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Nonholonomic Ricci Flows: III. Curve Flows and Solitonic Hierarchies

The geometric constructions are elaborated on (semi) Riemannian manifolds and vector bundles provided with nonintegrable distributions defining nonlinear connection structures induced canonically by metric tensors. Such spaces are called nonholonomic manifolds and described by two equivalent linear connections also induced in unique forms by a metric tensor (the Levi Civita and the canonical distinguished connection, d-connection). The lifts of geometric objects on tangent bundles are performed for certain classes of d-connections and frame transforms when the Riemann tensor is parametrized by constant matrix coefficients. For such configurations, the flows of non-stretching curves and corresponding bi-Hamilton and solitonic hierarchies encode information about Ricci flow evolution, Einstein spaces and exact solutions in gravity and geometric mechanics. The applied methods were elaborated formally in Finsler geometry and allows us to develop the formalism for generalized Riemann-Finsler and Lagrange spaces. Nevertheless, all geometric constructions can be equivalently re-defined for the Levi Civita connections and holonomic frames on (semi) Riemannian manifolds.


Introduction
Both the theory of Ricci flows and the theory of integrable partial differential equations have deep links to the geometry of curves and surfaces, generalized Riemann-Finsler spaces and geometric analysis: Originally, the Ricci flow theory has addressed geometrical and topological issues, and certain applications in physics, for Riemannian manifolds [1,2,3,4,5] (we cite here some reviews on Hamilton-Perelman theory [6,7,8,9]). In parallel, it was found that various classes of solitonic equations (for instance, the sine-Gordon, SG, and modified Korteveg-de Vries, mKdV, equations) and along with their hierarchies of symmetries, conservation laws and associated recursion operators can be encoded into the geometry of flows of non-stretching curves in Riemannian symmetric spaces and related Lie algebras and Klein spaces [10,11,12,13,14,15,16,17], see also reviews and new results in Refs. [18,19].
In [20], it was proven that a more general class of (pseudo) Riemannian spaces can be encoded into bi-Hamilton structures and related solitonic hierarchies. The key construction was to deform nonholonomically the frame and linear connection structures in order to get constant matrix curvature coefficients, with respect to certain classes of nonholonomic frames. Such frames are adapted to a nonlinear connection (N-connection) structure induced by some generic off-diagonal metric coefficients . We also concluded that having generated the corresponding solitonic hierarchies for the so-called canonical distinguished connection (d-connection), we can re-define equivalently the geometric objects, conservation laws and basic equations and solutions in terms of the usual Levi Civita connection.
The formalism applied in [20], based on the geometry of moving nonholonomic frames with associated N-connection structure, was originally developed in Finsler-Lagrange geometry and generalizations [21,22,23]. Our idea [24] was to apply it to usual (semi) Riemannian spaces, or to Riemann-Cartan ones (with nontrivial torsion), prescribing certain nonholonomic distributions arising naturally if we constrain partially some degrees of freedom. For such systems, there are defined certain classes of preferred frames and symmetries for the gravitational and matter field interactions. Following this approach, it was possible to construct various classes of exact solutions in Einstein and string gravity modelling Finsler like locally anisotropic structure, possessing noncommutative symmetries and defining generically off-diagonal metrics and nonlinear interactions of pp-waves, two and three dimensional gravitational solitons and spinor fields [24,26,27].
Nevertheless, if realistic theories of gravitational and gauge field interactions and/or generalized Finsler models of geometric mechanics are intro-duced into consideration, the solitonic encoding of metric, connection and frame structures is possible for certain effective generalized Lagrange spaces. In this case, we model the geometric constructions on couples of symmetric Riemannian spaces provided with nonholonomic distributions. The work [25] concluded an approach when different theories of gravity and geometric mechanics are treated in a unified geometric way as nonholonomic (semi) Riemannian manifolds, or vector bundles. It was also proven that the data for geometric objects and fundamental physical equations (their symmetries and conservation laws) can be encoded into bi-Hamilton structures and correspondingly derived solitonic hierarchies.
Integrable and nonintegrable (i.e. holonomic and nonholonomic / anholonomic) flows of geometric structures were also recently considered in a series of works on nonholonomic Ricci flows [28,29] and applications in physics [30,31,32,33]. Some important results of those works were the proofs that constrained Ricci flows of (semi) Riemannian metrics result in Finsler like metrics and generalizations and, inversely, Finsler-Lagrange type geometrical objects can be described equivalently by flows on Riemann (in general, Riemann-Cartan) spaces.
The goal of this paper, the third one in a series defined together with [28,29], is to prove that solitonic hierarchies can be generated by any (semi) Riemannian metric g ij on a manifold V of dimension dim V = n ≥ 2 if the the geometrical objects are lifted in the total space of the tangent bundle T V, or of a vector bundle E = (M, π, E), dim E = m ≥ n, by defining such frame transforms when constant matrix curvatures are defined canonically with respect to certain classes of preferred systems of reference. We shall also define the criteria when families of bi-Hamilton structures and solitonic hierarchies encode (in general, nonholonomic) Ricci flow evolutions of geometric objects and/ or exact solutions of gravitational field equations.
The paper is organized as follows: In section 2 we outline the geometry of nonholonomic manifolds and vector/ tangent bundles provided with nonlinear connection structure. We emphasize the possibility to define fundamental geometric objects induced by a (semi) Riemannian metric on the base space when the Riemannian curvature tensor has constant coefficients with respect to a preferred nonholonomic basis. We also present some results on evolution equations of nonholonomic Ricci flows and exact solutions in gravity.
In section 3 we consider Ricci flow families of curve flows on nonholonomic vector bundles. We sketch an approach to classification of such spaces defined by conventional horizontal and vertical symmetric (semi) Riemannian subspaces and provided with nonholonomic distributions defined by the nonlinear connection structure. It is constructed a corresponding family of non-holonomic Klein spaces for which the bi-Hamiltonian operators are defined by canonical distinguished connections, adapted to the nonlinear connection structure, for which the distinguished curvature coefficients are constant. Section 4 is devoted to the formalism of distinguished bi-Hamiltonian operators and vector soliton equations for arbitrary (semi) Riemannian spaces. Then we consider the properties of cosympletic and sympletic operators adapted to the nonlinear connection structure. We define the basic equations for nonholonomic curve flows and parametrize their possible Ricci flows.
Section 5 is devoted to formulation of the Main Result: a proof that for any nonholonomic Ricci flow system, one can be defined a natural family of N-adapted bi-Hamiltonian flow hierarchies inducing anholonomic solitonic configurations. There are constructed in explicit form the solitonic hierarchies corresponding to the bi-Hamiltonian anholonomic curve flows. Finally, there are speculated the conditions when from solitonic hierarchies we can extract solutions of the Ricci flow and/or field equations.
We summarize and discuss the results in section 6. For convenience, we outline the necessary definitions and formulas from the geometry of nonholonomic manifolds in Appendix A. Then, in Appendix B, we consider the geometry of N-anholonomic Klein spaces. A proof of the Main Theorem is sketched in Appendix C.

Notation remarks:
There are considered two types of flows of geometrical objects on manifolds of necessary smooth class, induced by 1) non-stretching curve flows γ(τ, l), defined by real parameters τ and l, and 2) Ricci flows of metrics g ij (χ), parametrized by a real χ. The non-stretching flows of a curve are constrained by the condition g ij (γ τ , γ l ) = 1, which under Ricci flows transforms into a family of such conditions, g ij (γ τ , γ l , χ) = 1. For Ricci flows, we get evolutions of families of non-stretching curves parametrized by hypersurfaces γ(τ, l,χ) χ γ(τ, l). It is convenient to use in parallel two types of denotations for the geometric objects subjected to both curve and Ricci flows: by emphasized all dependencies on parameters τ, l and χ or by introducing "up/low" labels like χ γ = γ(..., χ) or χ A = A(..., χ).
We shall also write "boldface" symbols for geometric objects and spaces adapted to a noholonomic/ nonlinear connection structure, like V, E, ... and write V, E, ... if the nonholonomic structure became trivial, i.e. integrable/ holonomic. In order to investigate the properties of curve and Ricci flow evolution equations it is convenient to use both abstract/global denotations and coefficient formulas with respect to coordinate or nonholonomic bases.
A nonholonomic distribution with associated nonlinear connection structure splits the manifolds into conventional horizontal (h) and vertical (v) subspaces. The geometric objects, for instance, a vector X can be written in abstract form as X = (hX, vX) = ( h X, v X), or in coefficient forms as X α = (X i , X a ) = (X i , X a ), where X = X α e α = X i e i + X a e a = X i ∂ i + X a ∂ a can be equivalently decomposed with respect to a general nonholonomic frame e α = (e i , e a ) or coordinate frame ∂ α = (∂ i , ∂ a ) for local h-and v-coordinates u = (x, y), or u α = (x i , y a ) when ∂ α = ∂/∂u α and ∂ i = ∂/∂x i , ∂ a = ∂/∂x a , when indices will be underlined if it is necessary to emphasize certain decompositions are defined for coordinate bases. The h-indices i, j, k, ... = 1, 2, ...n will be used for base/ nonholonomic vector objects and the v-indices a, b, c... = n + 1, n + 2, ...n + m will be used for fiber/ holonomic vector objects. Greek indices of type α, β, ... will be used as cumulative ones.
Finally, we note that we shall omit labels, indices and parametric/ coordinate dependencies for some formulas if it does not result in ambiguities.

Nonholonomic Lifts and Ricci Flows
In this section, we prove that for any family of (semi) Riemannian metrics g ij (χ) on a manifold V, parametrized by a real parameter χ, it is possible to define lifts to the tangent bundle T V provided with canonical nonlinear connection (in brief, N-connection), Sasaki type metrics and (linear) canonical distinguished connection (d-connection) structures. We also outline some important formulas for nonholonomic Ricci flow evolution equations of geometric structures. The reader is recommended to consult Refs. [24,26,21,22,28,29] and Appendix A on details on N-connection geometry and recent developments in modern gravity and Ricci flow theory.

N-connections induced by families of Riemannian metrics
Let E = (E, π, F, M) be a (smooth) vector bundle over base manifold M, dim M = n and dim E = (n + m), for n ≥ 2, and m ≥ n being the dimension of the typical fiber F. It is defined a surjective submersion π : E → M. In any point u ∈ E, the total space E splits into "horizontal", M u , and "vertical", F u , subspaces. We denote the local coordinates in the form u = (x, y), or u α = (x i , y a ) , with horizontal indices i, j, k, . . . = 1, 2, . . . , n and vertical indices a, b, c, . . . = n + 1, n + 2, . . . , n + m. 1 The summation rule on the same "up" and "low" indices will be applied. The base manifold M is provided with a family of (semi) Riemannian metrics, nondegenerate second rank tensors, 2 hg(χ) = g ij (x, χ)dx i ⊗ dx j , for 0 ≤ χ ≤ χ 0 ∈ R. We also introduce a family of vertical metrics vg(χ) = g ab (x, χ)dy a ⊗ dy b by completing the matrices g ij (x, χ) diagonally with ±1 till any nondegenerate second rank tensor g ab (x, χ) if m > n and then subjecting to any frame transforms. This way, we define certain families of metrics g(χ) = [hg(χ), vg(χ)] (we shall also use the notation g αβ (χ) = [g ij (χ), g ab (χ)]) on E. Considering frame (vielbein) transforms, where g αβ (x, χ) is written in equivalent form g αβ (x, χ), we can deform the metric structures, g αβ → g αβ = [g ij , g ab ] (we shall omit dependencies on coordinates and parameters if it does not result in ambiguities). The coefficients e α α (x, y, χ) will be defined below (see formula (6)) from the condition of generating curvature tensors with constant coefficients with respect to certain preferred systems of reference.
For any g ab (χ) from the set g αβ (χ), we can construct a family of effective generation functions L(x, y, χ) = g ab (x, y, χ)y a y b inducing families of vertical metrics g ab (x, y, χ) = 1 2 which is "weakly" regular if det |g ab | = 0. By straightforward computations, we prove 3 : 1 In a particular case, we have a tangent bundle E=T M, when n = m; for such bundles both type of indices run the same values but it is convenient to distinguish the horizontal and vertical ones by using different groups of small Latin indices. Here one should be noted that on T M we are able to contract the vertical indices with the corresponding horizontal ones, and inversely, but not on a general nonholonomic manifold V, or E.
2 in physical literature, one uses the term (pseudo) Riemannian/Euclidean space 3 see Refs. [21,22] for details of a similar proof; here we note that in our case, in general, e which are equivalent to a corresponding family of "nonlinear" geodesic equations (2).
The Theorem 2.1 states the possibility to geometrize the regular Lagrange mechanics by geometric objects on nonholonomic spaces and inversely: Proof. We sketch a proof by defining the coefficients of N-connection, with g ah (χ) andg ij (χ) defined respectively by formulas (1) and (2). The families of N-adapted partial derivative and differential operators, see Appendix for more general formulas (A.2) and (A.3), are defined by the N-connection coefficients (3) and may be denoted respectivelyẽ ν (χ) = (ẽ i (χ), e a ) andẽ µ (χ) = (e i ,ẽ a (χ)).
For any metric structure g on a manifold, there is the unique metric compatible and torsionless Levi Civita connection ∇ for which ∇ T α = 0 and ∇g = 0. This connection is not a d-connection because it does not preserve under parallelism the N-connection splitting (A.1). One has to consider less constrained cases, admitting nonzero torsion coefficients, when a d-connection is constructed canonically for a d-metric structure. A simple minimal metric compatible extension of ∇ is that of canonical d-connection D, with T i jk = 0 and T a bc = 0 but, in general, nonzero T a ji and T a bi , see (A.9). The coefficient formulas for such connections are given in Appendix, see (A.15) and related discussion. It should be noted that on tangent bundle T M it is possible to define the torsionless canonical d-connection (A.16) which is completely similar to the Levi Civita connection. For families of metrics g(χ), we get certain families of connections ∇(χ) and D(χ).
whereẽ i (χ) are elongated as in (A.3), but withÑ i j (χ) from (3). Then, we note that there are canonical d-connections on T M induced by g ij (x, χ) : we can construct them in explicit form by introducingg ij (χ) andg ab (χ) in formulas (A. 16), in order to compute the coefficientsΓ α βγ (χ) = (L i jk (χ),C a bc (χ)). The corresponding curvature curvature tensor can be computed by introducing respectively the valuesg ij (χ),Ñ i j (χ) and e k (χ) into (A.16), definingΓ α βγ (χ) = L i jk (χ),C a bc (χ) and then into formulas (A.21). Here one should be noted that the constructions on T M depend on arbitrary vielbein coefficients e α α (x, y, χ) in (1). We can restrict such sets of coefficients in order to generate various particular classes of (semi) Riemannian geometries on T M, for instance, in order to generate symmetric Riemannian spaces with constant curvature, see Refs. [36,37,38]. Proof. For a given set g ij (x, χ) on M, in (1), we chose such coefficients e α α (x, y, χ) = {e a a (x, y, χ)} that whereg ab (χ) are metrics of symmetric Riemannian space (of constant curvature). Considering a prescribed setg ab (χ), we have to integrate two times on y e in order to find any solution for e a a (χ) defining a frame structure in the vertical subspace. The next step is to construct the d-metric g αβ (χ) = [g ij (χ),g ab (χ)] of type (5), in our case, with respect to a nonholonomic base elongated by Ni j (χ), generated by g ij (x, χ) andg ef =g ab (χ), like in (3) and (4). This defines a constant curvature Riemannian space of dimension n+n. The coefficients of the canonical d-connection, which in this case coincide with those for the Levi Civita connection, and the coefficients of the Riemannian curvature can be computed respectively by introducing g ef =g ab (χ) in formulas (A.16) and (A.21). Finally, we note that the induced symmetric Riemannian spaces contain additional geometric structures like the N-connection and anholonomy coefficients W γ αβ (χ), see (A.5). There are various possibilities to generate on T M nonholonomic Riemannian structures from a given set g ij (x, χ) on M. They result in different geometrical and physical models.

Remark 2.1
We can simplify substantially the geometric constructions if instead of families of constant coefficientsg ef (χ), we consider only one set of constant coefficientsg ef =g ef (χ 0 ). This is possible even g ab (x, χ) have quite general dependencies on χ but supposing that we can define such e a a (x, y, χ) when (6) can be solved for a fixed right side. For simplicity, in our furhter considerations we shall fix any setg ef and parametrize the dependencies on χ for g ij and e a a .
In this work, we emphasize the possibility of generating spaces with constant curvature because for such symmetric spaces it was elaborated a bi-Hamiltonian approach and corresponding solitonic hierarchies [15,16,18,19]. For general Riemannian and/or Finsler-Lagrange spaces it is not possible to get constant coefficient curvature coefficients for the Levi Civita connection but for the corresponding lifts to the canonical d-connection there are constructions generating constant curvature coefficients [20,25]. This will allow us to construct the corresponding solitonic hierarchies from which, by imposing the corresponding constraints, it will be possible to extract the information for very general classes of metrics.
Example 2.1 The simplest example when a Riemannian structure with constant matrix curvature coefficients is generated on T M is given by a d-metric induced byg ij = δ ij , i.e.
withẽ i defined byÑ i j in their turn induced by a given set g ij (x) on M. For families of geometric objects, we consider whenÑ i j (χ) are defined by a given set g ij (x, χ).
For more general nonholonomic configurations on T M, we can consider families of metrics of type g(χ) = g ij (x, y, χ) e i ⊗ e j + g ab (x, y, χ) e a (χ) ⊗ e b (χ), where g ij (x, y, χ) = η ij (x, y, χ)g ij (x, y, χ) and g ab (x, y, χ) = η ab (x, y, χ)g ab (x, y, χ) and e a (χ) are elongated by N i j (x, y, χ) = η i j (x, y, χ)Ñ i j (x, y, χ). We note that in the formulas defining the coefficients of the metrics (8) one does not consider summation on repeating indices which are not "cross" one, i.e. η ij g ij means a simple product η ij ×g ij between deformation function η ij and metric coefficientg ij , for any fixed values i, j, ... or a, b, ... It is possible to write down the metrics (8) in "generic off-diagonal forms", see (A.11) and (A.12), for any fixed value of χ.

Nonholonomic Ricci flows and Einstein spaces
In the theory of Ricci flows, the families of metrics (8) must satisfy certain evolution equations on parameter χ. For normalized (holonomic) Ricci flows [3,6,8,9], with respect to a coordinate base ∂ α = ∂/∂u α , the evolution equations are postulated in the form where the normalizing factor r = RdV /dV is introduced in order to preserve the volume V and the metric coefficients g αβ are parametrized in the form (A.12), and R αβ is the Ricci tensor for the the Levi Civita connection ∇. In Refs. [28,29,33] we discuss in details the N-anholonomic Ricci flows and prove that the nonholonomic version of (9) can be proven by a N-adapted calculus from the Perelman's functionals, where λ = r/5. We note that the equations (12) constrain nonholonomic Ricci flows to result in symmetric metrics and that we wrote them with respect to N-adapted frames. A simple class of solutions can be constructed for the families of N-anholonomic Einstein spaces when and Such equations define some effective Einstein metrics subjected to Ricci flows under evolution of the N-anholonomic structure N a i correlated with the evolutions of h-metric g ij but g ab stated for a fixed value of χ 0 .
For our further considerations, we need the results of two Corollaries (see Refs. [6,28,29] for detailed proofs and discussions both for holonomic and nonholonomic manifolds): αβ (u), is given by equations For simplicity, we omit formulas for h-and v-decomposition of (15).

for the canonical d-connection on T M satisfies the evolution equations
Proof. It is similar to that for the Levi Civita connection because on T M the coefficients of canonical d-connection with respect to N-adapted frames are the same as those for the Levi Civita but decomposed into h-and vcomponents. We note that on T M the Ricci d-tensor R αβ is symmetric which does not hold true for a general nonholonomic manifold or vector bundle, see formulas (A. 19). The evolution equations (16) consist a particular case of more general formulas for Ricci flows on N-anholonomic manifolds proved in Theorem 4.1 of Ref. [28] (on T M the distorsion tensor transforming the ∇ into D is zero).
Finally, in this section, we note that a number of geometric ideas and methods applied in this section were considered in the approaches to the geometry of nonholonomic spaces and generalized Finsler-Lagrange geometry elaborated by the schools of G. Vranceanu and R. Miron and by A. Bejancu in Romania [39,40,21,22,23,34]. We emphasize that this way it is possible to construct geometric models with metric compatible linear connections which is important for elaborating standard approaches in modern (non)commutative gravity and string theory [26,24]. For Finsler spaces with nontrivial nonmetricity, for instance, for those defined by the the Berwald and Chern connections, see details in [41], the physical theories with local anisotropy are not imbedded into the class of standard models. It is also a more cumbersome task to elaborate a theory of Ricci flows of noncompatible metrics and connection structures.

N-Adapted Curve Flows in Vector Bundles
We formulate the geometry of curve flows adapted to the nonlinear connection structures constructed by certain classes of canonical lifts from the base space and nonholonomic frame deformations resulting into constant curvature coefficients for the canonical d-connection. The case of tangent bundles will be emphasized as a special one when both ∇ and D can be torsionless.

Non-stretching and N-adapted curve flows
Let us consider a vector bundle where τ is a parameter and l is the arclength of the curve on E, is defined with such evolution d-vector Y = γ τ and tangent d-vector X = γ l that The curve γ(τ, l) swept out a two-dimensional surface in T γ(τ,l) V ⊂ T V.
If the geometric objects evolve as Ricci flows on parameter χ, we get an additional parameter for the geometric objects like connections and metrics and the non-stretching condition (17) transforms into g(X, X,χ) = 1 for a family of d-metrics χ g g(χ) = [g(χ), h(χ)] and N a i (χ) which can be satisfied by certain families of curves, γ(τ, l,χ) χ γ(τ, l) (briefly, we shall write only χ γ = γ(χ)) and related curve evolution and tangent vectors, We work with families of N-adapted bases (A.2) and (A.3) and the con- where "⌋" denotes the interior product and the indices are lowered and raised respectively by the d-metric g αβ = [g ij , h ab ] and its inverse g αβ = [g ij , h ab ]. 4 We note that D X(χ) = χ X α χ D α is the covariant derivation operator along curve γ(τ, l,χ). It is convenient to orient the N-adapted frames to be parallel respectively to curves χ γ e 1 hX, for i = 1, and vX, for a = n + 1, and e b a , where vg(vX, e b a ) =0, for i = 2, 3, ...n and a = n + 2, n + 3, ..., n + m. For such frames, the covariant derivatives of each "normal" d-vectors χ e b α result into the d-vectors adapted to χ γ, which holds for certain classes of functions χ ρ The formulas (18) and (20) are distinguished into h-and v-components for Along any curve γ(χ), we can move differential forms in a parallel Nadapted form. For instance, Γ αβ X X⌋Γ αβ which for families of d-objects is to be written χ Γ αβ X χ X(χ)⌋ χ Γ αβ . The algebraic characterization of such spaces, can be obtained if we perform a frame transform preserving the decomposition (A.1) to an orthonormalized basis e α ′ , when called orthonormal d-basis (family of d-bases). In this case, the coefficients of the d-metric (A.10) transform into the Euclidean ones, g α ′ β ′ = δ α ′ β ′ , (we can define such frame transform (6) when the the same constant coefficients for d-metric are generated for all values of parameter χ; for such configurations, we do not emphasize the labels/dependencies on Ricci flow parameter which are present in d-connection operators and N-connection coefficients). In distinguished form, we obtain families of two skew matrices we omitted the Ricci flow label performing the constructions according Remark 2.1, and The above presented row-matrices and skew-matrices show that locally an total space of a vector bundle of dimension n + m, with respect to distinguished orthonormalized frames are characterized algebraically by couples of unit vectors in R n and R m preserved respectively by the SO(n−1) and SO(m − 1) rotation subgroups of the local N-adapted frame structure group SO(n) ⊕ SO(m). The connection matrices χ Γ belong to the orthogonal complements of the corresponding Lie subalgebras and algebras, so(n − 1) ⊂ so(n) and so(m − 1) ⊂ so(m).
The torsion (A.8) and curvature (A.17) tensors can be in orthonormalized component form with respect to (19) mapped into a distinguished orthotnomalized dual frame (21), and define respectively the N-adapted orthonormalized frame row-matrix and the canonical d-connection skew-matrix in the flow directs, and is the curvature matrix. Both torsion and curvature components can be distinguished in h-and v-components like (A.9) and (A.18), by considering N-adapted decompositions of type Finally, we note that the matrices for torsion (22) and curvature (23) can be computed for any families, parametrized by χ, metric compatible linear connection like the Levi Civita and the canonical d-connection. For our purposes, in this work, we are interested to define such a frame of reference with respect to which the curvature tensor has constant coefficients and the torsion tensor vanishes.

N-anholonomic bundles with constant matrix curvature
For vanishing N-connection torsion and constant matrix curvature of the canonical d-connection, we get couples of holonomic Riemannian manifolds and the equations (22) and (23) directly encode couples of bi-Hamiltonian structures, see details in Refs. [20,25,16,18,19]. A well known class of Riemannian manifolds for which the frame curvature matrix constant consists of the symmetric spaces M = G/H for compact semisimple Lie groups G ⊃ H. A complete classification and summary of main results on such spaces are given in Refs. [36,37,38]. We suppose that the base manifold is a symmetric space M = hG/SO(n) with the isotropy subgroup hH = SO(n) ⊃ O(n) and the typical fiber space to be a symmetric space F = vG/SO(m) with the isotropy subgroup vH = SO(m) ⊃ O(m). This means that hG = SO(n + 1) and vG = SO(m + 1) which is enough for a study of real holonomic and nonholonomic manifolds and geometric mechanics models. 5 Our aim is to solder in a canonic way (like in the N-connection geometry) the horizontal and vertical symmetric Riemannian spaces of dimension n and m with a (total) symmetric Riemannian space V of dimension n + m, when V = G/SO(n + m) with the isotropy group H = SO(n + m) ⊃ O(n + m) and G = SO(n + m + 1). We note that for the just mentioned horizontal, vertical and total symmetric Riemannian spaces one exists natural settings to Klein geometry. For instance, the metric tensor hg = {g ij } on a symmetric Riemannian space M is defined by the Cartan-Killing inner product < ·, · > h on T x hG ≃ hg restricted to the Lie algebra quotient spaces hp =hg/hh, with T x hH ≃ hh, where hg =hh ⊕ hp is stated such that there is an involutive automorphism of hG under hH is fixed, i.e. [hh,hp] ⊆ hp and [hp,hp] ⊆ hh. In a similar form, we can define the group spaces and related inner products and Lie algebras, We parametrize the metric structure with constant coefficients on V = 5 it is necessary to consider hG = SU (n) and vG = SU (m) for the geometric models with spinor and gauge fields where u α are local coordinates and when trivial, constant, N-connection coefficients are computedN e j =h ebg jb for any given setsh eb andg jb , i.e. from the inverse metrics coefficients defined respectively on hG = SO(n+1) and by off-blocks (n×n)-and (m×m)-terms of the metricg αβ . As a result, we define an equivalent d-metric structure of type (A.10)g e i = dx i ,e a = dy a +N e i dx i defining a trivial (n + m)-splittingg =g⊕Nh because all nonholonomy coef-ficientsW γ αβ and N-connection curvature coefficientsΩ a ij are zero. In more general form, we can consider any covariant coordinate transforms of (26) preserving the (n + m)-splitting resulting in any W γ αβ = 0 (A.5) and Ω a ij = 0 (A.4). It should be noted that even such trivial parametrizations define algebraic classifications of symmetric Riemannian spaces of dimension n + m with constant matrix curvature admitting splitting (by certain algebraic constraints) into symmetric Riemannian subspaces of dimension n and m, also both with constant matrix curvature and introducing the concept of Nanholonomic Riemannian space of typeV = [hG = SO(n+1), vG = SO(m+ 1),N e i ]. One can be considered that such trivially N-anholonomic group spaces have possess a Lie d-algebra symmetry soN (n + m) so(n) ⊕ so(m).
The simplest generalization on a vector bundleE is to consider nonhlonomic distributions on V = G/SO(n + m) defined locally by families of N-connection coefficients N a i (x, y, χ) with nonvanishing W γ αβ (χ) and Ω a ij (χ) but with constant d-metric coefficients when This family of metric is very similar to (7) but with the coefficientsg ij andh ab induced by the corresponding Lie d-algebra structure soN (n + m). Such spaces transform into families of N-anholonomic Riemann-Cartan man-ifoldsV N = [hG = SO(n + 1), vG = SO(m + 1), N e i ] with nontrivial Nconnection curvature and induced d-torsion coefficients of the canonical dconnection (see formulas (A.9) computed for constant d-metric coefficients and the canonical d-connection coefficients in (A.15)). One has zero curvature for the canonical d-connection (in general, such spaces are curved ones with generic off-diagonal metric (27) and nonzero curvature tensor for the Levi Civita connection). 6 This allows us to classify the N-anholonomic manifolds (and vector bundles) as having the same group and algebraic structures of couples of symmetric Riemannian spaces of dimension n and m but nonholonomically soldered to the symmetric Riemannian space of dimension n+m. With respect to N-adapted orthonormal bases (21), with distinguished h-and v-subspaces, we obtain the same inner products and group and Lie algebra spaces as in (24).
The classification of N-anholonomic vector bundles is almost similar to that for symmetric Riemannian spaces if we consider that n = m and try to model tangent bundles of such spaces, provided with N-connection structure. For instance, we can take a (semi) Riemannian structure with the N-connection induced by a absolute energy structure like in (3) and with the canonical d-connection structure (A.15), forg ef =g ab , like in (6). A straightforward computation of the canonical d-connection coefficients 7 and of d-curvatures for •g ij and •Ñ i j proves that the nonholonomic Riemanian manifold (M = SO(n + 1)/SO(n), • L) possess constant both zero canonical d-connection curvature and torsion but with induced nontrivial N-connection curvature •Ωi jk . Such spaces, being tangent to symmetric Riemannian spaces, are classified similarly to the Riemannian ones with constant matrix curvature, see (24) for n = m but provided with a nonholonomic structure induced by generating function • L. We can introduce Ricci flows on parameter χ when for certain systems of coordinates the metric coefficients are constant but satisfy the evolution equations (13) and (14).

Basic Equations for N-anholonomic Curve Flows
Introducing N-adapted orthonormalized bases, for N-anholonomic spaces of dimension n + n, with constant curvatures of the canonical d-connection, we can derive bi-Hamiltonian and vector soliton structures similarly to [19,18,16]. In symbolic, abstract index form, the constructions for nonholonomic vector bundles are similar to those for the Riemannian symmetric-spaces soldered to Klein geometry. We have to distinguish the horizontal and ver-tical components of geometric objects and related equations. The previous bi-Hamiltonian and solitonic constructions were for an extrinsic approach soldering the Riemannian symmetric-space geometry to the Klein geometry [38]. For the N-anhlonomic spaces of dimension n + n, with constant dcurvatures, similar constructions hold true but we have to adapt them to the N-connection structure, see Appendix B.
There is an isomorphism between the real space so(n) and the Lie algebra of n×n skew-symmetric matrices. This allows us to establish an isomorphism between hp ≃ R n and the tangent spaces T x M = so(n + 1)/ so(n) of the Riemannian manifold M = SO(n + 1)/ SO(n) as described by the following canonical decomposition and hp T mean the transposition of the row hp. The Cartan-Killing inner product on hg is stated following the rule where tr denotes the trace of the corresponding product of matrices. This product identifies canonically hp ≃ R n with its dual hp * ≃ R n . In a similar form, we can consider and define the Cartan-Killing inner product vp·vp 1 2 tr{...}. In general, in the tangent bundle of a N-anholonomic manifold, we can consider the Cartan-Killing N-adapted inner product p · p =hp·hp+vp·vp.
Following the introduced Cartan-Killing parametrizations, we analyze the flow γ(τ, l) of a non-stretching curve in V N = G/SO(n)⊕ SO(m). Let us introduce a family of coframes e ∈ T * γ V N ⊗ (hp⊕vp), which is a Nadapted (SO(n)⊕SO(m))-parallel basis along γ, and its associated canonical d-connection 1-form χ Γ ∈ T * γ V N(χ) ⊗ (so(n)⊕so(m)). Such d-objects are respectively parametrized: The above parametrizations are fixed in order to preserve the SO(n) and SO(m) rotation gauge freedoms on the N-adapted coframe and canonical d-connection 1-form, distinguished in h-and v-components.
There are defined decompositions of horizontal SO(n + 1)/ SO(n) matrices like , into tangential and normal parts relative to e hX via corresponding decompositions of h-vectors hp = (hp ,h − → p ⊥ ) ∈R n relative to 1, − → 0 , when hp is identified with hp C and h − → p ⊥ is identified with hp ⊥ = hp C ⊥ . In a similar form, it is possible to decompose vertical SO(m + 1)/ SO(m) matrices, into tangential and normal parts relative to e vX via corresponding decompo- The canonical d-connection induces matrices decomposed with respect to the flow direction. In the h-direction, we parametrize , when e hY ∈ hp, he , h − → e ⊥ ∈ R n and h − → e ⊥ ∈ R n−1 , and where In the v-direction, we parametrize , The components he (χ) and h − → e ⊥ (χ) correspond to the decomposition χ e hY = hg( χ γ τ , χ γ l , χ) χ e hX + ( χ γ τ ) ⊥ ⌋he ⊥ (χ) into tangential and normal parts relative to χ e hX . In a similar form, one considers ve (χ)(χ) and v ← − e ⊥ (χ) corresponding to the decomposition χ e vY = vg(γ τ , γ l , χ) χ e vX + ( χ γ τ ) ⊥ ⌋ve ⊥ (χ).
Using the above stated matrix parametrizations, we get We can use formulas (29) and (30) in order to write the structure equations (22) and (23) in terms of N-adapted curve flow operators soldered to the geometry Klein N-anholonomic spaces using the relations (B.1). One obtains respectively the G-invariant N-adapted torsion and curvature generated by the canonical d-connection, (31) and where e X γ l ⌋e, e Y γ τ ⌋e, Γ X γ l ⌋Γ and Γ Y γ τ ⌋Γ. The formulas (31) and (32) are equivalent, respectively, to (A.9) and (A.18). In general, T(γ τ , γ l ) = 0 and R(γ τ , γ l )e can not be defined to have constant matrix coefficients with respect to a N-adapted basis. For N-anholonomic spaces with dimensions n = m, we have E T(γ τ , γ l ) = 0 and E R(γ τ , γ l )e defined by constant, or vanishing, d-curvature coefficients (see discussions related to formulas (A.21) and (A.16)). For such cases, we can consider the hand v-components of (31) and (32) in a similar manner as for symmetric Riemannian spaces but with the canonical d-connection instead of the Levi Civita one. One obtains, respectively, and Following the N-adapted curve flow parametrizations (29) and (30), the equations (33) and (34) are written The tensor and interior products, for instance, for the h-components, are defined in the form: ⊗ denotes the outer product of pairs of vectors (1 × n row matrices), producing n×n matrices , and ⌋ denotes multiplication of n × n matrices on vectors (1 × n row matrices); one holds the properties which is the transpose of the standard matrix product on column vectors, and Here we note that similar formulas hold for the v-components but, for instance, we have to change, correspondingly, n → m and − → A → ← − A . The variables e and Θ, written in h-and v-components, can be expressed corresponding in terms of variables − → v , − → ̟ , h − → e ⊥ and ← − v , ← − ̟ , v ← − e ⊥ (see respectively the first two equations in (35) and the last two equations in (36) Substituting these values, correspondingly, in the last two equations in (35) and in the first two equations in (36), we express contained in the h-and v-flow equations respectively on − → v and ← − v , considered as scalar components when D hY where the scalar curvatures of the canonical d-connection, − → R and ← − S are defined by formulas (A.20). For symmetric Riemannian spaces like SO(n + 1)/SO(n) ≃ S n , the value − → R is just the scalar curvature χ = 1, see [19]. On tangent bundles, it is possible that − → R and ← − S are certain zero or nonzero constants with the h-part equivalent to the base scalar curvature.
For Ricci flows of geometric objects, the curve flow evolution equations (37) contain additional dependencies on parameter χ, where the scalar curvatures evolve following formulas (16) for Ricci flows. The above presented considerations consist the proof of

Lemma 4.1 There are canonical lifts of Ricci flows from a (semi) Riemannian manfiold M to T M when certain families of constant constant curvature matrix coefficients for the canonical d-connections define families of N-adapted Hamiltonian sympletic operators,
and cosympletic operators For any fixed value χ = χ 0 , the formulas for this Lemma transform into similar ones from Ref. [20]. The properties of operators (38) and (39)

-operators defines the Hamiltonian form for the curve and Ricci flows equations on N-anholonomic tangent bundles with constant d-connection curvature: the h-flows are given by
the v-flows are given by where

the so-called Ricci flows of heriditary recursion d-operator has the respective h-and v-components
and D i D i − → R = D a D a ← − S = 0 and R ij R ij = const, S ab S ab = const for lifts to constant curvature matrices.
Proof. One follows from the Lemma 4.1, Corollaries 2.2 and 2.3 and (37). In a detailed form, for holonomic structures, it is given in Ref. [16] and discussed in [19].
Finally, we note that for any fixed value χ 0 we get a Theorem from [20], on curve flows in symmetric Riemannian spaces, which has generalizations for curve flows on generalized Lagrange and Finsler spaces [25].

Curve Flows and Solitonic Hierarchies for Ricci Flows
The final aim of this paper is to prove that for any nonholonomic Ricci flow system we can define naturally a family of N-adapted bi-Hamiltonian flow hierarchies inducing anholonomic solitonic configurations.

Formulation of the main theorem
Following a usual solitonic techniques generalized for N-anholonomic spaces, see details in Ref. [20,25,18,19], the recursion h-operators from (42), There are related hierarchies, generated by adjoint operators R * = (hR * , vR * ), of involuntive Hamiltonian h-covector fields − → ̟ The relations between different type families of hierarchies are established correspondingly by formulas where k = 0, 1, 2, .... All hierarchies (horizontal, vertical and their adjoint ones) have a typical mKdV scaling symmetry, for instance, l → λl and ⊥ (χ) and hH (k) (χ) have scaling weight 2+2k, while − → ̟ (k) (χ) has scaling weight 1+2k. Following the above presented considerations, we prove with families of horizontal Hamiltonians and v-evolution equations and with families of vertical Hamiltonians

. The Ricci flows of d-operators H(χ) and J (χ) are Nadapted and mutually compatible from which we can construct a family of alternative (explicit) Hamilton d-operators
The Main Result of this work is formulated in the form: There are families of +1 flows defined as Ricci flows of non-stretching mKdV maps Proof is outlined in Appendix C.

Nonholonomic mKdV and SG hierarchies
Let us consider some explicit constructions when families of solitonic hierarchies are derived following the conditions of Theorem 5.1.
The h-flow and v-flow equations resulting from (47) are when, respectively, The d-flow equations possess horizontal and vertical conservation laws This corresponds to In general, such laws are more sophisticate than those on (semi) Riemannian spaces because of nonholonomic constraints resulting in non-symmetric Ricci tensors and different types of identities. But for the geometries modelled for dimensions n = m with canonical d-connections, we get similar h-and vcomponents of the conservation law equations as on symmetric Riemannian spaces. 8 It is possible to rescale conformally the variable τ and obtain | (hγ) τ (χ)| 2 hg = 1 and (it can be for other rescale) | (vγ) τ (χ)| 2 vg = 1, i.e.
We can express he (χ) and h − → e ⊥ (χ) in terms of χ− → v and its derivatives and, similarly, we can express ve (χ) and v ← − e ⊥ (χ) in terms of χ← − v and its derivatives, which follows from (48). The N-adapted wave map equations describing the -1 flows reduce to a system of two independent nonlocal evolution equations for the h-and v-components, parametrized by χ, For N-anholonomic spaces of constant scalar d-curvatures, we can rescale the equations on τ to the case when the terms − → R 2 (χ) and ← − S 2 (χ) are constant, and the evolution equations transform into a system of hyperbolic d-vector equations, where χ D hX = ∂ hl and χ D vX = ∂ vl are usual partial derivatives on direction l =hl+vl with χ− → v τ and χ← − v τ considered as scalar functions for the covariant derivatives χ D hX and χ D vX defined by the canonical d-connection. It also follows that h − → e ⊥ (χ) and v ← − e ⊥ (χ) obey corresponding vector sine-Gordon (SG) equations and The above presented formulas and Corollary 5.1 imply with all such terms commuting with the -1 flow associated to the vector SG equation (50); the second one is vertical, with all such terms commuting with the -1 flow

associated to the vector SG equation (51).
In its turn, using the above Conclusion, we derive that the family of adjoint d-operators R * (χ) = χ J • H(χ) generates corresponding families of horizontal hierarchies of Hamiltonians (for simplicity, here we omit labels/dependences on χ), and of vertical hierarchies of Hamiltonians all of which are conserved densities for respective horizontal and vertical -1 flows and determining higher conservation laws for the corresponding hyperbolic equations (50) can be extracted from Ricci flows of solitonic hierarchies (such spaces with nonhomogeneous effective constants χ λ(u) were examined in details in Refs. [24,26], for exact solutions in gravity, and [29,30,31,32], for exact solutions and applications in physics of the nonholonomic Ricci flow theory).
and evolution of N-adapted frames stated by formulas Proof. The equations (60) folow from equations (40) and (41), see also (16), when hold true (59). The equation (61) is a consequence of (15). Similar proofs can be provided for different signatures of metrics.
The vacuum equations in Einstein gravity can be parametrized as solitonic hierarchies with constant on χ N-anholonomic frames, see (61). The corresponding SG hierarchies are defined as solutions of the equations (49) and (50) and (51), with constant scalar curvature and frame coefficients.

Conclusion
In summary, we have considered a geometric formalism of encoding general (semi) Riemannian metrics and their lifts to tangent bundles into families of nonholonomic hierarchies of bi-Hamiltonian structures and related solitonic equations derived for curve flows on tangent spaces. Towards this ends, we have applied a programme of study that is based on prior works on nonholonomic Ricci [28,29] and curve flows [20,25]. The premise of this methodology is that one can derive solitonic hierarchies for non-stretching curve flows on constant curvature Riemannian spaces [14,15,16,19]. The validity of this approach was substantiated by the encoding into solitonic hierarchies of arbitrary (semi) Riemannian and Finsler-Lagrange metrics [20,25].
Our analysis was completed by explicit constructions related to solitonic encoding of Ricci flow equations [1,2,3,4,5]. First of all, we note that it was not possible to perform such constructions working only with the Levi Civita connection because, in general, the curvature tensor for this connection can not be parametrized by constant coefficients. The idea was to re-define equivalently the geometric objects and basic Ricci flow and field equations for other classes of linear connections generated on vector/ tangent bundles and/or (semi) Riemannian manifolds enabled with certain classes of preferred frames with associated nonlinear connection (N-connection) structure. In particular, we elaborated such lifts of (semi) Riemannian metrics to the tangent bundle when in a canonical form there are defined metric structures, a class of N-connections and distinguished (d) connections when with respect to certain classes of N-adapted frames it is possible to get constant curvature coefficients and zero torsion. Such geometric models and their lifts to different classes of fibred spaces are related to effective Lagrangians and, inversely, any regular geometric mechanics can be encoded into geometric objects on nonholonomic Riemannian manifolds/ tangent bundles.
Secondly, we considered Ricci flows of geometric objects and fundamental evolution/field equations. We found that a nonholonomically constrained flow of (semi) Riemannian metrics can result in generalized Finsler-Lagrange configurations which motivates both application of such Finsler geometry methods for usual Riemannian spaces if moving frames are introduced into consideration and a deep study of nonholonomic structures with associated N-connections soldering couples of Klein spaces and/or constant curvature Riemannian spaces.
Third, we argued that the geometry Riemann and Finsler-Lagrange spaces can be encoded into bi-Hamilton structures and nonholonomic solitonic equations and anticipated that such curve flow -solitonic hierarchies can be constructed in a similar manner for exact solutions of Einstein-Yang-Mills-Dirac equations, derived following the anholonomic frame method, in noncommutative generalizations of gravity and geometry and possible quantum models based on nonholonomic Lagrange-Fedosov manifolds. In spite of the fact that there are a number of conceptual and technical difficulties (such as the physical meaning of the general N-connections, additionally to the preferred frame systems and nonlinear generalizations of the usual linear connections, explicit relations of the Ricci flows to renormalization group flows in quantum gravity models, cumbersome geometric analysis calculus...) the outcome of such approach is almost obvious that we can encode nonlinear fundamental field/evolution equations in terms of corresponding vector solitonic equations, their hierarchies and conservation laws.
Finally, we tried not only to encode some geometric information about metrics, connections and frames into solitons but also formulated certain criteria when from such solitonic equations we can extract vacuum gravitational spaces or certain more general solutions for the Ricci flow equations and related Einstein spaces or Euler-Lagrange equations in geometric mechanics. For more general classes of solutions and extensions to quantum gravity, noncommutative geometry, these are desirable purposes for further investigations.

A The Geometry of N-anholonomic Vector Bundles
We denote by π ⊤ : T E → T M the differential of map π : E → M defined by fiber preserving morphisms of the tangent bundles T E and T M. The kernel of π ⊤ is just the vertical subspace vE with a related inclusion mapping i : vE → T E.

Definition A.1 A nonlinear connection (N-connection) N on a vector bundle E is defined by the splitting on the left of an exact sequence
i. e. by a morphism of submanifolds N : T E → vE such that N•i is the unity in vE.
In an equivalent form, we can say that a N-connection is defined by a Whitney sum of conventional horizontal (h) subspace, (hE) , and vertical (v) subspace, (vE) , This sum defines a nonholonomic (equivalently, anholonomic, or nonitegrable) distribution of horizontal and vertical subspaces on T E. Locally, a N-connection is defined by its coefficients N a i (u), The well known class of linear connections consists on a particular subclass with the coefficients being linear on y a , i.e., N a i (u) = Γ a bj (x)y b . 9 Remark A.1 A bundle space, or a a manifold, is called nonholonomic if it provided with a nonholonomic distribution (see historical details and summary of results in [34]). In particular case, when the nonholonomic distribution is of type (A.1), such spaces are called N-anholonomic [24]. For any N-connection, we can introduce its N-connection curvature with the coefficients defined as the Neijenheuse tensor, The with (antisymmetric) nontrivial anholonomy coefficients W b ia = ∂ a N b i and W a ji = Ω a ij . The geometric objects can be defined in a form adapted to a N-connection structure, following decompositions being invariant under parallel transports preserving the splitting (A.1). In this case we call them to be distinguished (by the connection structure), i.e. d-objects. For instance, a vector field X ∈ T V is expressed X = (hX, vX), or X = X α e α = X i e i + X a e a , where hX = X i e i and vX = X a e a state, respectively, the adapted to the N-connection structure horizontal (h) and vertical (v) components of the vector (which following Refs. [21,22] is called a distinguished vector, in brief, d-vector). In a similar fashion, the geometric objects on V, for instance, tensors, spinors, connections, ... are called respectively d-tensors, d-spinors, d-connections if they are adapted to the N-connection splitting (A.1).

Definition A.2 A distinguished connection (in brief, d-connection) D = (hD, vD) is a linear connection preserving under parallel transports the nonholonomic decomposition (A.1).
The N-adapted components Γ α βγ of a d-connection D α = (e α ⌋D) are defined by the equations The N-adapted splitting into h-and v-covariant derivatives is stated by where, by definition, L i jk = (D k e j )⌋e i , L a bk = (D k e b )⌋e a , C i jc = (D c e j )⌋e i , C a bc = (D c e b )⌋e a . The components Γ γ αβ = L i jk , L a bk , C i jc , C a bc completely define a d-connection D on E.
The simplest way to perform N-adapted computations is to use differential forms. For instance, starting with the d-connection 1-form, with the coefficients defined with respect to N-elongated frames (A.3) and (A.2), the torsion of a d-connection, is characterized by (N-adapted) d-torsion components, For d-connection structures on T M, we have to identify indices in the form i ⇆ a, j ⇆ b, ... and the components of N-and d-connections, for instance, N a i ⇆ N j i and L i jk ⇆ L a bk , C i ja ⇆ C b ca ⇆ C i jk .

Definition A.3 A distinguished metric (in brief, d-metric) on a vector bun-
dle E is a usual second rank metric tensor g =g⊕ N h, equivalently, adapted to the N-connection decomposition (A.1).
With respect to a coordinate basis, the metric g (A.10) can be written in the form i.e. D j g kl = 0 and D a g kl = 0. There are two types of preferred linear connections uniquely determined by a generic off-diagonal metric structure with n+m splitting, see g = g⊕ N h: For E = T M, the canonical d-connectionD = (hD, vD) can be defined in torsionless form 10 with the coefficients Γ α βγ = (L i jk , L a bc ), The curvature of a d-connection D, bc − e c C a bd + C e bc C a ed − C e bd C a ec . Contracting respectively the components, R αβ R τ αβτ , one computes the h-v-components of the Ricci d-tensor (there are four N-adapted components) The scalar curvature is defined by contracting the Ricci d-tensor with the inverse metric g αβ , If E =T M, there are only three classes of d-curvatures, bc − e c C a bd + C e bc C a ed − C e bd C a ec , where all indices a, b, ..., i, j, ... run the same values and, for instance, C e bc → C i jk , ...
With respect to N-adapted orthonormalized frames, the geometry of Nanholonomic manifolds is defined algebraically, on their tangent bundles, by couples of horizontal and vertical Klein geometries considered in [38] and for bi-Hamiltonian soliton constructions in [18]. The N-connection structure induces a N-anholonomic Klein space stated by two left-invariant hgand vg-valued Maurer-Cartan form on the Lie d-group G = (hG, vG) is identified with the zero-curvature canonical d-connection 1- For trivial N-connection structure in vector bundles with the base and typical fiber spaces being symmetric Riemannian spaces, we can consider that hG L i ′ j ′ k ′ and vG C i ′ j ′ k ′ are the coefficients of the Cartan connections hG L and vG C, respectively for the hG and vG, both with vanishing curvatures, i.e. with and h-and v-components, d hG L + 1 2 [ hG L, hG L] = 0 and d vG C + 1 2 [ vG C, vG C] = 0, where d denotes the total derivatives on the d-group manifold G = hG ⊕ vG or their restrictions on hG or vG. We can consider that G Γ defines the so-called Cartan d-connection for nonintegrable N-connection structures, see details and supersymmetric/ noncommutative developments in [26,24].
We generate a G( = hG ⊕ vG)-invariant d-derivatives D whose restriction to the tangent space T V N for any N-anholonomic curve flow γ(τ, l,χ) in V N = G/SO(n)⊕ SO(m) is defined via χ D X χ e = [ χ e, γ l ⌋ χ Γ] and χ D Y χ e = [ χ e, γ τ ⌋ χ Γ] , (B.1) admitting further h-and v-decompositions. The derivatives χ D X and χ D Y in (B.1) are equivalent to those considered in (18) and obey the Cartan structure equations (22) and (23). For the canonical d-connections, a large class of N-anholonomic spaces of dimension n = m, the d-torsions are zero and the d-curvatures are with constant coefficients. Let χ e α ′ = (e i ′ , χ e a ′ ) be a family of N-adapted orthonormalized coframes identified with the (hp ⊕ vp)-valued coframe e in a fixed orthonormal basis for p =hp⊕vp ⊂hg⊕vg. Considering the kernel/ cokernel of Lie algebra multiplications in the h-and v-subspaces, respectively, [e hX , ·] hg and [e vX , ·] vg , we can decompose the coframes into parallel and perpendicular parts with respect to e X . We write χ e = (e C = he C + ve C , e C ⊥ = he C ⊥ + ve C ⊥ ), for p( = hp ⊕ vp)-valued mutually orthogonal d-vectors e C and e C ⊥ , when there are satisfied the conditions [e X , e C ] g = 0 but [e X , e C ⊥ ] g = 0; such conditions can be stated in h-and v-component form, respectively, [he X , he C ] hg = 0, [he X , he C ⊥ ] hg = 0 and [ve X , ve C ] vg = 0, [ve X , ve C ⊥ ] vg = 0. One holds also the algebraic decompositions with p ⊆ p C and p C ⊥ ⊆ p ⊥ , where p , p C = 0, < p C ⊥ , p C >= 0, but p , p C ⊥ = 0 (i.e. p C is the centralizer of e X in p =hp ⊕ vp ⊂hg ⊕ vg); in hand v-components, one have hp ⊆ hp C and hp C ⊥ ⊆ hp ⊥ , where hp , hp C = 0, < hp C ⊥ , hp C >= 0, but hp , hp C ⊥ = 0 (i.e. hp C is the centralizer of e hX in hp ⊂hg) and vp ⊆ vp C and vp C ⊥ ⊆ vp ⊥ , where vp , vp C = 0, < vp C ⊥ , vp C >= 0, but vp , vp C ⊥ = 0 (i.e. vp C is the centralizer of e vX in vp ⊂vg). Using the canonical d-connection derivative D X of a d-covector perpendicular (or parallel) to e X , we get a new d-vector which is parallel (or perpendicular) to e X , i.e. D X e C ∈ p C ⊥ (or D X e C ⊥ ∈ p C ); in h-and vcomponents such formulas are written D hX he C ∈ hp C ⊥ (or D hX he C ⊥ ∈ hp C ) and D vX ve C ∈ vp C ⊥ (or D vX ve C ⊥ ∈ vp C ). All such d-algebraic relations can be written in N-anholonomic manifolds and canonical d-connection settings, for instance, using certain relations of type for some antisymmetric d-tensors χ v α ′ β ′ = − χ v β ′ α ′ . We get a N-adapted (SO(n) ⊕ SO(m))-parallel frame defining a generalization of the concept of Riemannian parallel frame on N-adapted manifolds whenever p C is larger than p . Substituting χ e α ′ = (e i ′ , χ e a ′ ) into the last formulas and considering h-and v-components, we define SO(n)-parallel and SO(m)-parallel frames (for simplicity we omit these formulas when the Greek small letter indices are split into Latin small letter h-and v-indices). The final conclusion of this section is that the Cartan structure equations on hypersurfaces swept out by nonholonomic curve flows on N-anholonomic spaces with constant matrix curvature for the canonical d-connection geometrically encode two O(n − 1)-and O(m − 1)-invariant, respectively, horizontal and vertical bi-Hamiltonian operators. This holds true if the distinguished by N-connection freedom of the d-group action SO(n) ⊕ SO(m) on χ e and χ Γ is used to fix them to be a N-adapted parallel coframe and its associated canonical d-connection 1-form is related to the canonical covariant derivative on N-anholonomic manifolds.

C Proof of the Main Theorem
We provide a proof of Theorem 5.1 for the horizontal Ricci and curve flows (similar results were published in [25] and [18], respectively, for Lagrange-Finsler and symmetric Riemannian spaces). The vertical constructions are similar but with respective changing of h-variables / objects into v-variables/ objects.
Finally, we note that the formulas for the v-components, stated by Theorem 5.1 can be derived in a similar form by respective substitution in the the above proof of the h-operators and h-variables into v-ones, for instance, − → R → ← − S , hD →vD, hR →vR,hJ →vJ ,...where, for simplicity, we omit parametric dependencies on χ.