Sub-Riemannian (2, 3, 5, 6)-Structures

We describe all Carnot algebras with growth vector (2, 3, 5, 6), their normal forms, an invariant that separates them, and a change of basis that transforms such an algebra into a normal form. For each normal form, Casimir functions and symplectic foliations on the Lie coalgebra are computed. An invariant and normal forms of left-invariant (2, 3, 5, 6)-distributions are described. A classification, up to isometries, of all left-invariant sub-Riemannian structures on (2, 3, 5, 6)-Carnot groups is obtained.

Sub-Riemannian structures [1] are stratified in depth, i.e., with respect to the minimum order of Lie brackets required for generating a tangent space from basis vector fields. The complexity of sub-Riemannian (SR) structures grows substantially with increasing step. At present, SR structures of step at most three have been examined in detail [2][3][4][5][6][7][8]. Accordingly, a task of great interest is to systematically investigate SR structures of step 4. The study of the simplest of these structures, namely, a nilpotent SR structure with growth vector (2,3,5,8) (see Example 2 below) was initiated in [9][10][11]. Below, we obtain a complete classification of nilpotent SR structures and distributions with growth vector (2,3,5,6). It is shown that all such structures are quotient structures of a nilpotent SR structure with growth vector (2,3,5,8).

SUB-RIEMANNIAN QUOTIENT STRUCTURES
In this paper, all Lie algebras are considered over the field . Definition 1. A nilpotent Lie algebra is called a Carnot algebra if (i) is graded: (2) The corresponding connected simply connected Lie group is called a Carnot group.
Conditions (i) and (ii) are equivalent to the condition Definition 2. The growth vector of a Carnot algebra is defined as Here, is the growth vector of a left-invariant distribution on the Lie group of the Lie algebra generated by the subspace . Let be a smooth manifold. A sub-Riemannian structure on M [1] is a pair consisting of a vector distribution and a scalar product g in Δ. Let G be a Lie group and be its Lie algebra. A leftinvariant SR structure on G consists of a left-invariant distribution on G and a left-invariant scalar product in the distribution. This structure is specified by a subspace and a scalar product g in . In this case, we say that is an SR structure in the algebra . Left-invariant SR structures on Carnot groups arise as nilpotent approximations of general SR structures on smooth manifolds [1]. Definition 3. Let be an SR structure in a Lie algebra , and let be an ideal such that = {0}. Let be a quotient algebra and : be a g = . g g (1) Lie( )

CONTROL THEORY
a Ailamazyan Program Systems Institute, Russian Academy of Sciences, Pereslavl-Zalessky, Yaroslavl oblast, 152020 Russia *e-mail: yusachkov@gmail.com canonical projection. Define and , π(Y)) = g(X, Y) for . Then ( , ) is an SR structure in , which will be called a quotient structure of the SR structure .

Example 1.
Let be a free nilpotent Lie algebra of step 3 with two generators (Cartan algebra); this is the Carnot algebra with growth vector (2,3,5). There exists a basis in which the nonzero Lie brackets are Consider an SR structure in with an orthonormal frame [8]. Sequentially choosing the subspaces , , and as an ideal , we obtain SR quotient structures in the Engel algebra (growth vector (2, 3, 4)) [7], Heisenberg algebra (growth vector (2, 3)) [2], and the two-dimensional commutative algebra (growth vector (2)).

Example 2.
Let be a free nilpotent Lie algebra of step 4 with two generators; this is the Carnot algebra with growth vector (2,3,5,8), which will be called a nilpotent (2, 3, 5, 8)-algebra. There exists a basis in which the nonzero Lie brackets are given by Consider an SR structure in with an orthonormal frame [9][10][11]. It is easy to see that this SR structure is unique, up to a Lie algebra automorphism, SR structure in of rank 2 satisfying the condition ; we call it a nilpotent SR (2, 3, 5, 8)-structure. In what follows, we will use a dual basis in the Lie coalgebra : The goal of this paper is, given a structure , to describe all SR quotient structures for two-dimensional ideals . These are exactly nilpotent SR structures with growth vector (2,3,5,6).
It is easy to see that a two-dimensional subspace is an ideal if and only if = span(X 6 , . Therefore, any quotient algebra by the two-dimensional ideal is a Carnot algebra with growth vector (2,3,5,6). Let us describe such algebras.  (2,3,5,6), together with nonzero Lie brackets in the corresponding basis. Recall that the notation for these algebras was used in [12,13]. A classification of nilpotent Lie algebras of dimension ≤7 was obtained in [12], and all Carnot algebras of dimension ≤7 were described in [13].
(ii) The number is an invariant of the Carnot algebra ; s is called the signature of .
(ii) The algebras are separated by the signature : Thus, the signature is an invariant separating three classes of isomorphism of the algebras . Remark 2. Item (i) in Theorem 2 was first proved in [13]. We proved it independently, together with an algorithm for reducing the multiplication table in the (2, 3, 5, 6)-algebra to the normal form in Examples 3-5.
To find a basis change in the algebra that transforms the basis into one of the normal forms indicated in Examples 3-5, it suffices to reduce the quadratic form Q(x, y) = to a sum of squares, to apply this change to the basis of the space , and to normalize the vector X 6 generating the space .  X X X X X X X X X X X X X X : :  ( ) X X g (1) g (4) g Δ, ( ) g g Δ = , Δ = , g dim 2 Lie( ) or, equivalently, the equality = . The corresponding plane is called a nilpotent (2, 3, 5, 6)-distribution in .
The vector (2, 3, 5, 6) is the growth vector of the left-invariant distribution on the Lie group G of the Lie algebra defined by the plane . Theorem 3. Let be a Carnot algebra with growth vector (2,3,5,6) of signature . For any nilpotent (2, 3, 5, 6)-distribution , there exists a frame such that Theorem 4. Let be a Carnot algebra with growth vector (2,3,5,6). For any nilpotent SR (2, 3, 5, 6)structures in , there exists an orthonormal frame and a number for which (11) The number in (12) is called the canonical parameter of the SR structure ( . It is easy to see that ν is equal to the ratio of the smaller (in absolute value) eigenvalue of Q to the larger eigenvalue. Moreover, . Theorem 5. The canonical parameter is an invariant of the nilpotent SR (2, 3, 5, 6)-structure.
Let be an SR structure on a manifold . The corresponding SR distance d [1] transforms M into a metric space. Recall that an isometry between metric spaces and is a mapping : such that The metric group [15] is a Lie group with a leftinvariant distance inducing a manifold topology on this group. In particular, any Carnot group is a metric group. It was proved in [15] that any isometry between metric groups is an analytic mapping; any isometry between connected nilpotent metric groups is an affine mapping (i.e., the composition of a left shift and an isomorphism).
The following theorem provides a classification of left-invariant SR (2, 3, 5, 6)-structures up to isometries of corresponding Carnot groups. Theorem 6. Let and be nilpotent Carnot (2, 3, 5, 6)-algebras, and let and be corresponding Carnot groups. Let and be nilpotent SR (2,3,5,6)structures in g and , and be corresponding canoni- cal parameters, and and be corresponding leftinvariant SR metrics on and .
The metric spaces and are isometric if and only if .

Theorem 7.
Let be the (2, 3, 5, 8)-algebra from Example 2 with a basis according to (3), (4), and let ( be an SR structure in with orthonormal frame . Let with be any 2-dimensional subspace of . Then the SR quotient structure is one of the SR (2, 3, 5, 6)-structures in : (i) in the parabolic case s = 0, we have ; (ii) in the hyperbolic case , we have ; (iii) in the elliptic case s = 1, we have . Conversely, any nilpotent (2, 3, 5, 6)-structure in each of these algebras is realized as a quotient structure of a nilpotent (2, 3, 5, 8)-structure . Remark 3. An SR structure in the elliptic algebra with canonical parameter ν = 1 was considered in [14]. For this structure, it was proved that the vertical subsystem of the Hamiltonian system in the Pontryagin maximum principle [1] is Liouville integrable. This Hamiltonian system was integrated in [11]   , . In each of the cases presented below, the symplectic leaf is a connected component of common level surfaces of the Casimir functions on corresponding submanifolds in .

CONCLUSIONS
The following results were obtained in this paper. We described the Carnot algebras with growth vector (2, 3, 5, 6), i.e., quotient algebras of a free nilpotent Carnot algebra of step 4 with two generators. Previously, it was known that there are three normal forms of such algebras: parabolic, hyperbolic, and elliptic. An invariant separating these algebras was found, namely, the signature . Additionally, a change of basis transforming the multiplication table into one of three normal forms was described.
For each Carnot algebra with growth vector (2, 3, 5, 6), the rank of the Poisson bracket, Casimir functions, and a symplectic foliation in the Lie coalgebra were calculated.