Nonholonomic Ricci Flows of Riemannian Metrics and Lagrange-Finsler Geometry

A series of the most remarkable results in mathematics are related to Grisha Perelman’s proof of the Poincare Conjecture [1-3] built on geometrization (Thurston) conjecture [4,5] for three dimensional Riemannian manifolds, and R. Hamilton’s Ricci flow theory [6,7] see reviews and basic references explained by Kleiner [8-11]. Much of the works on Ricci flows has been performed and validated by experts in the area of geometrical analysis and Riemannian geometry. Recently, a number of applications in physics of the Ricci flow theory were proposed, by Vacaru [12-16].Some geometrical approaches in modern gravity and string theory are connected to the method of moving frames and distributions of geometric objects on (semi) Riemannian manifolds and their generalizations to spaces provided with nontrivial torsion, nonmetricity and/or nonlinear connection structures [17,18]. The geometry of nonholonomic manifolds and non–Riemannian spaces is largely applied in modern mechanics, gravity, cosmology and classical/quantum field theory expained by Stavrinos [19-35]. Such spaces are characterized by three fundamental geometric objects: nonlinear connection (N–connection), linear connection and metric. There is an important geometrical problem to prove the existence of the ” best possible” metric and linear connection adapted to a N– connection structure. From the point of view of Riemannian geometry, the Thurston conjecture only asserts the existence of a best possible metric on an arbitrary closed three dimensional (3D) manifold. It is a very difficult task to define Ricci flows of mutually compatible fundamental geometric structures on non–Riemannian manifolds (for instance, on a Finsler manifold). For such purposes, we can also apply the Hamilton’s approach but correspondingly generalized in order to describe nonholonomic (constrained) configurations. The first attempts to construct exact solutions of the Ricci flow equations on nonholonomic Einstein and Riemann–Cartan (with nontrivial torsion) manifolds, generalizing well known classes of exact solutions in Einstein and string gravity, were performed and explanied by Vacaru [13-16].


Introduction
A series of the most remarkable results in mathematics are related to Grisha Perelman's proof of the Poincare Conjecture [1][2][3] built on geometrization (Thurston) conjecture [4,5] for three dimensional Riemannian manifolds, and R. Hamilton's Ricci flow theory [6,7] see reviews and basic references explained by Kleiner [8][9][10][11]. Much of the works on Ricci flows has been performed and validated by experts in the area of geometrical analysis and Riemannian geometry. Recently, a number of applications in physics of the Ricci flow theory were proposed, by Vacaru [12][13][14][15][16].Some geometrical approaches in modern gravity and string theory are connected to the method of moving frames and distributions of geometric objects on (semi) Riemannian manifolds and their generalizations to spaces provided with nontrivial torsion, nonmetricity and/or nonlinear connection structures [17,18]. The geometry of nonholonomic manifolds and non-Riemannian spaces is largely applied in modern mechanics, gravity, cosmology and classical/quantum field theory expained by Stavrinos [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Such spaces are characterized by three fundamental geometric objects: nonlinear connection (N-connection), linear connection and metric. There is an important geometrical problem to prove the existence of the " best possible" metric and linear connection adapted to a Nconnection structure. From the point of view of Riemannian geometry, the Thurston conjecture only asserts the existence of a best possible metric on an arbitrary closed three dimensional (3D) manifold. It is a very difficult task to define Ricci flows of mutually compatible fundamental geometric structures on non-Riemannian manifolds (for instance, on a Finsler manifold). For such purposes, we can also apply the Hamilton's approach but correspondingly generalized in order to describe nonholonomic (constrained) configurations. The first attempts to construct exact solutions of the Ricci flow equations on nonholonomic Einstein and Riemann-Cartan (with nontrivial torsion) manifolds, generalizing well known classes of exact solutions in Einstein and string gravity, were performed and explanied by Vacaru [13][14][15][16].
We take a unified point of view towards Riemannian and generalized Finsler-Lagrange spaces following the geometry of nonholonomic manifolds and exploit the similarities and emphasize differences between locally isotropic and anisotropic Ricci flows. In our works, it will be shown when the remarkable Perelman-Hamilton results hold true for more general non-Riemannian configurations. It should be noted that this is not only a straightforward technical extension of the Ricci flow theory to certain manifolds with additional geometric structures. The problem of constructing the Finsler-Ricci flow theory contains a number of new conceptual and fundamental issues on compatibility of geometrical and physical objects and their optimal configurations.There are at least three important arguments supporting the investigation of nonholonomic Ricci flows: 1) The Ricci flows of a Riemannian metric may result in a Finsler-like metric if the flows are subjected to certain nonintegrable constraints and modelled with respect to nonholonomic frames (we shall prove it in this work). 2) Generalized Finsler-like metrics appear naturally as exact solutions in Einstein, string, gauge and noncommutative gravity, parametrized by generic off-diagonal metrics, nonholonomic frames and generalized connections and methods explained by Vacaru S [33][34][35]. It is an important physical task to analyze Ricci flows of such solutions as well of other physically important solutions (for instance, black holes, solitonic and/pp-waves solutions, Taub NUT configurations [13][14][15] resulting in nonholonomic geometric configurations. 3) Finally, the fact that a 3D manifold establishes an appropriate Riemannian metric, which implies certain fundamental consequences (for instance) for our spacetime topology, allows us to consider other types of "also not bad" metrics with possible local anisotropy and nonholonomic gravitational interactions. What are the natural evolution equations for such configurations and how can we relate them to the topology of nonholonomic manifolds? We shall address such questions here (for regular Lagrange systems) and in further works. The notion of nonholonomic manifold was introduced independently by G. Vranceanu [36] and Horak [37] as there was a need for geometric interpretation of nonholonomic mechanical systems modern approaches, criticism and historical remarks explained by Vacaru [34,38,39]. A pair ( , ) M  , where M is a manifold and  is a nonintegrable distribution on M, is called a nonholonomic manifold. Three well known classes of nonholonomic manifolds, where the nonholonomic distribution defines a nonlinear connection (N-connection) structure, are defined by the Finsler spaces [40][41][42] and their generalizations as Lagrange and Hamilton spaces [34,43] (usually such geometries are modelled on the tangent bundle TM) More recent examples, related to exact off-diagonal solutions and nonholonomic frames in Einstein/string/gauge/ noncommutative gravity and nonholonomic Fedosov manifolds [33,34,44] also emphasize nonholonomic geometric structures.Let us now sketch the Ricci flow program for nonholonomic manifolds and Lagrange-Finsler geometries. Different models of "locally anisotropic" spaces can be elaborated for different types of fundamental geometric structures (metric, nonlinear and linear connections). In general, such spaces contain nontrivial torsion and nonmetricity fields. It would be a very difficult technical task to generalize and elaborate new proofs for all types of non-Riemannian geometries. Our strategy will be different: We shall formulate the criteria to determine when certain types of Finsler like geometries can be "extracted" (by imposing the corresponding nonholonomic constraints) from "well defined" Ricci flows of Riemannian metrics. This is possible because such geometries can be equivalently described in terms of the Levi Civita connections or by metric configurations with nontrivial torsion induced by nonholonomic frames. By nonholonomic transforms of geometric structures, we shall be able to generate certain classes of nonmetric geometries and/or generalized torsion configurations.The aim of this paper (the first one in a series of works) is to formulate the Ricci flow equations on nonholonomic manifolds and prove the conditions under which such configurations (of Finsler-Lagrange type and in modern gravity) can be extracted from well defined flows of Riemannian metrics and evolution of preferred frame structures. Further works will be dedicated to explicit generalizations of Perelman results [1][2][3] for nonholonomic manifolds and spaces provided with almost complex structure generated by nonlinear connections. We shall also construct new classes of exact solutions of nonholonomic Ricci flow equations, with noncommutative and/or Lie algebroid symmetry, defining locally anisotropic flows of black hole, wormhole and cosmological configurations and developing the results from work of Vacaru [13][14][15][33][34][35]. The paper is organized as follows: We start with preliminaries on geometry of nonholonomic manifolds provided with nonlinear connection (N-connection) structure in Section 2. We show how nonholonomic configurations can be naturally defined in modern gravity and the geometry of Riemann-Finsler and Lagrange spaces in Section 3. Section 4 is devoted to the theory of anholonomic Ricci flows: we analyze the evolution of distinguished geometric objects and speculate on nonholonomic Ricci flows of symmetric and nonsymmetric metrics. In Section 5, we prove that the Finsler-Ricci flows can be extracted from usual Ricci flows by imposing certain classes of nonholonomic constraints and deformations of connections. We also study regular Lagrange systems and consider generalized Lagrange-Ricci flows. The Appendix outlines some necessary results from the local geometry of N-anholonomic manifolds.

Notation remarks
We shall use both the free coordinate and local coordinate formulas which are both convenient to introduce compact denotations and sketch some proofs. The left up/lower indices will be considered as labels of geometrical objects, for instance, on a nonholonomic Riemannian of Finsler space. The boldfaced letters will be used to denote that the objects (spaces) are adapted (provided) to (with) nonlinear connection structure.

Preliminaries: Nonholonomic Manifolds
We recall some basic facts in the geometry of nonholonomic manifolds provided with nonlinear connection (N-connection) structure. The reader can refer to the concepts explained by Etayo [33,34,38,44] for details and proofs (for some important results we shall sketch the key points for such proofs). On nonholonomic vectors and (co-) tangent bundles and related Riemannian-Finsler and Lagrange-Hamilton geometries [34, 41,42].

N-connections
Consider a (n+m)-dimensional manifold V, with (for a number of physical applications, it is equivalently called to be a physical and/or geometric space). In a particular case, M n is a vector bundle on M, with total space E. In a general case, we can consider a manifold V provided with a local fibred structure into conventional "horizontal" and "vertical" directions. The local coordinates on V are denoted in the form = ( , ), defined by fiber preserving morphisms of the tangent bundles TV and TM. The kernel of π Τ is only the vertical subspace vV with a related inclusion mapping :

Definition 2.1: A nonlinear connection (N-connection) N on a manifold V is defined by the splitting on the left of an exact sequence
i. e. by a morphism of submanifolds Locally, a N-connection is defined by its coefficients Globalizing the local splitting, one proves: The sum (2) states on TV a nonholonomic (equivalently, anholonomic, or nonintegrable) distribution of horizontal and vertical subspaces. The well known class of linear connections consists of a particular subclass with the coefficients being linear on , a y i.e.
The geometric objects on V can be defined in a form adapted to a N-connection structure, following certain decompositions being invariant under parallel transports preserving the splitting (2). In this case, we call them to be distinguished (by the N-connection structure), i.e. d-objects. For instance, a vector field In local form, we have for (3) Any N-connection N may be characterized by an associated frame (vierbein) structure = ( , ), These vielbeins are called respectively N-adapted frames and coframes. In order to preserve a relation with the previous denotations [33,34] we emphasize that For simplicity, we shall work with a particular class of nonholonomic manifolds:

Definition 2.3: A manifold V is N-anholonomic if its tangent space
TV is enabled with a N-connection structure (2). In a similar manner, we can consider different types of (super) spaces, Riemann or Riemann-Cartan manifolds, noncommutative bundles, or superbundles, provided with nonholonomc distributions (2) and preferred systems [33,34].

Torsions and curvatures of d-connections and d-metrics
One can be defined N-adapted linear connection and metric structures: Definition 2.4: A distinguished connection (d-connection) D on a N-anholonomic manifold V is a linear connection conserving under parallelism the Whitney sum (2).
For any d-vector X, there is a decomposition of D into h-and vcovariant derivatives, The symbol " "  in (8) denotes the interior product. We shall write conventionally that One has a N-adapted decomposition Considering h-and v-projections of (10) and taking into account that for any d-vectors X,Y By straightforward calculations, one check the properties for any for any d-vectors X, Y, Z.  In general, a metric structure is not adapted to a N-connection structure.  (12) adapted to a given N-connection structure.
Proof: e introduce

(Non) adapted linear connections
For any metric structure g on a manifold V, there is the unique metric compatible and torsionless Levi Civita connection ∇ for which The d-metric structure g on RC V is of type (14) and satisfies the metricity conditions (15). With respect to a local coordinate basis, the metric g is parametrized by a generic off-diagonal metric ansatz (2). For a particular case, we can take  D = D and treat the torsion  T as a nonholonomic frame effect induced by a nonintegrable Nsplitting. We conclude that a N-anholonomic Riemann manifold is with nontrivial torsion structure (9) (defined by the coefficients of N-connection (1), and d-metric (14) and canonical d-connection (15)). Nevertheless, such manifolds can be described alternatively, equivalently, as a usual (holonomic) Riemann manifold with the usual Levi Civita for the metric (1) with coefficients (2). We do not distinguish the existing nonholonomic structure for such geometric constructions.For more general applications, we have to consider additional torsion components, for instance, by the so-called H-field in string gravity [45]. As a result, the metric structure is transformed into a d-metric of type (14). We can say that V is equivalently re-defined as a N-anholonomic manifold V.

Theorem 2.5: The geometry of a (semi) Riemannian manifold V with prescribed (n+m)-splitting (nonholonomic h-and v-decomposition) is equivalent to the geometry of a canonical  .
It is also possible to compute the coefficients of canonical dconnection  D following formulas (15). We conclude that the geometry of a (semi) Riemannian manifold V with prescribed (n+m)-splitting can be described equivalently by geometric objects on a canonical N-anholonomic manifold  R V with induced torsion  T with the coefficients computed by introducing (15) into (9). The inverse construction also holds true: A d-metric (14) on  R V is also a metric on V but with respect to certain N-elongated basis (6). It can be also rewritten with respect to a coordinate basis having the parametrization (2). From this Theorem, by straightforward computations with respect to N-adapted bases (6) and (5) be optimal to elaborate a N-adapted tensor and differential calculus for nonholnomic structures, i.e. to choose the canonical d-connection. With respect to N-adapted frames, the coefficients of one connection can be expressed via coefficients of the second one, see formulas (16) and (15). Both such linear connections are defined by the same offdiagonal metric structure. For diagonal metrics with respect to local coordinate frames, the constructions are trivial.
Having prescribed a nonholonomic n+m splitting on a manifold V, we can define two canonical linear connections ∇ and  .

D
Correspondingly, these connections are characterized by two curvature tensors, (7) and (10)) and

Metrization procedure and preferred linear connections
On a N-anholonomic manifold V, with prescribed fundamental geometric structures g and N, we can consider various classes of dconnections D, which, in general, are not metric compatible, i.e.    The geometry and classification of metric-affine manifolds and related generalized Finsler-affine spaces is considered in Part I of monograph explained by Vacaru [34]. From Theorems 2.6, 2.7 and 2.5, follows Conclusion 2.1: The geometry of any manifold ma V can be equivalently modelled by deformation tensors on Riemann manifolds provided with preferred frame structure. The constructions are elaborated in N-adapted form if we work with the canonical dconnection, or not adapted to the N-connection structure if we apply the Levi Civita connection.
Finally, in this section, we note that if the torsion and nonmetricity fields of ma V are defined by the d-metric and N-connection coefficients (for instance, in Finsler geometry with Chern or Berwald connection, see below section 5.1) we can equivalently (nonholonomically) transform ma V into a Riemann manifold with metric structure of type (1) and (2).

Gravity and Lagrange-Finsler Geometry
We study N-anholonomic structures in Riemmann-Finsler and Lagrange geometry modelled on nonholonomic Riemann-Cartan manifolds.

Generalized lagrange spaces
If a N-anholonomic manifold is stated to be a tangent bundle, V=TM the dimension of the base and fiber space coincide, = , n m and we obtain a special case of N-connection geometry. For such geometric models, a N-connection is defined by Whithney sum with local coefficients i.e.
( ) One calls = ( , ) a b ab h x y y y ε to be the absolute energy associated to a ab h of constant signature.

Theorem 3.1: For nondegenerated Hessians
where Proof: ne has to consider local coordinate transformation laws for some coefficients a i N preserving splitting (16). We can verify that c a i N satisfy such conditions. The sketch of proof is given and expained by Vacaru [34] for TM. We can consider any nondegenerated quadratic y y x y and = .
Proof: t follows from formulas (19), (20), (17) and (19) and adapted d-connection (21)   Our approach to the geometry of N-anholonomic spaces (in particular, to that of Lagrange, or Finsler, spaces) is based on canonical d-connections. It is more related to the existing standard models of gravity and field theory allowing to define Finsler generalizations of spinor fields, noncommutative and supersymmetric models, discussed in by Vacaru [33,34]. Nevertheless, a number of schools and authors on Finsler geometry prefer linear connections which are not metric compatible (for instance, the Berwald and Chern connections, see below Definition 5.1) which define new classes of geometric models and alternative physical theories with nonmetricity field, see details in [34, [40][41][42]. From a geometrical point of view [46,47], all such approaches are equivalent. It can be considered as a particular realization, for nonholonomic manifolds, of the Poincare's idea on duality of geometry and physical models stating that physical theories can be defined equivalently on different geometric spaces [48].
where the source ¡ reflects any contributions of matter fields and corrections from, for instance, string/brane theories of gravity. In a physical model, the equations (22) have to be completed with equations for the matter fields and torsion (for instance, in the Einstein-Cartan theory one considers algebraic equations [49] for the torsion and its source). It should be noted here that because of nonholonomic A very important class of models can be elaborated when which defines the so-called Nanholonomic Einstein spaces with "nonhomogeneous" cosmological constant (various classes of exact solutions in gravity and nonholonomic Ricci flow theory were constructed and analyzed in [13][14][15]33,34].

Holonomic Ricci flows
where ∆ is the Laplace operator defined by .
g Usually, one considers normalized Ricci flows defined by where the normalizing factor = There are unique solutions for such linear ordinary differential equations for all time 0 0, ). τ τ ∈ Using the equations (24), (25) and (26), one can define the evolution equations under Ricci flow, for instance, for the Riemann tensor, Ricci tensor, Ricci scalar and volume form stated in coordinate frames (see, for example, the Theorem 3.13 in [10]. In this section, we shall consider such nonholnomic constraints on the evolution equation where the geometrical object will evolve in N-adapted form; we shall also model sets of N-anholnomic geometries, in particular, flows of geometric objects on nonholonomic Riemann manifolds and Finsler and Lagrange spaces.    (26), we can extract a set of preferred frame structures with associated N-connections, with respect to which we can perform the geometric constructions in N-adapted form.

Ricci flows and N-anholonomic distributions
We shall need a formula relating the connection Laplacian on contravariant one-tensors with Ricci curvature and the corresponding deformations under N-anholonomic maps. Let A be a d-tensor of rank k. Then we define 2 ∇ A, for ∇ being the Levi Civita connection, to be a contravariant tensor of rank k+2 given by This defines the (Levi Civita) Laplacian connection where the deformation d-tensor of the Laplacian, , ∆  is defined canonically by the N-connection and d-metric coefficients.
Proof: e sketch the method of computation . ∆  Using the formula (17), we have is defined for any α X with γ αβ Z  computed following formulas (17); all such coefficients depend on N-connection and d-metric coefficients and their derivatives, i.e. on generic offdiagonal metric coefficients (2) and their derivatives. Introducing (33) into (29) and (30), and separating the terms depending only on  In a similar form as for Proposition 4.2, we prove

Proposition 4.3: The curvature, Ricci and scalar tensors of the Levi Civita connection ∇ and the canonical d-connection  D are defined by formulas
for  R computed following formula (11) and In the theory of Ricci flows, one considers tensors quadratic in the curvature tensors, for instance, for any given , Using the connections , ∇ or  D, we similarly define and compute the values    [10].

Conclusion 4.1: The evolution of corresponding d-objects on Nanholonomic Riemann manifolds can be canonically extracted from the evolution under Ricci flows of geometric objects on Riemann manifolds.
In the sections 5.3 and 5.1, we shall consider how Finsler and Lagrange configurations can be extracted by more special parametrizations of metric and nonholonomic constraints.

Nonholonomic ricci flows of (non) symmetric metrics
The Ricci flow equations were introduced by Hamilton [6] in a heuristic form similarly to how A. Einstein proposed his equations by considering possible physically grounded equalities between the metric and its first and second derivatives and the second rank Ricci tensor. On (pseudo) Riemannian spaces the metric and Ricci tensors are both symmetric and it is possible to consider the parameter derivative of metric and/or correspondingly symmetrized energymomentum of matter fields as sources for the Ricci tensor.On Nanholonomic manifolds there are two alternative possibilities: The first one is to postulate the Ricci flow equations in symmetric form, for the Levi Civita connection, and then to extract various N-anholonomic configurations by imposing corresponding nonholonomic constraints. The bulk of our former and present work is related to symmetric metric configurations.
In the second case, we can start from the very beginning with a nonsymmetric Ricci tensor for a non-Riemannian space. In this section, we briefly speculate on such geometric constructions: The nonholonomic Ricci flows even beginning with a symmetric metric tensor may result naturally in nonsymmetric metric tensors Nonsymmetric metrics in gravity were originally considered by Einstein [50] and Eisenhart [51], see modern approaches [52]. or any parameter or extra dimension coordinate.

Theorem 4.2: With respect to N-adapted frames, the canonical nonholonomic Ricci flows with nonsymmetric metrics defined by equations
Proof. t follows from a redefinition of equations (24) with respect to N-adapted frames (by using the frame transform (4) and (5)), and considering respectively the canonical Ricci d-tensor (12) constructed from [ , ]. Here we note that normalizing factor r is considered for the symmetric part of metric.
One follows:

. y
We constructed and investigated various types of exact solutions of the nonholonomc Einstein equations and Ricci flow equations [33][34][35] and [13][14][15]. They are parametrized by ansatz of type (39) which positively constrains the Ricci flows to be with symmetric metrics. Such solutions can be used as backgrounds for investigating flows of Eisenhart (generalized Finsler-Eisenhart geometries) if the constraints (38) are not completely imposed. We shall not analyze this type of Nanholonomic Ricci flows in this series of works.

Generalized Finsler-Ricci Flows
The aim of this section is to provide some examples illustrating how different types of nonholonomic constraints on Ricci flows of Riemannian metrics model different classes of N-anholonomic spaces (defined by Finsler metrics and connections, geometric models of Lagrange mechanics and generalized Lagrange geometries).   One generates sets of geometric objects on pull-back cotangent bundle T M π * * and its tensor products:

Finsler-Ricci flows
a corresponding family of Cartan tensors i.e. the torsion free condition; i.e. the almost metric compatibility condition.
Proof: t follows from straightforward computations. For any fixed value 0 = , τ τ it is just the Chern's Theorem 2.4.1. from, In order to elaborate a complete geometric model on TM, which also allows us to perform the constructions for N-anholonomic manifolds, we have to extend the above considered forms with nontrivial coefficients with respect to ( ).
which is similar to formulas (21) but for Einstein and string gravity and in noncommutative gravity. It should be emphasized that the models of Finsler geometry with Chern, Berwald or Hashiguchi type d-connections are with nontrivial nonmetricity field [33,34]. So, in general, a family of Finsler fundamental metric functions ( ) F τ may generate various types of N-anholonomic metricaffine geometric configurations, see Definition 2.10, but all components of such induced nonmetricity and/or torsion fields are defined by the coefficients of corresponding families of generic off-diagonal metrics of type (1), when the ansatz (2) is parametrized for c a i N τ Applying the results of Theorem 2.7, we can transform the families of "nonmetric" Finsler geometries into corresponding metric ones and model the Finsler configurations on N-anholonomic Riemannian spaces, see Conclusion 2.1. In the "simplest" geometric and physical manner (convenient both for applying the former Hamilton-Perelman results on Ricci flows for Riemannian metrics, as well for further generalizations to noncommutative Finsler geometry, supersymmetric models and so on...), we restrict our analysis to Finsler-Ricci flows with canonical d-connection of Cartan type when (43) and (44). This provides a proof for

Lemma 5.1: A family of Finsler geometries defined by ( ) F τ can be characterized equivalently by the corresponding canonical d-connections (in N-adapted form) and Levi Civita connections (in not N-adapted form) related by formulas
where Z γ αβ  is computed following formulas (18) for nd satisfying the equations (for instance, for normalized flows) with where F g α β is inverse to (46) and F R β γ  is the Ricci tensor constructed from the Levi Civita coefficients of (46).
Proof. e have to introduce the metric and N-connection coefficients (42) and (41), defined by ( ), F τ into (4). The equations (48) are similar to (26), but in our case for the N-adapted frames (47). We note that the evolution of the Riemann and Ricci tensors and scalar curvature defined by the Cartan d-connection, i.e. the canonical dconnection,  ,  (46) and (45).Finally, in this section, we conclude that the Ricci flows of Finsler metrics can be extracted from Ricci flows of Riemannian metrics by corresponding metric ansatz, nonholonomic constraints and deformations of linear connections, all derived canonically from fundamental Finsler functions.

Ricci flows of regular lagrange systems
There were elaborated different approaches to geometric mechanics. We follow those related to formulations in terms of almost symplectic geometry [27] and generalized Finsler and Lagrange geometry [43]. We note that Lagrange-Finsler spaces can be equivalently modelled as almost Kähler geometries (see formulas (17) defining the almost complex structure) and, which is important for applications of the theory of anholonomic Ricci flows, modelled as nonholonomic Riemann manifolds, see Conclusion 3.1.
and satisfying the equations (for instance, normalized)

Generalized Lagrange-Ricci flows
We have the result that any mechanical system with a regular Lagrangian ( , ) L x y can be geometrized canonically in terms of nonholonomic Riemann geometry, see Conclusion 3.1, and for certain conditions such configurations generate exact solutions of the gravitational field equations in the Einstein gravity and/or its string/ gauge generalizations, see Result 6.2 and Theorem 6.1. In other words, for any symmetric tensor are defined for nondegenerated Hessians For any fixed value of τ, the existence of fundamental geometric objects (49), (50) and (51) and satisfying the equations (for instance, normalized) , it was proven that certain types of gravitational interactions can be modelled as generalized Lagrange-Finsler geometries and inversely, certain classes of generalized Finsler geometries can be modelled on N-anholonomic manifolds, even as exact solutions of gravitational field equations. The approach elaborated by Romanian geometers and physicists [33][34][35] originates from Vranceanu G and Horac Z works [36,37] on nonholonomic manifolds and mechanical systems, see a review of results and recent developments explained by Bejancu [38]. Recently, there were proposed various models of " analogous gravity", a review [53], which do not apply the methods of Finsler geometry and the formalism of nonlinear connections.

Local Geometry Of N-Anholonomic Manifolds
Let us consider a metric structure on N-anholonomic manifold V,  (14)) is a special type of a manifold provided with a global splitting into conventional "horizontal" and "vertical" subspaces (2) induced by the "off-diagonal" terms The simplest way to perform computations with d-connections is to use N-adapted differential forms like with the coefficients defined with respect to (6) and (5). For instance, torsion can be computed in the form Locally it is characterized by (N-adapted) d-torsion coefficients By a straightforward d-form calculus, we can find the N-adapted components of the curvature It should be noted that this tensor is not symmetric for arbitrary d-connections D. The scalar curvature of a d-connection is = , defined by a sum the h-and v-components of (12) and d-metric (14).
The Einstein tensor is defined and computed in standard form There is a minimal extension of the Levi Civita connection ∇ to a canonical d-connection  D which is defined only by a metric g  are defined by the coefficients of d-metric (14) and N-connection (1), or equivalently by the coefficients of the corresponding generic offdiagonal metric (2).
defining the generalized Christoffel symbols, where (for simplicity, we omitted the left up labels ( ) L for N-adapted bases).
We conclude that any regular Lagrange mechanics can be geometrized as a nonholonomic Riemann manifold V equipped with canonical N-connection (19) and adapted d-connection (21) and dmetric structures (20) all induced by a ( , ).

L x y
Let us show how N-anholonomic configurations can defined in gravity theories explained by Vacaru [33,34]. In this case, it is convenient to work on a general manifold , dim = n m + V V enabled with a global N-connection structure, instead of the tangent bundle  . TM

Result 6.2: Various classes of vacuum and nonvacuum exact solutions of (22) parametrized by generic off-diagonal metrics, nonholonomic vielbeins and Levi Civita or non-Riemannian connections in Einstein and extra dimension gravity models define explicit examples of Nanholonomic Einstein-Cartan (in particular, Einstein) spaces.
It should be noted that a subclass of N-anholonomic Einstein spaces was related to generic off-diagonal solutions in general relativity by such nonholonomic constraints when where  D is the canonical d-connection and ∇ is the Levi-Civita connection.
A direction in modern gravity is connected to analogous gravity models when certain gravitational effects and, for instance, black hole configurations are modelled by optical and acoustic media. Following our approach on geometric unification of gravity and Lagrange regular mechanics in terms of N-anholonomic spaces, one holds