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## Complex Spaces in Finsler, Lagrange and Hamilton Geometries

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## Nonholonomic Ricci Flows of Riemannian Metrics and Lagrange-Finsler Geometry

Proof: ne has to consider local coordinate transformation laws for some coefficients preserving splitting We can verify that satisfy such conditions. The sketch of proof is given and expained by Vacaru [ 34 ] for TM. We can consider any nondegenerated quadratic form on if we redefine the v—coordinates in the form and. Theorem 3. Proof: t follows from formulas 19 , 20 , 17 and 19 and adapted d—connection 21 and d—metric structures 20 all induced by a.

The class of Lagrange—Finsler geometries is usually defined on tangent bundles but it is possible to model such structures on general N—anholonomic manifolds, for instance, in pseudo Riemannian and Riemann—Cartan geometry, if nonholonomic frames are introduced into consideration [ 33 , 34 ]. Let us consider two such important examples when the N—anholonomic structures are modelled on TM.

Example 3. The Hessian 19 is defined. The notion of Lagrange space was introduced by Kern [ 43 ] and elaborated as a natural extension of Finsler geometry. In a more particular case, we have. 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. From a geometrical point of view [ 46 , 47 ], all such approaches are equivalent. Conclusion 3. In equivalent form, such Lagrange— Finsler geometries can be described by the same metric and N— anholonomic distributions but with the corresponding not adapted Levi Civita connections.

The Einstein equations are. 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. This imposes a more sophisticated form of conservation laws on such spaces with generic "local anisotropy" [ 34 ], a similar situation arises in Lagrange mechanics when nonholonomic constraints modify the definition of conservation laws.

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A very important class of models can be elaborated when which defines the so—called N— anholonomic Einstein spaces with "nonhomogeneous" cosmological constant various classes of exact solutions in gravity and nonholonomic Ricci flow theory were constructed and analyzed in [ 13 - 15 , 33 , 34 ]. Usually, one considers normalized Ricci flows defined by. It is important to study the evolution of tensors in orthonormal frames and coframes on nonholonomic manifolds. Let be a Ricci flow with and consider the evolution of basis vector fields. We evolve this local frame flows according to the formula.

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. 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.

On manifold V, the equations 24 and 25 describe flows not adapted to the N—connections For a prescribed family of such N—connections, we can construct from the corresponding set of d—metrics and the set of N—adapted frames on The evolution of such N—adapted frames is not defined by the equations 26 but satisfies the. Proposition 4. 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. Using the formula 17 , we have. Introducing 33 into 29 and 30 , and separating the terms depending only on we get The rest of terms with linear or quadratic dependence on and their derivatives define. In the theory of Ricci flows, one considers tensors quadratic in the curvature tensors, for instance, for any given and D.

There are d—objects d—tensors, d—connections with N—adapted evolution completely defined by solutions of the Ricci flow equations Definition 4. Theorem 4. The evolution equations from Theorem 4. The coefficients of d—objects are defined with respect to evolving N—adapted frames 27 and One holds. Conclusion 4.

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In the sections 5. 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 N— anholonomic 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 ].

The system of equations 35 , 36 and 38 , for "symmetric" nonholonomic Ricci flows, was introduced and analyzed in [ 13 , 14 ]. Example 4.