for example Proposition 1 of paper [27]) that there exists a reductive decomposition g = m ⊕ k, where the subspace m is an (Ad K)-invariant complement to the subalgebra k. Indeed, if G is a closed subgroup of Isom M , then the subgroup K is compact, so there exists a (Ad K)invariant scalar product on g and an invariant complement m for the invariant subspace k. If G is not closed, then we consider the compact closure Ad K. Note that g and k are (Ad K)-invariant subspaces of the Lie algebra of the group Isom M . So, there exists an invariant complement m ⊂ g for k with respect to an (Ad K)-invariant scalar product. This subspace m is (Ad K)-invariant as well.
We will naturally identify the subspace m with the tangent space T o M via the map ϕ : m → T o M such that ϕ(ξ) = d dt t=0 exp (tξ)o for ξ ∈ m. It is easy to see that ϕ((Ad k)ξ) = kϕ(ξ) for ξ ∈ m and k ∈ K. We will denote the subspace ϕ −1 (∆ o ) by the same letter ∆ bearing in mind the identification above. Also we denote by B the scalar product transferred from ∆ o . Note that the subspace ∆ ⊂ m is (Ad K)-invariant. Furthermore we assume that the group G acts on the manifold M effectively, i.e., there are no nontrivial group elements that fix all points of the manifold M .

Sub-Riemannian geodesics
Here we describe sub-Riemannian geodesics in terms of Hamiltonian vector fields on the cotangent bundle T * G via Pontryagin maximum principle. Simultaneously we introduce some necessary notation.
First of all lift our sub-Riemannian problem from the homogeneous space M = G/K to the Lie group G. There exists a theory developed by E. Grong [21] of lifting Hamiltonian systems via an Ehresmann connection. Here we do it in an explicit way. Namely we consider a distribution on the Lie group G obtained by the left shifts of the subspace ∆ ⊂ m ⊂ g equipped with the scalar product B. (The left shifts of the subspace m form an Ehresmann connection on the bundle G → M mentioned above.) We have an optimal control problem to find an optimal admissible curve g : [0, T ] → G going from one left coset to another, see below (2.1). Due to the left-invariance we may assume that the starting left coset is just the subgroup K. So, the problem is to find a control u ∈ L ∞ ([0, T ], ∆) and a Lipschitz curve g u : [0, T ] → G for a given g 1 ∈ G such that It is equivalent to minimize the energy functional 1 2 T 0 B(u(t), u(t)) dt instead of the functional (2.1) because of Cauchy-Bunyakowski-Schwartz inequality. Below we use the energy functional.
Remark 2.1. Notice that problem (2.1) is G-left and K-right invariant. Indeed, for any k ∈ K and g ∈ G we have R k L g ∆ = L gk (Ad k −1 )∆ = L gk ∆.
Consider the following family of functions on the cotangent bundle T * G depending on the parameters u ∈ ∆ and ν ∈ R: where π : T * G → G is the natural projection and · , · is the natural pairing of elements from T g G and T * g G. A necessary condition of optimality is given by the Pontryagin maximum principle [3,39].
Definition 2.3. A curve λ : [0, T ] → T * G satisfying the conditions of Theorem 2.2 is called an extremal. Its projectiong = π(λ) to the group G is called an extremal curve. If the parameter ν = 0, then corresponding extremals and extremal curves are called normal. If ν = 0, then these curves are called abnormal. The further projection of a normal extremal curve to the homogeneous space M = G/K is called a normal sub-Riemannian geodesic.
Below we will consider only normal geodesics.
By the transversality condition of the Pontryagin maximum principle we get It is easy to see that the maximum condition in the normal case gives us It follows thatũ(t) = B −1 (p(t)| ∆ ), where B : ∆ → ∆ * maps X ∈ ∆ to B(X, · ) ∈ ∆ * and by p| ∆ we denote the restriction of a linear function p ∈ g * to the subspace ∆ ⊂ g. Thereby the left-invariant normal maximized Hamiltonian of the Pontryagin maximum principle reads as H(p) = 1 2 B(p| ∆ , p| ∆ ) as a function of variable p ∈ k • ⊂ g * . (Here we transfer the quadratic form B from ∆ to ∆ * .) In the left trivialization the Hamiltonian system of the Pontryagin maximum principleλ(t) = H(λ(t)) takes the form (see, for example, Section 18.3.2 of book [3]) Note that the first equation of system (2.2), also called the vertical part of Pontryagin's Hamiltonian system, does not depend on the second variable g ∈ G. So, this subsystem can be studied independently. Finally, note that every sub-Riemannian geodesic is determined by its initial momentum p(0) = λ(0) ∈ g * = T * e G.

Homogeneous geodesics
Let M be a sub-Riemannian manifold, and let a Lie group G ⊂ Isom M be a subgroup of isometries that acts on the manifold M transitively and effectively. Therefore, M = G/K, where K is an isotropy subgroup. Definition 3.1. A sub-Riemannian geodesic m : [0, T ] → M is called a homogeneous geodesic if it is an orbit of one-parametric subgroup of isometries, i.e., m(t) = exp (tX)m(0) for some element X ∈ g that is called a geodesic vector.
Remark 3.2. An important property of homogeneous geodesics in view of optimal synthesis (or, generally speaking, in view of geometric control theory) is an equioptimality. This means that the time of loss of optimality (the cut time) for a geodesic does not depend on a starting point on this geodesic. In other words, the cut time as a function of an initial momentum of a geodesic is constant on the trajectories of the vertical part of Pontryagin's Hamiltonian system. Yu. L. Sachkov [43] made an amazing observation that for all completely solved sub-Riemannian problems geodesics are equioptimal. The reason of this phenomenon is unknown, but homogeneous geodesics deliver a clear example of such situation. At the same time, not every equioptimal geodesic is homogeneous [42].
There is a well known criterion for homogeneous geodesics in the Riemannian case, which appears first in works of B. Kostant [26],È. B. Vinberg [48] and later in paper of O. Kowalski and L. Vanhecke [28].
The main difference of sub-Riemannian situation is that a geodesic passing through a fixed point is determined by its initial momentum instead of an initial geodesic vector (tangent vector). Therefore, we need a criterion in terms of initial momenta. We apply a Hamiltonian version of the Geodesic Lemma obtained by G. Z. Tóth [46] to the sub-Riemannian Hamiltonian. Let us formulate this condition in a more geometric way.
Lemma 3.4. The following conditions are equivalent for a sub-Riemannian structure.
(1) A geodesic with an initial momentum p ∈ k • is homogeneous.
(3) The trajectory of the vertical part of the Pontryagin Hamiltonian system that corresponds to a geodesic lies in an (Ad * K)-orbit on k • .
Remark 3.5. In particular, any fixed point p 0 ∈ k • of the vertical part of the Hamiltonian vector field corresponds to a homogeneous geodesic, since p 0 ∈ (Ad * K)p 0 .
Consider the corresponding extremal λ : [0, T ] → T * G. Its points are elements of an orbit of the one-parametric subgroup shifted by some elements of the subgroup K acting on the right. Thus, there exists a curve k : Via left trivialization we have So, p(t) = (Ad * k(t) −1 )p(0). Therefore, the trajectory of the vertical subsystem p : [0, T ] → k • lies in an Ad * Korbit.
(3)⇒(2) Assume that p(t) = (Ad * k(t))p(0) for some curve k : [0, T ] → K. Take a derivative d dt | t=0 of this equality. We getṗ(0) = −(ad * Z)p(0) where Z =k(0) ∈ k. But by the first equation of system (2.2) we havė We claim thatλ(0) coincides with the Hamiltonian vector field at the point λ(0). Indeed, first consider the vertical component of the curve λ Its tangent vector at the point p(0) equals d dt | t=0 (Ad * exp (tZ))p(0) = −(ad * Z)p(0). This is equal toṗ(0) by the assumption. Second, the horizontal projection of the tangent vector to λ(t) for t = 0 is d p H ∈ m. So, λ(0) = H(λ(0)). Recall that the Hamiltonian vector field H is invariant under the left action of the group G and the right action of the group K (Rem. 2.1). Thus, by definition of the curve λ, it is a trajectory of the Hamiltonian vector field. Then the corresponding sub-Riemannian geodesic, i.e., the projection of λ to the homogeneous space G/K is an orbit of the one-parametric subgroup {exp t(d p H + Z)} ⊂ G.   Example 3.8. Consider the Lie group M of proper isometries of a sphere (or a hyperbolic plane). This group is isomorphic to the Lie group SO 3 or PSL 2 (R), respectively. Any left-invariant Riemannian metric is defined by a positive definite quadratic form B on m. Let I 1 , I 2 , I 3 > 0 be its eigenvalues and X 1 , X 2 , X 3 be corresponding eigenvectors. The vertical part of the Pontryagin Hamiltonian system is just Euler's equation for momenta of a rigid body with a fixed point.
If I 1 = I 2 = I 3 , then the corresponding metric is called axisymmetric. In this case G = Isom M = M × SO 2 , and as a homogeneous space M = (M × SO 2 )/SO 2 , where the isotropy subgroup is an anti-diagonal subgroup SO 2 ⊂ M × SO 2 . The solution of the vertical part of the Hamiltonian system is (see [37,38]) where p 3 is the third component of the covector p in the basis dual to the basis X 1 , X 2 , X 3 and κ is a constant depending on the eigenvalues of the form B. Thus, by Lemma 3.4 all geodesics are homogeneous. See Figure 1 for the phase portrait of the vertical part of the Hamiltonian system on the level surface of the Hamiltonian H = 1 2 . In the general case of all different eigenvalues I 1 , I 2 , I 3 the isometry group is M × Z 3 . The vertical part of the Hamiltonian system has 6 fixed points (4 stable and 2 unstable ones), see Figure 2. These points are the initial momenta of 6 homogeneous geodesics passing through the point e.
The shortest arcs of these sub-Riemannian structures were described by V. N. Berestovskii and I. A. Zubareva [9,12] and earlier U. Boscain and F. Rossi [17] found the cut loci. The solution of the vertical part of the Pontryagin Hamiltonian system has the same form as in Example 3.8, see (3.1). The corresponding phase portrait is similar to Figure 1 except that the level surface of the Hamiltonian is a cylinder instead of a sphere. Hence, every geodesic is homogeneous. Note, that it was shown by explicit formulae in papers [9,12].
Remark 3.10. The homogeneous spaces (3.2) are weakly symmetric. This means that for every pair of points there exists an isometry that swaps these points. This notion was introduced by A. Selberg and the second space of (3.2) is his original example. It is known that Riemannian weakly symmetric spaces are geodesic orbit, this is a result of J. Berndt, O. Kowalski and L. Vanhecke [14], later V. N. Berestovskii and Yu. G. Nikonorov proved it using another method [10]. For sub-Riemannian spaces this question was stated in paper [13], we give an answer in Section 5. The sub-Riemannian structure is defined by the distribution ∆ = p and the restriction of the Killing form to ∆. A. A. Agrachev [1], R. W. Brockett [18] and V. Jurdjevic [23] noted that geodesics of this structure are products of two one-parametric subgroups This means that this manifold is geodesic orbit, since the isometry group is G = M × K, and (3.3) is an orbit of a one-parametric subgroup of G.
Example 3.12. Consider the optimal control problem for rolling of a sphere on a plane with twisting but without slipping. I. Yu. Beschastnyi [15] studied it as a sub-Riemannian problem on the Lie group M = SO 3 × R 2 . Let V 1 , V 2 , V 3 be an orthonormal basis of the Lie algebra so 3 with respect to the Killing form, and e 1 , e 2 be a basis of the plane R 2 . The distribution is defined by the orthonormal frame e 1 − V 2 , e 2 + V 1 , V 3 . Below we will identify the Lie algebra m with the dual space m * via the basis V 1 , V 2 , V 3 , e 1 , e 2 . The vertical subsystem of Pontryagin's Hamiltonian system is as follows [15]: wherep ∈ so 3 is the tangent vector to the one-parametric group of rotations around the vector (p 1 , p 2 , 0) ∈ R 3 (here p 1 , p 2 are the components of the vector p). I. Yu. Beschastnyi [15] obtained geodesics as products of two one-parametric subgroups But despite this, and also despite the fact that the rotations exp tˆ are symmetries of system (3.4), this sub-Riemannian structure is not geodesic orbit. Let us prove this by contradiction. Assume that there is an isotropy group K acting on m by conjugation. By Lemma 3.4, since any automorphism of the Lie algebra so 3 is inner, there exists X ∈ m such that˙ = [X, ] andṗ = [X,p].
But from (3.4) we know that˙ = [p, ] andṗ = 0. Thus, there are three possible cases. 1. If p = 0, then we can take X = 0. 2. If p = 0 and ,p are collinear, then we can take X = 0 as well.
3. If p = 0 and ,p are not collinear, then X =p.
In the first two cases the geodesic with the initial momentum ( , p) is homogeneous, since vertical subsystem (3.4) is trivial. In the third case we get that exp X is a rotation around a horizontal axis. Hence, exp X does not preserve the distribution ∆ in contradiction with the assumption that exp X is an isotropy.
Notice that in the first case the corresponding geodesic is a rotation of the sphere around vertical axis at a fixed point. The second case corresponds to a rolling of the sphere along a straight line (this is a non strict abnormal geodesic [15]).

Nilpotent geodesic orbit sub-Riemannian manifolds
Nilpotent Lie groups play an important role in sub-Riemannian geometry because of existence of nilpotent approximation [4]. In this section we show that left-invariant sub-Riemannian structures on the two step free nilpotent Lie groups are geodesic orbit and prove that any left-invariant sub-Riemannian structure on a Carnot group of step more than 2 cannot be geodesic orbit.
Definition 4.1. A Lie algebra g is called a Carnot algebra of step s if there is a decomposition g = g 1 ⊕ · · · ⊕ g s , [g 1 , g i ] = g i+1 for i = 1, . . . , s, [g 1 , g s ] = 0, such that g 1 generates the Lie algebra g. The corresponding connected and simply connected nilpotent Lie group is called a Carnot group. The left shifts of the subspace g 1 give a non-holonomic distribution on a Carnot group. Any scalar product on g 1 gives rise to a sub-Riemannian structure. The rank of sub-Riemannian structure on a Carnot group is equal to the dimension of the subspace g 1 .

Proposition 4.2.
A left-invariant sub-Riemannian structure on a Carnot group of step 2 is geodesic orbit.
Proof. Consider the case of a free Carnot group of step 2 and rank r and the following model of such group.
Endow the set M with a product rule as follows: The corresponding tangent algebra is m = V ⊕ Λ 2 V and the sub-Riemannian distribution is defined by the subspace ∆ = V ⊕ 0. The vertical part of the Pontryagin Hamiltonian system for normal geodesics reads as (see [40]) V. Kivioja  Indeed, we can take X = . So, by Lemma 3.4 any normal geodesic of a free two-step Carnot group is homogeneous.
Any two-step nilpotent Lie algebra g is a factor of a free nilpotent Lie algebra g of the same rank by an ideal i g laying at the second layer of the Lie algebra i ⊂ g 2 . Since g * ∼ = i • ⊂ g * , we obtain that any geodesic γ on the Carnot group G can be lifted to a geodesic γ on the free Carnot group G with an initial momentum from the subspace i • . It follows from Lemma 3.4 that if this lifted geodesic γ is homogeneous, then the geodesic γ is homogeneous as well.
Remark 4.3. The sub-Riemannian problems for free Carnot groups of step 2 and small ranks are completely solved. For rank 2 the corresponding sub-Riemannian problem is equivalent to the classical variational Dido problem [44], the complete discovering of this sub-Riemannian structure was made by A. M. Vershik and V. Ya. Gershkovich [47]. For rank 3 O. Myasnichenko [34] constructed the cut locus and A. Montanari, D. Morbidelli [33] derived the cut time and described singularities of the distance function. L. Rizzi, U. Serres obtained some upper bounds for the cut time for ranks more than 3 [40]. H.-Q. Li and Ye Zhang in recent preprints [29,30] introduced formulae for sub-Riemannian distance for Heisenberg type groups.
Remark 4.4. Of course, Carnot groups of step 1 (i.e., commutative groups) are geodesic orbit since the corresponding sub-Riemannian geometry is just Euclidian geometry.
It turns out that sub-Riemannian structures on Carnot groups of step more than 2 cannot be geodesic orbit. This is a generalization of C. S. Gordon's result [20] obtained for nilpotent Riemannian manifolds. Proof. Let us begin with considering a more general situation. Assume that G is a group of isometries acting transitively on a sub-Riemannian geodesic orbit manifold M and K is an isotropy subgroup. Consider a reductive decomposition g = m ⊕ k. The sub-Riemannian distribution is generated by the subspace ∆ ⊂ m. This subspace is Ad K-invariant. Let ∆ ⊥ ⊂ m be its Ad K-invariant complement. Continue the non-degenerate quadratic form B on ∆ (that defines the sub-Riemannian structure) to a non-degenerate quadratic form B on m such way that m = ∆ ⊕ ∆ ⊥ is the orthogonal decomposition with respect to the form B and the form B is Ad K-invariant.
Following Proposition 1.8 of paper [20] we claim that for any X ∈ ∆ the operator π ∆ ⊥ (ad X) is skewsymmetric on the subspace ∆ ⊥ with respect to the form B, where π ∆ ⊥ : m → ∆ ⊥ is the orthogonal projection.
Identify the tangent space m with the dual space m * by the map B : m → m * , where m ξ → B(ξ, · ). Since the sub-Riemannian structure is geodesic orbit, applying Lemma 3.4, we get that for any since H(p) = 1 2 B(X, X) and d p H = X in the sub-Riemannian case. This is equivalent to In particular, In particular, So, the second term of equation (4.2) equals zero, since X ∈ ∆ and ∆ is orthogonal to ∆ ⊥ . We obtain B(X, [X + Z, Y ]) = 0, i.e., the first term of equation (4.1) is zero. It follows that the second term of equation (4.1) is zero as well: The second summand in equation (4.3) equals zero, because the operator ad Z is skew-symmetric for Z ∈ k. We get This means that the operator π ∆ ⊥ (ad X) is skew-symmetric as well. Finally, by Theorem 1.2.ii of V. Kivioja and E. Le Donne [25] if N is a nilpotent Lie group with a left-invariant sub-Riemannian structure, then G = N K, where K is an isotropy subgroup. We have an Ad K-invariant decomposition n = ∆ ⊕ [n, n]. We proved that for any X ∈ ∆ the operator π [n,n] (ad X) is skew-symmetric. On the other hand, this operator is nilpotent. It follows that it is equal to zero. Since ∆ generates the Lie algebra n, we obtain [n, [n, n]] = 0. So, the step of our Carnot group is less than 3.
Example 4.6. Consider the Cartan group, i.e., the free Carnot group of rank 2 and step 3. The corresponding Carnot algebra g = g 1 ⊕ g 2 ⊕ g 3 , where g 1 = span {X 1 , X 2 }, g 2 = span {X 3 }, g 3 = span {X 4 , X 5 }, has the following non-zero commutators of the basis elements: The isotropy subgroup of the group of isometries is SO 2 acting tautologically on g 1 , g 3 and trivially on g 2 , see Statement 4.1 of paper [41]. There are 3 independent Casimir functions: where h i = X i , · for i = 1, . . . , 5 are linear functions on g * . These functions together with the Hamiltonian H = 1 2 (h 2 1 + h 2 2 ) from a system of 4 first integrals of the vector field H in the 5-dimensional space g * . Consider trajectories of the vertical part of the Hamiltonian vector field. In the case of h 4 = h 5 = 0 the corresponding trajectories are the same as for the Heisenberg group, i.e., the free Carnot group of rank 2 and step 2. In the case h 2 4 + h 2 5 = 0 the corresponding trajectories are intersections of two quadrics. Hence, by Lemma 3.4 the only homogeneous geodesics are geodesics with initial momenta such that h 4 = h 5 = 0.
Remark 4.7. It would be interesting to find all the homogeneous geodesics for Carnot groups. We can just say that there are some nontrivial homogeneous geodesics, i.e., geodesics corresponding to nontrivial trajectories of the vertical part of the Hamiltonian vector field H vert . Note, that for a free Carnot group the isotropy subgroup is SO(g 1 ). For a free Carnot group there exist homogeneous geodesics that appear from the free Carnot group of the same rank and lower step due to the natural factorization. Besides this, for step 3 free Carnot groups there exist series of two-dimensional coadjoint orbits that are organized as coadjoint orbits of the Heisenberg group [36]. So, geodesics with initial momenta that lie in this kind of coadjoint orbits are homogeneous by Lemma 3.4.

Some general properties of geodesic orbit sub-Riemannian manifolds
Here we prove some general facts about geodesic orbit sub-Riemannian manifolds and obtain a series of important examples. First we get the following criterion. It follows from Lemma 3.4 that the sub-Riemannian manifold M is geodesic orbit if and only if any trajectory of the vertical part of the normal Pontryagin Hamiltonian system is tangent to Ad * K-orbits in m * . It is equivalent to the following fact: any function that is constant on Ad * K-orbits has zero derivative in the direction of H vert .
From Proposition 5.1 it follows that the geodesic flow of a geodesic orbit sub-Riemannian manifold is integrable in noncommutative sense. Recall some necessary definitions.
Remark 5.3. Roughly speaking, this means that there is a set of dim M + k independent functions on T * M , but instead of Liouville integrable condition we have that the matrix of the Poisson brackets of these function is nonzero and has rank 2k.
Recall the definition of the momentum map and some of its properties.
Definition 5.4. For ξ ∈ g take the corresponding velocity field H ξ of the action of the group G on the manifold M : where g(t) ∈ G is the one-parametric subgroup such thatġ(0) = ξ. Consider this velocity field H ξ as a linear function on fibers of the cotangent bundle T * M . Define the momentum map µ : T * M → g * , µ(λ)(ξ) = H ξ (λ), for λ ∈ T * M, ξ ∈ g.
Remark 5.5. Notice that H ξ is a velocity field ξ * of the G-action on T * M : By definition of a symplectic gradient we obtain the following expression for the differential of the momentum map where ω is the canonical symplectic structure on T * M . This means that the orbits of the left G-action on T * M and the fibers of the momentum map are orthogonal with respect to the symplectic structure, for details see, for example, [32] or [49]. Remark 5.11. This result is consistent with the fact that the geodesic flow for sub-Riemannian structures on 2-step Carnot groups is integrable in trigonometric functions [34,40,47]. Recall that these sub-Riemannian structures are geodesic orbit, see Proposition 4.2. But sub-Riemannian structures on Carnot groups of step more than 2 cannot be geodesic orbit. Actually, it was shown numerically in paper [16] that the geodesic flow of sub-Riemannian structures on 3-step Carnot groups of rank more than 2 is not Liouville integrable and as a consequence is not integrable in noncommutative sense. (For 3-step Carnot group of rank 2, i.e., the Cartan group, the sub-Riemannian geodesic flow is integrable in elliptic functions [41].) The sub-Riemannian geodesic flow for Carnot groups of step more than 3 is not Liouville integrable as it was proved by L. V. Lokutsievskii and Yu. L. Sachkov [31]. We know near to nothing about dynamics of sub-Riemannian geodesic flow on Carnot groups of step more than 2, except the Cartan group [41]. There are only particular results on an asymptotic behaviour of extremal controls [35,36]. It is well known that weakly symmetric spaces (and in particular symmetric spaces) are weakly commutative [5,49]. But there are no symmetric spaces with an invariant sub-Riemannian structure [8].
See Remark 3.10 above for references on the Riemannian version of this theorem.

Existence of a homogeneous geodesic
In this section we prove that under some broad conditions there exists a homogeneous geodesic passing through an arbitrary point of a homogeneous sub-Riemannian manifold. Moreover, we prove that there exists a fixed point of the vertical part of the corresponding Hamiltonian system in this case. Denote by r = Ker K the kernel of the Killing form. Note that r is a solvable ideal in the Lie algebra g. Since the Killing form is non degenerate on k (see Prop. 2 of paper [27]), we obtain r ⊂ m. There are two cases: r = m and r m.
In the fist case following Proposition 3 of [27] we can assume that a solvable group of isometries acts on the manifold M transitively. Let us denote this group by the same letter G. Then [g, g] = g. So, there exists p ∈ g * such that p([g, g]) = 0 and the conditions of Lemma 3.4 are satisfied automatically.
Consider now the second case r m. The Lie group corresponding to the ideal r acts on the manifold M . We will reduce our sub-Riemannian structure to a sub-Riemannian structure on the factor manifold by this action (see Lem. 6.2 below). Any geodesic of reduced sub-Riemannian structure lifts to the geodesic of original sub-Riemannian structure. Herewith homogeneous geodesics lift to homogeneous geodesics.
After that in Lemma 6.4 we consider a semisimple group of isometries of the new sub-Riemannian structure such that the restriction of the Killing form to the sub-Riemannian distribution is non trivial. Then Theorem 6.1 follows from Lemmas 6.2, 6.4. Lemma 6.2. Let i g be an ideal such that it does not contain the distribution ∆. There exists an invariant sub-Riemannian structure on G/ K such that any geodesic of this structure lifts to a geodesic of the sub-Riemannian structure on G/K, where K ⊂ G is a Lie subgroup with the tangent algebra k + i.
Proof. Consider the subspace i ∆ = ∆ ∩ i and its orthogonal complement i ⊥ B ∆ in ∆ with respect to the form B. Let ∆ = π(i ⊥ B ∆ ) be the image of this orthogonal complement under the factorization map π : g → g/i. Notice that the subspace ∆ and the subalgebra π(k) generate the Lie algebra g/i since i is an ideal and ∆ + k generates g. Moreover, there exists a scalar product on ∆ that is inherited from the scalar product B on ∆. So, we can define a sub-Riemannian structure on G/ K.
Any extremal of the sub-Riemannian problem on G/ K is an extremal of the sub-Riemannian problem on G/K with an initial momentum from the subspace (k + ∆ ∩ i) • ⊂ k • . This means that the corresponding geodesic on G/ K lifts to a geodesic on G/K. Remark 6.3. Note that in this case the sub-Riemannian structure is a kind of generalization of a Chaplygin system [6]. Lemma 6.4. Assume that the restriction of the Killing form to the subspace ∆ is non trivial. Then there exists a homogeneous sub-Riemannian geodesic.
Proof. The kernel r of the Killing form does not contain the subspace ∆ since K| ∆ = 0. According to Lemma 6.2 we can assume that r = 0.
Denote by ∆ ⊥ K the orthogonal complement of ∆ with respect to the Killing form. Take the subspace Γ ⊂ ∆ that is the orthogonal complement of ∆ ∩ ∆ ⊥ K in ∆ with respect to the form B. Obviously, Γ ⊥ K ⊃ ∆ ⊥ K .
According to the main result of paper [27] by O. Kowalski and J. Szenthe for the Riemannian metric that is defined by the scalar product B there exists a homogeneous geodesic. We will show that the initial momentum of this geodesic is also an initial momentum of a homogeneous geodesic of our sub-Riemannian structure. In other words, we will find a geodesic vector that is one of the main axes of the forms K and B at the same time and we prove that this vector belongs to the subspace Γ ⊂ ∆.
Note that Γ is an invariant subspace of the operator A. From the proof of Theorem 6.1 immediately follows Corollary 6.5.
(1) If a solvable group of isometries acts transitively on a sub-Riemannian manifold, then there is a homogeneous geodesic passing through an arbitrary point.
(2) If a semisimple group of isometries acts transitively on a sub-Riemannian manifold and the restriction of the Killing form to the sub-Riemannian distribution is non trivial, then there is a homogeneous geodesic passing through an arbitrary point.

Conclusion
We generalized several facts known about Riemannian homogeneous geodesics to the sub-Riemannian case. We used the Hamiltonian approach to formulate an analogue of the Geodesic Lemma, i.e., a criterion for geodesic to be homogeneous.
We studied several properties of geodesic orbit sub-Riemannian manifolds. In particular, the corresponding geodesic flow is integrable in non-commutative sense. Since we obtain that Carnot groups of step more than 2 cannot be geodesic orbit, this agrees with previously known facts about the integrability of geodesic flows on Carnot groups.
Finally, we proved that there exists at least one homogeneous geodesic for a sub-Riemannian manifold with a semisimple group of isometries such that the restriction of the Killing form to the distribution is non trivial. However, it is still unknown if this condition is essential. It seems that it is possible to avoid this condition using another method for proof.
Nevertheless, several important questions remain open. How to describe all geodesic orbit sub-Riemannian manifolds? How to describe all homogeneous geodesics of a given sub-Riemannian structure? It requires the knowledge of the group of isometries, that is a separate problem.