Constantin Carathéodory axiomatic approach and Grigory Perelman thermodynamics for geometric flows and cosmological solitonic solutions
aa r X i v : . [ phy s i c s . g e n - ph ] M a r Constantin Carathéodory axiomatic approach and Grigory Perelmanthermodynamics for geometric flows and cosmological solitonic solutions
Iuliana Bubuianu ∗ Radio Iaşi, 44 Lascˇar Catargi, Iaşi, 700107, Romania and
Sergiu I. Vacaru † Physics Department, California State University at Fresno, Fresno, CA 93740, USA; andDep. Theoretical Physics and Computer Modelling, 101 Storozhynetska street, Chernivtsi, 58029, Ukraine
March 19, 2020
Abstract
We elaborate on statistical thermodynamics models of relativistic geometric flows as generalizations ofG. Perelman and R. Hamilton theory centered around C. Carathéodory axiomatic approach to thermo-dynamics with Pfaffian differential equations. The anholonomic frame deformation method, AFDM, forconstructing generic off–diagonal and locally anisotropic cosmological solitonic solutions in the theory ofrelativistic geometric flows and general relativity is developed. We conclude that such solutions can notbe described in terms of the Hawking–Bekenstein thermodynamics for hypersurface, holographic, (anti)de Sitter and similar configurations. The geometric thermodynamic values are defined and computed fornonholonomic Ricci flows, (modified) Einstein equations, and new classes of locally anisotropic cosmologicalsolutions encoding solitonic hierarchies.
Keywords: axiomatic relativistic geometric flow thermodynamics, nonholonomic Ricci solitons, Grig-ory Perelman, Constantin Carathéodory, Pfaffian differential equations, geometric methods for constructingexact solutions, W–entropy for cosmological solitonic solutions.MSC 2010: 53C50, 53E20, 82D99, 83F99, 83C15, 83D99, 37J60PACS 2010: 02.40.Vh, 02.90.+p, 04.20.Cv, 04.20.Jb, 04.90.+e, 05.90.+m
Contents ∗ email: [email protected] † emails: [email protected] and [email protected] ; Address for post correspondence in 2019-2020 as a visitor senior researcher at YF CNU Ukraine is:
37 Yu. Gagarina street, ap.3, Chernivtsi, Ukraine, 58008 Decoupling and integrability of cosmological solitonic flow equations 12
B.1 Target d-metrics with geometric evolution of polarization functions . . . . . . . . . . . . . . . 31B.2 Off-diagonal and diagonal parameterizations of prime d-metrics . . . . . . . . . . . . . . . . . 32B.3 Approximations for flows of target d-metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Thermodynamics is a fundamental physical theory with various branches and applications in modernphysics, engineering, biology, chemistry, information theory and mathematics, see reviews [1, 2, 3, 4, 5]and references therein. The equilibrium thermodynamics originated from the study of heat engines whenthe combination of mechanical and thermal concepts was done in an empirical way with further essentialdevelopments and contributions to statistical physics and ergodic theory, modern gravity, cosmology etc.Thermodynamic ideas and methods were developed and applied in black hole, BH, physics [6, 7, 8, 9],using the Bekenstein–Hawking entropy, and (using different concepts and constructions) in the proof of the2hurston–Poincaré conjecture due to Grigory Perelman [10], see original fundamental physical and mathe-matical works and reviews of geometric analysis and topological results in [11, 12, 13, 14, 15, 16]. In a seriesof our and co-authors works, we studied possible implications of the approach elaborated by G. Perelman forgeometric flow statistical thermodynamics in certain directions of modified and Einstein gravity and cosmol-ogy and astrophysics [17, 18, 19, 20] and classical and quantum geometric information flow theory [21, 22, 23].It was exploited the idea that the concept of Perelman W-entropy and associated statistical models presentsmore general mathematical and physical possibilities comparing to those elaborated for theories and solutionswith Bekenstein–Hawking and another area–holographic type entropies.Constantin Carathéory formulated the first systematic and axiomatic formulation of equilibrium thermo-dynamics [24, 25]. In such an approach (see [5, 26, 27, 28] on further extensions), the geometry of thermody-namics is symplectic and analogous to the structure of Hamilton mechanics and can expressed through Pfaffforms and related systems of first order partial differential equations. The axiomatic treatment of thermody-namics caught the attention of a number of famous and well-known scientists [29, 30, 31, 32, 33, 34, 5, 35] whorecognized and in one case criticized [36] Carathéodory’s papers; for brief reviews, we cite [37, 38, 39, 40]. In the present work, we will not get into the details of C. Carathéodory and G. Perelman achievements; seeabove cited works and [41, 42, 43, 44, 37, 38], on life and contributions in mathematics, physics, and education.We shall study only how the mathematic tools of axiomatic thermodynamics can be applied to relativisticgeneralizations of geometric flow theory and compute geometric thermodynamic values for locally anisotropiccosmological solutions. There will be considered also certain applications in modern cosmology. There arethree main purposes of this article: 1) To show how the C. Carathéory axiomatic approach to thermodynamicsin the language of Pfaff forms can be extended in order to include in the scheme generalizations of theG. Perelman thermodynamics for relativistic geometric flows; 2) To consider possible applications of theanholonomic frame deformation method, AFDM, and study main properties of geometric evolution flowsof locally anisotropic cosmological models (in particular with generic off-diagonal solitonic deformations ofthe Friedman-Lemaître-Robertson-Walker, FLRW, metrics; 3) To provide explicit examples how geometricflow thermodynamic values are computed for cosmological solitonic solutions which can not be described bythermodynamic concepts elaborated in the framework of Bekenstein–Hawking entropy and generalizations.This paper is organized as follows: In section 2, we present an introduction into the theory of relativisticnonholonomic flows with modified F- and W–functionals and elaborate on respective statistical thermodynamicmodels. How the Carathéodory axiomatic approach can be extended in order to include some classes ofgeneralized Ricci flows and solitons is considered. Then, in section 3, we develop the AFDM and show that theimportant general decoupling and integration properties of geometric evolution flow and Ricci soliton equationsare preserved for cosmological solitonic spaces. Possible locally anisotropic cosmological parameterizations aresummarized in Table 1. We provide Table 2 summarizing the AFDM for generating such locally anisotropiccosmological solutions. In section 4, we show how exact and parametric cosmological solitonic solutions canbe constructed for relativistic geometric flow evolution equations. There are analyzed explicit examples ofcomputing respective W-entropy, thermodynamic values and Pfaffians. Finally, conclusions and perspectivesare considered in section 5. In Appendix A, we outline some results on Pfaffian differential equations. AppendixB contains necessary parameterizations for flows of cosmological solitonic metrics.
In this section, we provide a brief introduction to the theory of relativistic geometric flows and analogousstatistical thermodynamics for nonholonomic Einstein systems, NESs, see details in [17, 18, 19, 20] and M. Planck and some other authors criticism "targeting quick results" was about the difficulty to provide a simple physicalpicture of the Carathéodory method and the concept of entropy together with sophisticate geometric methods unknown at thattime to the bulk of physicists and mathematicians. At present, the functional analysis, measure theory and topology techniquesare familiar to researchers publishing works in mathematical physics and geometry and physics.
We consider a relativistic spacetime as in general relativity, GR. Geometrically, it is defined by a (pseudo)Riemannian manifold V with a conventional splitting of dimension, dim V = 4 = 2 + 2 , and two dimensionalhorizontal, h, and two dimensional vertical, v, components (such a decomposition will be useful for constructingexact solutions of systems of important physical equations). This induces diadic decompositions of local basesand corresponding tangent bundles T V and, its dual, T ∗ V . Being enabled with a metric g = ( h g , v g ) of a local pseudo-Euclidean signature (+ + + − ) and postulating local causality conditions as in specialrelativity theory, we model a curved spacetime as a Lorentzian manifold. We can always consider thatsuch spacetimes are endowed with a double nonholonomic 2+2 and 3+1 splitting (the first splitting will beused for elaborating new methods of constructing exact solutions and the second splitting will be necessaryfor elaborating thermodynamical models).In this work, we say that a Lorentz manifold V is nonholonomic (in literature, there are also used equivalentterms like anholonomic, or non-integrable) if it is endowed with a h- and/or v-splitting defined by a Whitneysum defining a nonlinear connection, N-connection, structure N : T V = h V ⊕ v V , where T V is the tangentbundle on V . Such a geometric structure is a fundamental one for elaborating various models of Finsler–Lagrange–Hamilton geometry which are determined in complete form if there are prescribed three fundamentalgeometric objects/structures (a nonlinear quadratic line element, a nonlinear connection and a distinguishedconnection which is adapted to a h-v-splitting). N–connections can be introduced also in (pseudo) Riemanniangeometry when (in local form) N = N ai ( u ) dx i ⊗ ∂ a is determined by for a corresponding set of coefficients { N ai } which can be related to certain off-diagonal terms of metrics in certain local frames of coordinates. Corresponding subclasses of N-adapted (co) frames allow, for instance, nonholonomic diadic decompositionsof geometric and physical objects. Together with a so-called canonical nonholonomic deformations of linearconnection structures (we shall use "hats" on geometric and physical objects adapted to such canonicalnonholonomic frames) this allows to integrate (modified) Einstein and geometric flow equations in very generalforms depending, in principle, on all spacetime coordinates and a geometric evolution parameter.We shall use two important linear connections which can be constructed using the same metric structure: g → (cid:26) ∇ : ∇ g = 0; ∇ T = 0 , the Levi–Civita, LC, connection; b D : b D g = 0; h b T = 0 , v b T = 0 . the canonical d–connection. (1)In these formulas (see [17, 18, 19, 20] for details on computing coefficients with respect to N-adapted and/orcoordinate frames), the distinguished connection (d–connection) b D = ( h b D , v b D ) preserves under parallelism thedecomposition N and b T [ g , N ] is the corresponding torsion d-tensor. We use the terms d-tensor, d-connectionetc. for geometric objects adapted to a N–connection h-v-splitting. The LC–connection ∇ can be introducedwithout any N–connection structure but the zero torsion condition in the case of generic–off diagonal metricsdo not allow to prove a decoupling property and explicit integration of physically important systems ofnonlinear partial differential equations, PDEs. Nevertheless, any geometric data ( g , ∇ ) can be distorted tosome canonical ones, ( g , b D ) , with decoupling of (modified) Einstein equations and encoding in general formvarous classes of physically important solutions) b D [ g , N ] = ∇ [ g , N ] + b Z [ g , N ] , (2) We parameterize the coordinates as u µ = ( x i , y a ) , in brief, u = ( x, y ) , where i, j, ... = 1 , and a, b = 3 , , with small Greekindices α, β, ... = 1 , , , , when u = y = t is the time like coordinate. We shall summarise on "up-low" repeating indices anduse boldface symbols for spaces and geometric objects adapted to a N-connection splitting. For a double 2+2 and 3+1 splitting,the local coordinates are labeled u α = ( x i , y a ) = ( x ` ı , u = t ) for ` ı, ` j, ` k = 1 , , . The nonholonomic distributions can be N-adaptedform for any open region U ⊂ V covered by a family of 3-d spacelike hypersurfaces Ξ t with a time like parameter t. b Z is the distortion d-tensor determined in standard algebraic form by the torsion tensor b T [ g , N ] of b D . These values are completely defined by the metric d-tensor g = ( h g , v g ) adapted to a prescribed N . Thevalues h b T and v b T denote respective torsion components which vanish on conventional h- and v–subspaces.There are also nontrivial components hv b T defined by certain anholonomy (equivalently, nonholonomic/non-integrable) relations. All geometric constructions on a Lorentz manifold V can be performed in a not adaptedN-connection form with ∇ and/or in N-adapted form using b D from (1) or other type d-connections. Thecoresponding torsion d-tensor, b T γαβ ; Ricci d-tensor, b R βγ ; scalar curvature s b R := g αβ b R βγ ; and Einsteind-tensor, b E βγ := b R βγ − g βγ s b R, are defined and computed in standard forms as in metric–affine geometryand related via distortion formulas to respective values determined by ∇ . In the theory of Ricci flows of geometric objects on V , it is considered an evolution positive parameter τ, ≤ τ ≤ τ , which for thermodynamic models can identified with the temperature, or chosen to be propor-tional to a temperature parameter. For the geometric flow evolution of Riemannian metrics and respectivestatistical thermodynamic models, this was considered in G. Perelman’s famous preprint [10]. For evolutionof (generalized) pseudo–Riemannian configurations we can elaborate on two classes of (effective) geometrictheories, when families of metrics 1] g ( τ ) := g ( τ, u ) are labelled by a conventional evolution (relativistic tem-perature parameter) or 2] g ( τ ) := g ( τ, x ` ı ) for an imaginary time like coordinate u α = ( x i , y a ) = ( x ` ı , u = ict ) , where i = − and, for simplicity, there used unities when the fundamental speed of light is c = 1 . Hereafter,we shall write in brief only the dependence on evolution parameter, without spacetime or space coordinates ifthat will not result in ambiguities. In this work, we study only theories of class 1] when the evolution modelsare relativistic, encode solitonic waves for pseudo-Riemannian metric signatures and can be characterized byrelativistic thermodynamic models, see details in [17, 18, 19] and references therein. In such geometric andthermodynamic theories, we consider also flows of N–connections N ( τ ) = N ( τ, u ) , canonical d-connections b D ( τ ) = b D ( τ, u ) . On V , we can introduce also families of Lagrange densities g L ( τ ) , for gravitational fieldsin a MGT or GR (when g L ( τ ) = s R [ ∇ ( τ )] ), and tot L ( τ ) = tot L [ g ( τ ) , b D ( τ ) , ϕ ( τ )] , as total Lagrangiansfor effective and matter fields which will be defined below for certain cosmological models with scalar fields ϕ ( τ ) = ϕ ( τ, u ) . For any region U ⊂ V with a 2+2 splitting ( N , g ) , we consider an additional structure of 3-d hypersurfaces Ξ t parameterized by time like coordinate y = t for coordinates u α = ( x i , y a ) = ( x ` ı , t ) . The families of metricscan be represented as d-metrics with 3+1 splitting and N–adapted geometric evolution, g ( τ ) = g α ′ β ′ ( τ, u ) d e α ′ ( τ ) ⊗ d e β ′ ( τ ) (3) = q i ( τ, x k ) dx i ⊗ dx i + q ( τ, x k , y a ) e ( τ ) ⊗ e ( τ ) − [ q N ( τ, x k , y a )] e ( τ ) ⊗ e ( τ ) , for e µ ( τ ) = ( e i = dx i , e a ( τ ) = dy a + N ai ( τ ) dx i ) . (4)In (3), there are considered geometric flows of "shift" coefficients q ` ı ( τ ) = ( q i ( τ ) , q ( τ )) related to flows of a 3-dmetric q ij ( τ ) = diag ( q ` ı ( τ )) = ( q i ( τ ) , q ( τ )) on a hypersurface Ξ t if q ( τ ) = g ( τ ) and [ q N ( τ )] = − g ( τ ) , where q N ( τ ) is a family of lapse functions. Here, it should be noted that we follow notations which aredifferent from those in [45] for GR. In this work, it is used a left label q in order to avoid ambiguities withthe notations for the coefficients N ai of a N-connection. There are considered flows of N-adapted frames (4)determined by the flow evolution of N-connection coefficients.In nonholonomic canonical variables, the relativistic versions of G. Perelman functionals (originally definedin [10] for flows of Riemannian metrics), in this work encoding also the geometric evolution of matter fields,are postulated [17, 18, 19] in the form b F ( τ ) = Z (4 πτ ) − e − b f p | g | d u ( s b R + tot L + | b D b f | ) and (5) c W ( τ ) = Z b µ p | g | d u [ τ ( s b R + tot L + | h b D b f | + | v b D b f | ) + b f − . (6)5n such formulas, a normalizing function b f ( τ, u ) can be a convenient one for elaborating certain topological/geometric / physical models or subjected to the conditions b V ( τ ) = Z b µ p | g | d u = Z t t Z Ξ t b µ p | g | d u = 1 , (7)for a classical integration measure b µ = (4 πτ ) − e − b f (a version of Carathéodory measure); and the Ricci scalar s b R is taken for the Ricci d-tensor b R αβ of a d-connection b D . There is a series of arguments for writing the b F -functional (5) and c W -functional (6) in above forms:1. Fixing variations of such functionals on a d-metric and respective matter field evolution scenarios, andconsidering self-similar configurations for a τ = τ , we obtain systems of nonlinear PDEs for relativisticRicci solitons, which are equivalent to the gravitational field equations in nonholonomic variables, b R αβ = b Υ αβ (8)for nonholonomic Einstein systems, NESs, if the normalizing function b f is correspondingly chosen andthe nonholonomic constraints for extracting Levi–Civita configurations are imposed to extract LC–configurations b D | b T =0 = ∇ . The sources b Υ αβ ( τ ) = [ b Υ ij ( τ ) , b Υ ab ( τ )] with coefficients defined withrespect to N-adapted frames in (8) are of type b Υ µν = e b Υ µν + m b Υ µν , where e b Υ µν are effective sourcesdetermined by distortions of the linear connections and effective Lagrangians for gravitational fields.Such a source is not zero even in GR if there are nonzero distortions (2) from b D to ∇ . A source formatter field, m b Υ µν , can be constructed using a N–adapted variational calculus for a Lagrande density m L ( g , b D , A ϕ ) , when m b Υ µν = κ ( m b T µν − g µν m b T ) → κ ( m T µν − g µν m T ) for [coefficients of b D ] → [coefficients of ∇ ]. In such formulas, we consider m b T = g µν m b T µν for theN-adapted energy–momentum tensor m b T αβ := − p | g µν | δ ( p | g µν | m L ) δ g αβ . (9)For simplicity, we shall consider only Lagrange densities m L = φ L ( g , b D , φ ) determined by a scalar field φ ( x, u ) and/or geometric evolution of scalar fields φ ( τ ) = φ ( τ, x, u ) , when m b T αβ = φ b T αβ .2. For three dimensional, 3-d, Riemannian metrics, there are obtained respective Lyapunov type functionalsas it was postulated in [10] and used for the proof of the Thurston–Poincaré conjecture.3. The functional c W (6) defines a nonholonomic canonical and relativistic generalization of the so-calledW-entropy introduced in [10]. Various types of 4-d - 10-d W –entropies and associated statistical andquantum thermodynamics values are used for elaborating models of classical and (commutative andnoncommutative/ supersymmetric) quantum geometric flows and geometric information flows, see [16,17, 18, 19, 20, 21, 22, 23] and references therein.4. The functionals b F and c W result in generalized R. Hamilton equations [12] considered earlier in physics byD. Friedan [11] (respective proofs for and N-adapted variational calculus are presented in [17, 18, 19, 20]): ∂ g ij ∂τ = − (cid:16) b R ij − b Υ ij (cid:17) ; ∂ g ab ∂τ = − (cid:16) b R ab − b Υ ab (cid:17) ; (10) b R ia = b R ai = 0; b R ij = b R ji ; b R ab = b R ba ; ∂ τ b f = − b (cid:3) b f + (cid:12)(cid:12)(cid:12) b D b f (cid:12)(cid:12)(cid:12) − s b R + b Υ aa , (11)6here b (cid:3) ( τ ) = b D α ( τ ) b D α ( τ ) is used for the geometric flows of the d’Alambert operator. In nonholonomiccanonical variables with b D , such systems of nonlinear PDEs can be integrated in very general forms andrestricted to describe the geometric evolution and Ricci soliton configurations of NESs, see details andproofs in references from the previous paragraph (point).5. The measure b µ p | g | d u = (4 πτ ) − e − b f p | g | d u consists an explicit example of a Carathéodory typemeasure which allows to construct geometric and statistical thermodynamic models. Such a model waselaborated for the flow evolution of 3-d Riemannian metrics by G. Perelman and considered in theproof of the Thurston–Poincaré conjecture. Thermodynamic measures have more rich implications invarious branches of topology and applications and provide a natural tool to understand the difficulties(ergodicity, approach to equilibrium, irreversibility etc.) in the foundations of statistical physic andnon–equilibrium thermodynamics, see discussions and references in [46, 47, 48, 38, 39, 40]. Fixing a nor-malizing function b f , we prescribe an evolution scenarios with respective scales and phase space integralproperties determined by geometric and physical data ( g , b D , A ϕ ) . We can consider a geometric evolutionmodel without m L but with a re-defined functional measure µ ▽ p | g | d u = (4 πτ ) − e − b f ( f ▽ ) p | g | d u, where f ▽ is chosen to be a solution of this system of PDEs: tot L + | b D b f | = | b D ( f ▽ ) | and τ ( tot L + | h b D b f | + | v b D b f | ) + b f = τ ( | h b D ( f ▽ ) | + | v b D ( f ▽ ) | ) + f ▽ . (12)The solutions for such a f ▽ ( b f ) and/or b f ( f ▽ ) can be found in an explicit form (usually, such a normal-izing function can be approximated to a constant) for a large class of generic off-diagonal or diagonalsolutions of systems (10) and (11). In such constructions, usually there are prescribed respective val-ues of some generating functions and sources and effective cosmological constants subjected to certainnonlinear symmetry conditions, see details in [16, 17, 18, 19, 20, 21, 22, 23] and section 4.4. For variousphysical applications, it is enough to find a class of solutions of generalized R. Hamilton equations (10)and to consider that the geometric evolution is normalized by some functions a f ▽ ( b f ) and/or b f ( f ▽ ) subjected to conditions (11) and (12). In many cases, such normalizations can be performed withcertain integration constants or for series expansions on a small parameter.Using formulas (5), (6), (10), (11), we can elaborate on statistical thermodynamic models for geometricflows determined by data ( g , b D , A ϕ, f ▽ ) and applying concepts and formulas from statistical thermodynamics.It is considered a canonical ensemble at temperature β − = T , in this work T is proportional to τ , with parti-tion function Z = R exp( − βE ) dω ( E ) , where a measure ω ( E ) is defined as a density of states. In standard form,there are computed such important thermodynamical values: average flow energy, E = h E i := − ∂ log Z/∂β ; flow entropy, S := β h E i + log Z ; flow fluctuation, η := D ( E − h E i ) E = ∂ log Z/∂β . A geometric thermody-namic model can be constructed if we associate to (6) a respective thermodynamic generating functions (inthis work, flows of tot L are encoded into f ▽ ( b f , tot L ) subjected to the conditions (12)), b Z [ g ( τ ) , f ▽ ] = Z (4 πτ ) − e − b f ( f ▽ ) p | g | d u ( − f ▽ + 2) , for V . (13)Hereafter we shall not write functional dependencies on g ( τ ) and f ▽ if it will not result in ambiguities.Applying a similar variational calculus similar to that presented in details in [10, 14, 15, 16] (in N-adaptedform for frames (4) and d-connections b D ) to (13) and (6) and respective 3+1 parameterizations of d-metrics(3), we define and compute analogous thermodynamic values for geometric evolution flows of NES, b E ( τ ) = − τ Z (4 πτ ) − e − b f ( f ▽ ) q | q q q ( q N ) | δ u ( s b R + | b D f ▽ | − τ ) , (14) b S ( τ ) = − Z (4 πτ ) − e − b f ( f ▽ ) q | q q q ( q N ) | δ u h τ (cid:16) s b R + | b D f ▽ | (cid:17) + f ▽ − i , b η ( τ ) = − τ Z (4 πτ ) − e − b f ( f ▽ ) q | q q q ( q N ) | δ u [ | b R αβ + b D α b D β f ▽ − τ g αβ | ] .
7n these formulas, δ u contains N-elongated differentials and the data on matter fields and nonlinear sym-metries are encoded in f ▽ ( b f , tot L ) . For fixed self-similar Riemannian configurations, the values b E and b S provide an equilibrium thermodynamic description of Ricci solitons. Such concepts can be considered alongany causal curve on a Lorentz manifold when fixing certain normalization functions and nonlinear symmetriesthe conventional thermodynamic description holds true for evoluion models of NESs. The fluctuation b η ( τ ) allows to include into consideration small perturbations of metrics and corresponding distortion values. Sucha description defines a relativistic thermodynamic model which is irreversible and describes various types ofnonlinear self-organizing, pattern forming, kinetic and/or stochastic processes, see examples and references in[16, 17, 18, 19, 20, 21, 22, 23].After Carathéodory had completed the proof of Poincaré recurrence theorem [49], he was the first who sawthat measure theory is the natural language to discuss the problems of statistical physics and thermodynamics.The Poincaré hypothesis was formulated on topological properties of three dimensional hypersurfaces endowedwith Riemannian metrics. The proof of its generalized form as the Thurston–Poincaré conjecture was possibleby introducing measures of type c M = b µ p | g | d u = (4 πτ ) − e − b f p | g | d u (15)for Riemannian versions of functionals b F (5) c W (6), when with b D | b T =0 = ∇ , and a respective thermodynamicvalues (14). If p | g | is considered for pseudo-Riemannian metrics (and various, for instance, Finsler-Lagrange-Hamilton generalizations [16, 20]), we can speculate on respective generalized Poincaré recurrence theoremwhich can be reformulated in this form: The volume preserving dynamical transformations of an (effective)phase space with a measure b µ p | g | d u have the property that almost all points in any region of positivevolume (excepting possible subsets of zero volume) will return back into their region after some finite time. For relativistic configurations, we consider 3+1 splitting and a Lorentz type causality on respective spacetimeand/or phase space (co) tangent Lorentz bundles. Of course, the return time for each point connected by acausal curve, is different but can be computed for a respective exact/parametric solution of the relativisticgeometric flows and/or (modified) gravity theory (i.e. nonholonomic Ricci soliton configuration).Following the Carathéodory measure theoretic idea [49] and Birkhoff’s approach to ergodicity [50, 51],we can clarify the relation between ergodic and recurrent systems. In the case of geometric flows, we candefine ergodicity by replacing Boltzmann’s sets with sets of of non zero volume measure which was generalizedfor nonholonomic manifolds and generalized Finsler spaces in [52, 53, 54, 17]. Here we note that geometricflows as ergodic systems are recurrent but not vice versa. Non–ergodic systems decompose into time-invariantergodic sub-systems and this property can be extended to relativistic flows defined along causal curves.Mixing (it means that the statistical correlations decay and results in statistical regularity) of geometricflows implies ergodicity and is compatible with recurrence. Let us explain how it characterizes geometricevolution of NESs relating certain subsets
A, B, Y ⊂ V , when S t : Y → Y is the geometric (gradient) flowevolution on the phase space Y and there are satisfied non-zero measure conditions, c M [ A ] = 0 , c M [ B ] = 0 , for c M [ Y ] = 1 considered as a probability measure determined by (15), when lim t →±∞ c M [ A ∩ S t B ] c M [ A ] = c M [ B ] c M [ Y ] . Suchconditions for geometric evolution of NES determined by on families of 3-d hypersurfaces Ξ t mean that anyset B spreads over the phase space Y so that for any (fixed set/hypersurface/window) the fraction of B in A approaches the fraction of B over the whole phase space Y (this is an uniform mixture of d-metrics).Relativistic thermodynamic systems (14) with measure theoretic definitions of ergodicity and mixing forcorresponding classes of solutions of (10) possess such properties on an open spacetime region U ⊂ V ,
1. as ergodic systems they have an unique equilibrium distribution defined by g ( τ );
2. as mixing systems they approach an equilibrium state by g ( τ );
3. the rates of approach to equilibrium is determined by the rates of decay of correlations which can becomputed, for instance, for locally anisotropic cosmological solutions;8. using exact solutions, we can study irreversibility as an unidirectional spontaneous evolution from presentto future; this issue can addressed using operator theory and functional analysis (in this article, we donot consider such issues in Hilbert space and convex spaces, see references in [38]).Nonholonomically deformed G. Perelman functionals (5) and (6) determine both the relativistic dynamicaland thermodynamic (in general, with irreversible and non-equilibrium configurations) properties of NESs.
Following the first seminal Carathéodory’s work [24], we state the main definitions of the concepts ofstates, equilibrium, energy and entropy, and thermodynamic coordinates. A geometric flow state is givenby any data { g αβ = [ q q q ( q N )] , N ,f ▽ } defining a solution of the nonholonomic Ricci flow equations (10)for a fixed normalizing function f ▽ ( b f ) and/or b f ( f ▽ ) subjected to some conditions (11) and (12). Theevolution parameter τ can be identified with the temperature T in some conventional systems of referencesand chosen physical unities (in general, we can consider any convenient T ( τ ) ; we use a "cal" symbol in orderto avoid ambiguities when the capital letter T is used for torsion (1) or the energy-momentum tensor (9)).For such geometric data, we can always compute the statistical thermodynamic values b E ( τ ) and b S ( τ ) , see(14). To elaborate on analogous thermodynamic models we can consider a family of volumes b V ( τ ) = 1 (7)when the condition b V ( τ ) = 1 can be imposed by a corresponding f ▽ ( b f ) but d b V ( τ ) = 0 . This allows us tointroduce a conventional pressure P and external work A, and postulate that for any fixed τ the first law ofthermodynamics for geometric flows d b E = − P d b V ( τ ) . (16)Stating τ i and τ f for respective initial, and final states, we formulate Axiom 2.1
For any geometric flow thermodynamics of NESs, b E ( τ f ) − b E ( τ i ) + A = 0 . This axiom can be considered as the fist postulate of the relativistic thermodynamics of Ricci flows.Reversibility for such systems can be introduced for self-similar configurations for a fixed τ , i.e. for relativis-tic Ricci solitons defined equivalently by generalized Einstein equations (8). As dynamical equations, such(modified) gravitational and matter field equations possess reversible (at least in certain regions) solutions. Nevertheless, a general geometric flow evolution is described by irreversible equations (10) and (11).After that we can state the second axiom for relativistic geometric flows : Axiom 2.2
In neighborhood of any self-similar configurations for a fixed τ , i.e. of a nonholonomic Riccisoliton, there exists states/geometric data { g αβ = [1 q q ( q N )] , N ,f ▽ } which are inaccessible as nonholonomic A mathematical project usually starts as an axiomatic system starting with an ensemble of declarations/ statements. Thiscontains certain constructions, solutions of equations, and proofs of theorems. In the case of Euclidean geometry, the axioms areconsidered to be self-evident but various motivations and fundamental/ experimental arguments are put forward for advancedtheories related to physics and applications. As a typical axiomatic approach to modern thermodynamics can be considered[28, 55]. The axioms and certain definitions and "rules of interference" provide the basis for proving theorems. The word"postulate" is used in many cases instead of "axiom". Here we explain that in mathematics and logics the axioms are consideredas general statements accepted without proofs. In their turns, postulates are used for some specific cases and can not be consideredas "very general" statements. In many papers in non-mathematical journals oriented to mathematical physics and applicationsthe axioms, definitions and rules of interference are not cite and related rules of interference are not sited but certain proofs andsolutions are provided using corresponding mathematical tools. Such a geometric and PDE theory style will be used in this work. For standard thermodynamic systems, i.e. not for the Ricci flows, this is just the internal energy and external work conser-vation law, i.e. the first postulate of thermodynamics. Following Carathéodory (see also discussions and references in [37]), for standard thermodynamic systems the English versionof such a famous second axiom is "In the neighborhood of any equilibrium state of a system (of any number of thermodynamiccoordinates), there exists states that are inaccessible by reversible adiabatic processes". This axiom is better understood if itis used the Kelvin’s formulation of the second law of (standard, not geometric) thermodynamics "no cycle can exist whose neteffect is a total conversion of heat into work". icci soliton systems for a fixed f ▽ ( b f ) but as accessible for some τ ≥ τ if there are nontrivial solutions ofgeneralized Hamilton equations (10) and (11) relating τ as an initial state and a final state with τ . In principle, one could be certain "un-physical" processes which may connect two geometric thermodynamicNES or general d-metric systems when b E ( τ ) and b S ( τ ) are computed for data { g αβ = [ q q q ( q N )] , N ,f ▽ } which are not solutions of (10). Here we note that in the axioms and definitions of the Carathéodory methodthere is no mention of heat, temperature or entropy because heat is regarded as derived value (and not afundamental quantity) that appears as soon as the adiabatic restriction is removed. That was the weaknessand strength of the approach to standard thermodynamics developed by the aid of the theory of Pfaffianequations. M. Born [29] was the first who centered the attention to the elegance of the new method but M .Planck [36] sharply criticized Carathéodory’s method considering that the Thomson-Clausious treatment wasmore reliable being much nearer to experimental evidence, i.e. to natural processes. For details, discussionsand main references, we cite [37] and [38].Theories of geometric Ricci flows are not elaborated similarly to typical thermodynamical systems char-acterizing engines and cycles with heat, or chemical reactions and can not be studied in a phenomenologicalmanner with engineering methods. The strength of axiomatic methods is that introducing in terms of geo-metric objects (measure defined by a metric, linear connection and corresponding curvature scalar and Riccitensor) the concept of W-entropy, and related functionals and statistical thermodynamic constructions, G.Perelman was able to prove the Thurston–Poincaré conjecture. Such geometric methods are even more ad-vanced and sophisticated than those used by C. Carathéodory and M. Planck’s criticism on mathematical"harshness" is not relevant for such geometric thermodynamic theories. Fundamental values similar to energyand entropy of thermodynamic systems can be defined and computed in rigorous mathematical forms (14) andone of the main goals of this paper is to show that the Carathéodory approach with Pfaff forms can be natu-rally extended (even in an almost phenomenological manner) to geometric flow models of NES and generalizedgravity systems. We shall show that the Carathéodory method can be extended in such forms that to includein a relativistic form the original constructions with Riemannian metrics and statistical thermodynamic ideas. The axiomatic thermodynamics of Carathéodory is based on the theory of Pfaffian differential equations(fist studied by J. J. Pfaff, who proposed a general method of integrating PDEs of first order in 1814-1815), see [37], for a brief review and references, and Appendix A. We can elaborate on analogous "adiabatic"transformation of an "ideal" gas of evolution flows of relativistic systems { g αβ = [1 q q ( q N )] , N ,f ▽ } defininga solution for respective nonholonomic deformations of Einstein equations (8).In the approximation of ideal gas of NESs, for a nonholonomic Ricci soliton in a τ , we state an equationof thermodynamic state, P b V ( τ ) = ρτ for b V ( τ ) = const, (17)when the normalization (7) is not imposed, and (16), and write d b E = − P d b V ( τ ) = C v dτ = dA . Consideringabove formulas in local form, all values like R, P, C v , etc. depend also on spacetime coordinates u α or onspace like coordinates x ´ ı for cosmological configurations if we do not perform integration as in (14). Usingthe last two equations, we obtain C v τ dτ + ρ b V ( τ ) d b V = 0 . (18)For constant coefficients, such a Pfaff equation is exact (see Appendix A), ∂∂ b V ( C v τ ) | τ = τ = ∂∂τ ( ρ b V ) | τ = τ = 0 , i.e.the Schwarz equation (A.2) is satisfied. For such geometric flow and thermodynamic configurations, there isa solution of (18) as a function φ ( τ , b V ) = Z C v τ dτ | τ = τ + Z ρ b V ( τ ) d b V = const. τ b V γ − = const, for γ = C p /C v and ρ = C p − C v . This is not surprising because all constructions arederived for corresponding approximations in the statistical thermodynamic energy of relativistic Ricci flows b E (14) computed for nonholonomic Ricci soliton configurations. For general geometric flows, the analogous systems became asymmetric which can be characterized by anequation of type δQ = d b E − δA, where Q is a conventional "heat" related to nonholonomic Ricci flow evolution (the term is introduced as forthe usual thermodynamic systems which are not adiabatic). In such cases, not each term of this equation canbe a state function. If we consider the approximation of ideal gas (17) for NESs, we can write δQ = C v dτ + ρτ b V ( τ ) d b V . (19)Because for this Pfaff form the Schwarz condition (A.2) is not satisfied, we conclude that Q is not a statethermodynamic function.The value δA is not a state function because the equations (A.2) are not satisfied for δA = ρτ b V ( τ ) d b V + 0 · dτ .Nevertheless, b E is a state function because the Schwarz relation holds for d b E = C v dτ + 0 · d b V . The Schwartz condition (A.2) fails for δφ = ρτ P dP − ρdτ . Using an integrating factor K = − /P, we satisfythe condition (A.3) which allows to find a solution (with possible total differential) of b V = b V ( τ , P ) = ρτ /P. In a similar manner, we can find an integrating factor τ = τ ( T ) for (19) when the Schwartz condition(A.3) is fulfilled. This allows us to define a new state function, the thermodynamic entropy S ( T ) , when dS ( T ) := δQτ ( T ) = d b E + P d b V τ ( T ) . In the ideal gas approximation for NESs, we obtain an exact differential dS ( T ) := C v τ ( T ) dτ + ρ b V ( T ) d b V .It should be emphasized that the thermodynamic entropy S ( T ) , in general, is different from the statisticalgeometric thermodynamic one b S ( τ ) (14). For certain nonholonomic Ricci soliton configuration and flowevolution of NESs, we can chose such effective values τ ( T ) , C v , ρ etc. in order to have S ( T ) = b S ( τ ) for certain well defined models with respective normalization functions and relativistic causal structures.The values b E ( τ ) and b S ( τ ) are defined by a generalized W-entropy from an axiomatic approach to Ricciflows but extended to relativistic geometric flow. The method with Pfaff forms can be applied to suchstatistical/geometric thermodynamic models (which are different from standard thermodynamic ones) whichallow to state the conditions when generalizations of G. Perelman thermodynamics can be described followingthe Carathéodory axiomatic approach.Finally, we note that using homogeneous Pfaff forms as in [39] we can elaborate on relativistic modelsof Carathéodory–Gibbs–Perelman thermodynamics. In [40], a study of the Bekenstein–Hawking black holethermodynamics [6, 7, 8, 9] in Carathéodory approach was performed (such constructions can be extendedto and black hole/cosmological/ holographic models with conventional horizons). The geometric and statis-tical thermodynamic methods involving G. Perelman W–entropy are more general ones [17, 18, 19, 20, 23]because provide an unified approach to various classes of flow evolution and dynamical field theories when thethermodynamic ideas are not limited to horizon type configurations of some exact and parametric solutions.11 Decoupling and integrability of cosmological solitonic flow equations
The goal of this section is to apply the anholonomic frame deformation method, AFDM, in order to showa general decoupling and integration property of the system of nonlinear PDEs (10) for locally anisotropiccosmological configurations encoding solitonic hierarchies. The geometric/ physical objects for such effectivestatistical thermodynamical systems and corresponding Pfaff equations are determined by generating andintegration functions and (effective) matter sources encoding geometric flow evolutions and solitonic config-urations, see proofs and examples in [56, 57, 58, 59]. Locally anisotropic cosmological solutions in gravitytheories and corresponding inflation and dark matter and dark energy models were studied in [60, 61, 23].
We model geometric evolution of a a ’prime’ cosmological metric, ˚g , into a family ’target’ d-metrics g ( τ ) (3), when the nonholonomic deformations ˚g → g ( τ ) a modelled by η -polarization functions, g ( τ ) = η α ( τ, x k , t )˚ g α e α [ η ] ⊗ e α [ η ] = η i ( τ, x k )˚ g i dx i ⊗ dx i + η a ( τ, x k , t )˚ h a e a [ η ] ⊗ e a [ η ] , e α [ η ] = ( dx i , e a = dy a + η ai ˚ N ai dx i ) . (20)The target N-connection coefficients are parameterized in the form N ai ( τ, x k , t ) = η ai ( τ, x k , t ) ˚ N ai ( τ, x k , t ) . Thevalues η i ( τ ) = η i ( τ, x k ) , η a ( τ ) = η a ( τ, x k , t ) and η ai ( τ ) = η ai ( τ, x k , t ) are gravitational polarization functions,or η -polarizations. Any target d-metric g ( τ ) is defines a solution of the N-adapted Hamilton equations incanonical variables (10), or for relativistic nonholonomic Ricci soliton equations (8) with τ = τ which areequivalent to the canonical nonholonomic deformations of Einstein equations.A cosmological prime metric ˚g = ˚ g αβ ( x i , y a ) du α ⊗ du β is parameterized in a general coordinate form withoff-diagonal N-coefficients and/or represented equivalently in N-adapted form ˚g = ˚ g α ( u ) ˚e α ⊗ ˚e β = ˚ g i ( x ) dx i ⊗ dx i + ˚ g a ( x, y ) ˚e a ⊗ ˚e a , (21)for ˚e α = ( dx i , e a = dy a + ˚ N ai ( u ) dx i ) , and ˚e α = ( ˚e i = ∂/∂y a − ˚ N bi ( u ) ∂/∂y b , e a = ∂/∂y a ) . In general, such a d-metric ˚g ( τ ) =˚ g α ( u ) can be, or not, a cosmological solution of gravitational field equationsin GR but we impose the condition that under geometric evolution it transforms into a target metric (20)which must be an exact or parametric solution. Let us associate a non–stretching curve γ ( τ, l ) on a Einstein manifold V to geometric evolution of a d-metric g ( τ ) , where τ can be identified with the geometric flow parameter of temperature type and l is thearclength of the curve, see details in [53, 58, 59]. Such a curve is characterized by an evolution d–vector Y = γ τ and tangent d–vector X = γ l for which g ( X , X ) =1 . It also swepts out γ ( τ, l ) as a two–dimensionalsurface in T γ ( τ, l ) V ⊂ T V . On such nonholonomic configurations, we consider a coframe e ∈ T ∗ γ V N ⊗ ( h p ⊕ v p ) constructed as a N–adapted ( SO ( n ) ⊕ SO ( m )) –parallel basis along curve γ. For 4-d nonholonomic Lorentz manifolds and theirN-adapted flow evolution models, we consider that n = m = 4 and model the evolution of 4-d Lorentziand-metrics. The labels for respective dimension as n and m can be used in order to distinguish N-adapteddecompositions into h - and v -, or cv -components.Families of canonical d-connections b D ( τ ) can be associated with respective families of linear d-connection1–forms b Γ ( τ ) ∈ T ∗ γ V N ⊗ ( so ( n ) ⊕ so ( m )) . Similar families of 1-forms can be introduced for other types ofd-connections or for a LC-connection. We parameterize frame bases by 1-forms e X ( τ ) = e h X ( τ ) + e v X ( τ ) . Inany point τ and for (1 , −→ ∈ R n , −→ ∈ R n − and (1 , ←− ∈ R m , ←− ∈ R m − , for e h X = γ h X ⌋ h e = " , −→ − (1 , −→ T h , e v X = γ v X ⌋ v e = " , ←− − (1 , ←− T v . n + m splitting, b Γ ( τ ) = hb Γ h X ( τ ) , b Γ v X ( τ ) i , with b Γ h X ( τ ) = γ h X ⌋ b L = " , −→ − (0 , −→ T b L ∈ so ( n + 1) , where b L = (cid:20) −→ v −−→ v T h (cid:21) ∈ so ( n ) , −→ v ∈ R n − , h ∈ so ( n − and b Γ v X ( τ ) = γ v X ⌋ b C = " , ←− − (0 , ←− T b C ∈ so ( m + 1) , where b C = (cid:20) ←− v ) −←− v T v (cid:21) ∈ so ( m ) , ←− v ∈ R m − , v ∈ so ( m − . Using a family of canonical d–connections b D ( τ ) , we can define certain families of N-adapted matriceswhich are decomposed with respect to the flow direction: in the h–direction, e h Y = γ τ ⌋ h e = " (cid:0) h e k , h −→ e ⊥ (cid:1) − (cid:0) h e k , h −→ e ⊥ (cid:1) T h , when e h Y ∈ h p , (cid:0) h e k , h −→ e ⊥ (cid:1) ∈ R n , h −→ e ⊥ ∈ R n − , and b Γ h Y ( τ ) = γ h Y ⌋ b L = " , −→ − (0 , −→ T h̟ τ ∈ so ( n + 1) , where h̟ τ = (cid:20) −→ ̟ −−→ ̟ T h b Θ (cid:21) ∈ so ( n ) , −→ ̟ ∈ R n − , h b Θ ∈ so ( n − . Similar families of geometric objects and parameterizations can be constructed for the v–direction, e v Y = γ τ ⌋ v e = " (cid:0) v e k , v ←− e ⊥ (cid:1) − (cid:0) v e k , v ←− e ⊥ (cid:1) T v , when e v Y ∈ v p , (cid:0) v e k , v ←− e ⊥ (cid:1) ∈ R m , v ←− e ⊥ ∈ R m − , and b Γ v Y = γ v Y ⌋ b C = " , ←− − (0 , ←− T v b ̟ τ ∈ so ( m + 1) , where v̟ τ = (cid:20) ←− ̟ −←− ̟ T v b Θ (cid:21) ∈ so ( m ) , ←− ̟ ∈ R m − , v b Θ ∈ so ( m − . Adapting for general cosmological metrics the results proven in [53, 58, 59] for parameterizations related togeometric flows of 4-d Lorentzian metrics, we formulate such possibilities for generating solitonic hierarchieswith explicit dependence on geometric flow parameter and on a time like coordinate (and, for locally anisotropiccases, on space like coordinates): • The flows locally anisotropic cosmological spaces are convective (travelling wave) maps γ τ = γ l dis-tinguished as ( hγ ) τ = ( hγ ) h X and ( vγ ) τ = ( vγ ) v X . The classification of such maps depend on the typeof cosmological d-metrics and d-connection structures. • There are +1 flows defined as non–stretching mKdV maps describing geometric cosmological flows − ( hγ ) τ = b D h X ( τ, hγ ) h X + 32 | b D h X ( τ, hγ ) h X | h g ( hγ ) h X , − ( vγ ) τ = b D v X ( τ, vγ ) v X + 32 | b D v X ( τ, vγ ) v X | v g ( vγ ) v X , and the +2,... flows as higher order analogs. 13 Finally, the -1 flows are defined by the kernels of families of canonical recursion h–operator, h b R ( τ ) = b D h X ( τ ) (cid:16) b D h X ( τ ) + b D − h X ( τ ) ( −→ v · ) −→ v (cid:17) + −→ v ⌋ b D − h X ( τ ) (cid:16) −→ v ∧ b D h X ( τ ) (cid:17) , and families of canonical recursion v–operator, v b R ( τ ) = b D v X ( τ ) (cid:16) b D v X ( τ ) + b D − v X ( τ ) ( ←− v · ) ←− v (cid:17) + ←− v ⌋ b D − v X ( τ ) (cid:16) ←− v ∧ b D v X ( τ ) (cid:17) , inducing non–stretching maps b D h Y ( τ, hγ ) h X = 0 and b D v Y ( τ, vγ ) v X = 0 .The families of canonical recursion d-operator b R ( τ ) = ( h b R ( τ ) , v b R ( τ )) are respectively related to bi-Hamiltonian structures for families of cosmogical solitonic configurations. Such configurations are charac-terzied also by respective Carathéodory and/or Perelman thermodynamic values with explicit dependence ona temperature parameter and a time like coordinate. The geometric flow evolution of any cosmological d-metric on a Lorentz manifold can be encoded intosolitonic hierarchies. In this work, the geometric cosmological flow evolution is described by exact and para-metric solutions of type g ( τ ) = g ( τ, x i , t ) = [ h g ( τ, x i ) , v g ( τ, x i , t )] , with Killing symmetry on ∂ when inadapted coordinates the coefficients of such d-metrics do not depend on a space like coordinate y . In princi-ple, it is possible to construct more general classes of solutions with dependence on all spacetime coordinatesand a temperature like parameter but general formulas and classification are technically cumbersome andwe omit such consideratons. For certain cosmological configurations, we shall consider d-metrics of type g ( τ ) = g ( τ, t ) = [ h g ( τ ) , v g ( τ, t )] , see Appendix B. Cosmological solitonic waves:
We can consider nonlinear waves ι = ι ( x , x , y = t ) as solutions of solitonic 3-d equations ∂ ι + ǫ∂ ( ∂ ι + 6 ι∂ ι + ∂ ι ) = 0 , ∂ ι + ǫ∂ ( ∂ ι + 6 ι∂ ι + ∂ ι ) = 0 , (22) ∂ ι + ǫ∂ ( ∂ ι + 6 ι∂ ι + ∂ ι ) = 0 , ∂ ι + ǫ∂ ( ∂ ι + 6 ι∂ ι + ∂ ι ) = 0 ,∂ ι + ǫ∂ ( ∂ ι + 6 ι∂ ι + ∂ ι ) = 0 , ∂ ι + ǫ∂ ( ∂ ι + 6 ι∂ ι + ∂ ι ) = 0 , for ǫ = ± . These equations and their solutions can be redefined via frame/coordinate transforms fortemperature-time cosmological generating functions.
Generating nonlinear temperature-time solitonic waves:
Geometric flows of cosmological metrics for a Lorentz manifold can be characterized by 3-d solitonic waveswith explicit dependence flow parameter τ defined by functions ι ( τ, u ) as solutions of such nonlinear PDEs: ι = ι ( τ, x , t ) as a solution of ∂ ττ ι + ǫ∂ [ ∂ ι + 6 ι∂ ι + ∂ ι ] = 0; ι ( x , τ, t ) as a solution of ∂ ι + ǫ∂ [ ∂ τ ι + 6 ι∂ ι + ∂ ι ] = 0; ι ( τ, x , t ) as a solution of ∂ ττ ι + ǫ∂ [ ∂ ι + 6 ι∂ ι + ∂ ι ] = 0; ι ( x , τ, t ) as a solution of ∂ ι + ǫ∂ [ ∂ τ ι + 6 ι∂ ι + ∂ ι ] = 0; ι ( τ, t, x ) as a solution of ∂ ττ ι + ǫ∂ [ ∂ ι + 6 ι∂ ι + ∂ ι ] = 0; ι ( t, τ, x ) as a solution of ∂ ι + ǫ∂ [ ∂ τ ι + 6 ι∂ ι + ∂ ι ] = 0 . (23)Applying general frame/coordinate transforms on respective solutions of equations (23), we construct cosmo-logical solitonic waves parameterized by functions labeled in the form ι = ι ( τ, x i ) , = ι ( τ, x , t ) , or = ι ( τ, x , t ) .
14n a similar form, we can consider other types of cosmologic solitonic configurations determined, forinstance, by sine-Gordon and various types of nonlinear wave configurations characterized by geometric curveflows, see details in [53, 58, 59] and references therein. Any solitonic hierarchy configuration with nonlinearwaves of type ι ( τ, u ) (23) or ι = ι ( x i , t ) (22) can be can be used as generating functions for certain classes ofnonholonomic deformations of cosmological metrics. In this work, d-metrics of type g ( τ ) (3) and/or (20) ared-tensor functionals of type g ( τ ) = g [ ι ( τ, u )] = g [ ι ] =( g i [ ι ] , g a [ ι ]) (24)with polarization functions η i ( τ ) = η i ( τ, x k ) = η i [ ι ] , η a ( τ ) = η a ( τ, x k , y b ) = η a [ ι ] and η ai ( τ ) = η ai ( τ, x k , y b ) = η ai [ ι ] . A functional dependence [ ι ] can be considered for multiple solitonic hierarchies with mixing (for instance,on some different solutions of equations of type (23) and/or (22)). This can be written conventionally in theform [ ι ] = [ ι, ι, ... ] where the left label states the type of solitonic hierarchies. We shall construct in explicitform such cosmological solutions for geometric flows of NES in next section. In this work, we use brief notations of partial derivatives ∂ α q = ∂q/∂u α of a function q ( x k , y a ) , for instance, ∂ q = q • = ∂q/∂x , ∂ q = q ′ = ∂q/∂x , ∂ q = ∂q/∂y = ∂q/∂ϕ = q ⋄ , ∂ q = ∂q/∂t = ∂ t q = q ∗ . Second orderderivatives are written in the form ∂ q = q ⋄⋄ , ∂ = ∂ q/∂t = ∂ tt q = q ∗∗ . Partial derivatives on a flowparameter will be written in the form ∂ τ = ∂/∂τ. The τ -evolution of any d–metric g ( τ ) of type (3), (20) and(24) can be parameterized (using respective frame transforms) for respective local coordinates ( x k , y = t ) and a common geometric flow evolution and/or curve flows temperature like parameter τ,g i ( τ ) = e ψ ( τ,x i ) = e ψ [ ι ] , g a ( τ ) = ω ( τ, x i , y b ) h a ( τ, x i , t ) = ω [ ι ] h a [ ι ] ,N i ( τ ) = n i ( τ, x i , t ) = n i [ ι, ι ] , N i ( τ ) = w i ( τ, x i , t ) = w i [ ι ] . (25)For simplicity, we can consider ω = 1 for a large class of cosmological models with at least one Killingsymmetry, for instance, on ∂ . We can introduce effective sources for geometric flows of NES (10) which by corresponding nonholonomicframe transforms and tetradic (vierbein) fields are parameterized in N–adapted form, eff b Υ µν ( τ ) = e µ ′ µ ( τ ) e ν ′ ν ( τ )[ b Υ µ ′ ν ′ ( τ ) + 12 ∂ τ g µ ′ ν ′ ( τ )] = [ h b Υ ( τ, x k ) δ ij , b Υ ( τ, x k , y c ) δ ab ] . The values h b Υ ( τ ) and b Υ ( τ ) can be taken as functionals of certain solutions of nonlinear solitonic equationsand then considered as generating data for (effective) matter sources and certain forms compatible withsolitonic hierarchies for d-metrics (25). We write b ℑ [ ι ] = eff b Υ µν ( τ ) = [ h b ℑ [ ι ] = h b Υ ( τ, x i ) δ ij , v b ℑ [ ι ] = b Υ ( τ, x i , t ) δ ab ] . (26)There are used "hat" symbols in order to emphasize that such values are considered for systems of nonlinearPDEs involving a canonical d-connection.We can work using canonical nonholonomic variables with functional dependence of d-metrics and pre-scribed effective sources on some cosmological solitonic hierarchies. In such cases, the system of nonholonomicentropic R. Hamilton equations (10) can be written in a formal nonholonomic Ricci soliton form (equivalently,as a nonholonomic deformation of Einstein equtions when the geometric objects depend additionally on atemperature like parameter τ and for effective source (26)), b R αβ [ ι ] = b ℑ αβ [ ι ] . (27)Table 1 (see below) summarize the geometric data on nonholonomic 2+2 variables and correspondingansatz which allow us to transform relativistic geometric flow equations and/or nonholonomic Ricci solitons15nto respective systems of nonlinear ordinary differential equations, ODEs, and partial differential equations,PDEs, determined by cosmological solitonic hierarchies. Table 1: Geometric cosmological solitonic flows and modified Einstein eqs as systems of nonlinear PDEs and the Anholonomic Frame Deformation Method,
AFDM , for constructing generic off-diagonal exact and/or parametric cosmological solutions diagonal ansatz: PDEs → ODE s AFDM:
PDE s with decoupling; generating functions radial coordinates u α = ( r, θ, ϕ, t ) u = ( x, y ) : u α = ( x , x , y , y = t ); flow parameter τ LC-connection ˚ ∇ [connections] N : T V = hT V ⊕ vT V , locally N = { N ai ( x, y ) } canonical connection distortion b D = ∇ + b Z diagonal ansatz g αβ ( u )= ˚ g ˚ g ˚ g ˚ g ˚g ⇔ g ( τ ) g αβ ( τ ) = g αβ ( τ, x i , y a ) general frames / coordinates (cid:20) g ij + N ai N bj h ab N bi h cb N aj h ab h ac (cid:21) , g αβ ( τ ) = [ g ij ( τ ) , h ab ( τ )] , g ( τ ) = g i ( τ, x k ) dx i ⊗ dx i + g a ( τ, x k , y b ) e a ⊗ e b ˚ g αβ = ˚ g α ( t ) for FLRW [coord.frames] g αβ ( τ ) = g αβ ( τ, x i , y = t ) cosmological configurationscoord. transforms e α = e α ′ α ∂ α ′ ,e β = e ββ ′ du β ′ , ˚ g αβ = ˚ g α ′ β ′ e α ′ α e β ′ β ˚g α ( x k , y a ) → ˚ g α ( t ) , ˚ N ai ( x k , y a ) → . [N-adapt. fr.] (cid:26) g i ( τ, x i ) = g i [ ι ] , g a ( τ, x i , t ) = g a [ ι ] , d-metrics N i ( τ ) = w i [ ι ] , N i = n i [ ι, ι ] , N-connections ˚ ∇ , Ric = { ˚ R βγ } Ricci tensors b D , b R ic = { b R βγ } m L [ φ ] → m T αβ [ φ ] sources b ℑ [ ι ] = b Υ µν ( τ ) = e µµ ′ e ν ′ ν b Υ µ ′ ν ′ = [ h b ℑ [ ι ] δ ij , b ℑ [ ι ] δ ab ] , cosmological conf.trivial equations for ˚ ∇ -torsion LC-conditions b D | b T → = ∇ extracting new classes of solutions in GR In this paper, we study physically important cases when ˚g defines a cosmological metric in GR. Fordiagonalizable via coordinate transforms prime metrics, we can always find a coordinate system when ˚ N bi =0 . Non-singular nonholonomic deformations can be constructed for cosmological solutions with nontrivialfunctions η α = ( η i , η a ) , η ai , and nonzero coefficients ˚ N bi ( u ) . For a d-metric (20), we can analyze the conditionsof existence and properties of some target and/or prime cosmological solutions with solitonic waves andstructure formation when, for instance, η α → and N ai → ˚ N ai . The values η α = 1 and/or ˚ N ai = 0 can beimposed as some special nonholonomic constraints on temperature-time cosmological flows. In this subsection, we prove that the system of nonlinear PDEs (27) describing geometric flow evolutionof cosmological NES encoding solitonic hierarchies can be decoupled in general form.
We can chose certain systems of reference/ coordinates when coefficients of a d-metric (25) and derivedgeometric objects like the canonical d-connection and corresponding curvature and torsion d-tensors do notdepend on y = t with respect to a class of N-adapted frames. Using a d-metric ansatz with ω = 1 and asource b ℑ [ ι ] = [ h b ℑ [ ι ] , v b ℑ [ ι ]] (26) , we compute the coefficients of the Ricci d-tenso and write the geometricflow modified Einstein equations (27) in the form b R [ ι ] = b R [ ι ] = − h b ℑ [ ι ] i.e. g • g • g + ( g • ) g − g •• + g ′ g ′ g + ( g ′ ) g − g ′′ = − g g h b ℑ ; (28) b R [ ι ] = b R [ ι ] = − v b ℑ [ ι ] i.e. ( h ∗ ) h + h ∗ h ∗ h − h ∗∗ = − h h v b ℑ ; (29) b R k ( τ ) = h h n ∗∗ k + ( 32 h ∗ − h h h ∗ ) n ∗ k h = 0; (30) b R k ( τ ) = − w k "(cid:18) h ∗ h (cid:19) + h ∗ h h ∗ h − h ∗∗ h + h ∗ h ( ∂ k h h + ∂ k h h ) − ∂ k h ∗ h = 0 . (31)16et us explain the decoupling property of this system of nonlinear PDEs: From equation (28), we can find g (or, inversely, g ) for any prescribed functional of solitonic hierarchies encoded into a h-source h b ℑ [ ι ] andany given coefficient g ( τ, x i ) = g [ ι ] (or, inversely, g ( τ, x i ) = g [ ι ] ) when the solitonic hierarchies for thecoefficients of a h-metric in such coordinates depend on a temperature like parameter but not on time likecoordinates. We can integrate on time like coordinate y = t in (29) and define h ( τ, x i , t ) as a solution offirst order PDE for any prescribed v-source v b ℑ [ ι ] and given coefficient h ( τ, x i , t ) = h [ ι ] . Inversely, itis possible to define h ( τ, x i , t ) if h ( τ, x i , t ) = h [ ι ] is given for such solutions we have to solve a secondorder PDE. The coefficients of v-metrics involve, in general, different types of solitonic hierarchies even theyare related via corresponding formulas to another classes of solitonic hierarchies prescribed for the effectivev-source. We have to integrate two times on t in (30) in order to compute n k ( τ, x i , t ) = n k [ ι ] for any defined h and h . Introducing certain values of h and h in equations (31), we obtain a system of algebraic linearequations for w k ( τ, x i , t ) = w k [ ι ] . Here it should be also emphasized that the cosmological solitonic hierarchiesencoded in the coefficients of a N-connection are different (in general) from those encoded in the coefficientsof d-metric and nontrivial effective sources.Using the decoupling property of nonlinear off-diagoan cosmological solitonic systems (28)–(31), we canintegrate such PDEs step by step by prescribing respectively the effective sources, the h-coefficients, g i , andv–coefficients, h a , for geometric flowd of d-metrics. The geometric evolution of such solutions involves aprescribed nonholonomic constraint on ∂ τ g µ ′ ν ′ ( τ ) included in b ℑ [ ι ] . Let us define the the coefficients α i = h ∗ ∂ i ̟, β = h ∗ ̟ ∗ , γ = (cid:0) ln | h | / / | h | (cid:1) ∗ , where ̟ = ln | h ∗ / p | h h || (32)for nonsingular values for h ∗ a = 0 and ∂ t ̟ = 0 (we have to elaborate other methods if such conditions are notsatisfied), we transform the system of solitonic nonlinear PDEs (28)–(31) into ψ •• + ψ ′′ = 2 h b ℑ [ ι ] , ̟ ∗ h ∗ = 2 h h v b ℑ [ ι ] , n ∗∗ k + γn ∗ k = 0 , βw i − α i = 0 . (33)Such a system can be integrated in explicit and general forms (depending on the type of parameterizatons, seedetails in [53, 17] and some examples of cosmological solutions will be provided in next sections) if there areprescribed a generating function Ψ( τ ) = Ψ( τ, x i , t ) = Ψ[ ι ] := e ̟ and generating sources h b ℑ and v b ℑ . Here,we notat that Levi-Civita, LC, conditions for extracting cosmological solitonic solution with zero torsion, canbe transformed into a system of 1st order PDEs, ( ∂ i − w i ∂ t ) ln p | h | = 0 , ∂ t w i = ( ∂ i − w i ∂ t ) ln p | h | , ∂ t n i = 0 , ∂ i n k = ∂ k n i , ∂ k w i = ∂ i w k , (34)which are considered as additional constraints on off-diagonal coefficients of metrics of type (25).To generate exact and parametric solutions we have to solve a system of two equations for ̟ in (32) and(33) involving four functions ( h , h , v b ℑ , and Ψ) . We can check by corresponding computations that there isan important nonlinear symmetry which allows to redefine the generating function and the effective source andto introduce a family of effective cosmological constants Λ( τ ) = 0 , Λ( τ ) = const, not depending on spacetimecoordinates u α . Such nonlinear transforms (Ψ( τ ) , v b ℑ ( τ )) ⇐⇒ (Φ( τ ) , Λ( τ )) are defined by formulas Λ( Ψ [ ι ]) ∗ = | v b ℑ [ ι ] | (Φ [ ι ]) ∗ , or Λ Ψ [ ι ] = Φ [ ι ] | v b ℑ [ ι ] | − Z dt Φ [ ι ] | v b ℑ [ ι ] | ∗ (35)and allow us to introduce families of new generating functions Φ( τ, x i , t ) = Φ[ ι ] and families of (effective)cosmological constants . The families of constants Λ( τ ) can be chosen for certain physical models and thegeometric/physical data for Φ encode nonlinear symmetries both for the generating functions and sources and17espective cosmological solitonic hierarchies for v b ℑ , and Ψ . In result of nonlinear symmetries, we can describenonlinear systems of PDEs by two equivalent sets of generating data (Ψ , Υ) or (Φ , Λ) . But such symmetriesof solitonic cosmologia hierarchies are encoded into functionals with respective partial derivations ∂ t and/orintegration on dt. To generate certain classes of solutions, we can work with effective cosmological constantsbut for other ones we have to consider generating sources. Finaly, we note that modules in formulas (35)should be chosen in certain forms resulting in physically motivated nonlinear symmetries, relativistic causalmodels and thermodynamic values which are compatible with observational data in modern cosmology.
We study properties of some classes of generic off-diagonal cosmological solutions of (27) determined bygenerated functions and sources with solitonic hierarchies.
By straightforward computations we can verify that the system (28)–(31) represented in the form (33)(see similar details in [62, 61]) can be solved by if the cofficients of a d–metric and N–connection are computed g i ( τ ) = e ψ ( τ,x k ) as a solution of 2-d Poisson eqs. ψ •• + ψ ′′ = 2 h b ℑ [ ι ]; g [ ι ] = h ( τ, x i , t ) = h [0]3 ( τ, x k ) − Z dt (Ψ ) ∗ v b ℑ = h [0]3 ( τ, x k ) − Φ / τ ); (36) g [ ι ] = h ( τ, x i , t ) = − (Ψ ∗ [ ι ]) v b ℑ [ ι ]) h [ ι ] = − (Ψ ∗ ) v b ℑ ) (cid:16) h [0]3 ( τ, x k ) − R dt (Ψ ) ∗ / v b ℑ (cid:17) = − (Φ )(Φ ) ∗ h | Λ( τ ) R dt v b ℑ [Φ ] ∗ | = − [(Φ ) ∗ ] h [0]3 ( τ, x k ) − Φ / τ )] | R dt v b ℑ (Φ ) ∗ | ; N k [ ι ] = n k ( τ, x i , t ) = n k ( τ, x i ) + n k ( τ, x i ) Z dt (Ψ ∗ ) v b ℑ | h [0]3 ( τ, x i ) − R dt (Ψ ) ∗ / v b ℑ| / = n k ( τ, x i ) + n k ( τ, x i ) Z dt (Φ ∗ ) | Λ( τ ) R dt v b ℑ (Φ ) ∗ || h | / ; N i [ ι ] = w i ( τ, x i , t ) = ∂ i ΨΨ ∗ = ∂ i Ψ (Ψ ) ∗ = ∂ i [ R dt v b ℑ (Φ ) ∗ ] v b ℑ (Φ ) ∗ . In these formulas, there are considered different sets of solitonic hierarchies and respective integration func-tions h [0]3 ( τ, x k ) , n k ( τ, x i ) , and n k ( τ, x i ) encoding (non) commutative parameters and integration constantsbut also nonlinear geometric flow scenarios on τ and cosmological evolution. These data and symmetriesof solitonic hierarchies for generating geometric evolution data (Ψ , Υ) , or (Φ , Λ) , (all related by nonlineardifferential / integral transforms (35)) can be prescribed in explicit form following certain topology/ symme-try / asymptotic conditions and compotability with observational data. The coefficients (36) define genericoff-diagonal cosmological solitonic solutions with associated bi Hamilton structures if the corresponding an-holonomy coefficients are not trivial. In general, such geometric flow cosmological solutions are with nontrivialnonholonomically induced d-torsions and solitonic hierarchies determined by evolution of N-adapted coeffi-cients of d-metric structures. We can impose additional nonholonomic constraints (34) in order to extractLC-configurations for cosmological metrics under geometric flow evolution.18 .3.2 Quadratic line elements for off-diagonal cosmological solitonic hierarchies Instead of Ψ and/or Φ , we can consider as a generating function any coefficient h [ ι ] = h [0]3 − Φ / , h ∗ ( τ ) = 0 , and write formulas Φ ( τ ) = 4Λ (cid:16) h [ ι ] − h [0]3 (cid:17) , (Φ ) ∗ = 4Λ( h ) ∗ and (Φ ∗ ) = Λ( h ) ∗ ( h h [0]3 − . Using nonlinear symmetries (35), we find (Ψ ) ∗ = 4 (cid:12)(cid:12)(cid:12) v b ℑ [ ι ] (cid:12)(cid:12)(cid:12) ( h ) ∗ and Ψ = 4 (cid:12)(cid:12)(cid:12) v b ℑ (cid:12)(cid:12)(cid:12) h − Z dt (cid:12)(cid:12)(cid:12) v b ℑ (cid:12)(cid:12)(cid:12) ∗ h . Such formulas determine corresponding functionals Ψ[ v b ℑ , h , h [0]3 ] and Φ[Λ , h , h [0]3 ] . Introducing these valuesinto respective formulas for h a , N bi and v b ℑ in (36) and expressing the generating functions and the d–metric(20) with cosmological data (25) in terms of h , for respective integration functions and effective sources forgeometric evolution, we compute g [ ι ] = h ( τ, x i , t ) = h [0]3 ( τ, x k ) − Z dt (Ψ ) ∗ v b ℑ = h [0]3 ( τ, x k ) − Φ / τ ); g [ ι ] = h ( τ, x i , t ) = − (Φ )(Φ ) ∗ h | Λ( τ ) R dt v b ℑ (Φ ) ∗ | = 4 | ( h ) ∗ || R dt v b ℑ ( h ) ∗ | ; N k [ ι ] = n k ( τ, x i , t ) = n k ( τ, x i ) + n k ( τ, x i ) Z dt (Φ ∗ ) | Λ( τ ) R dt v b ℑ (Φ ) ∗ || h | / == n k ( τ, x i ) + e n k ( τ, x i ) Z dt ( h ) ∗ (1 − h /h [0]3 ) | Λ R dt v b ℑ ( h ) ∗ || h | / ; N i [ ι ] = w i ( τ, x i , t ) = ∂ i Ψ(Ψ) ∗ = ∂ i Ψ (Ψ ) ∗ = ∂ i [ R dt v b ℑ (Φ ) ∗ ] v b ℑ (Φ ) ∗ = ∂ i (cid:16)(cid:12)(cid:12)(cid:12) v b ℑ (cid:12)(cid:12)(cid:12) h − R dt (cid:12)(cid:12)(cid:12) v b ℑ (cid:12)(cid:12)(cid:12) ∗ h (cid:17)(cid:12)(cid:12)(cid:12) v b ℑ [ ι ] (cid:12)(cid:12)(cid:12) h ∗ . In result, we can express the quadratic line element corresponding to this class of cosmological flow solutionsin three equivalent forms: ds = e ψ ( τ,x k ) [( dx ) + ( dx ) ] (37) − h [ dy + ( n k ( τ, x i ) + e n k ( τ, x i ) R dt ( h ) ∗ (1 − h /h [0]3 ) | Λ R dt v b ℑ ( h ) ∗ || h | / ) dx k ] − | ( h ) ∗ || R dt v b ℑ ( h ) ∗ | [ dt + ∂ i ( | v b ℑ | h − R dt | v b ℑ | ∗ h ) | v b ℑ [ ι ] | h ∗ dx i ] , or gener. funct. h , source v b ℑ , or Λ; − [ dy + ( n k + n k R dt (Ψ ∗ ) v b ℑ ) | h [0]3 − R dt (Ψ2) ∗ v b ℑ | / ) dx k ]( h [0]3 − R dt (Ψ ) ∗ v b ℑ ) + (Ψ ) ∗ v b ℑ ) ( h [0]3 − R dt (Ψ2) ∗ v b ℑ ) [ dt + ∂ i Ψ ∂ Ψ dx i ] or gener. funct. Ψ , source v b ℑ ; − [ dy + ( n k + n k R dt [(Φ ) ∗ ] | R dt v b ℑ [(Φ) ] ∗ | | h [0]3 − Φ | − / ) dx k ]( h [0]3 − Φ ) − [(Φ ) ∗ ] | Λ R dt v b ℑ [(Φ) ] ∗ | ( h [0]3 − Φ24Λ ) [ dt + ∂ i [ R dt v b ℑ (Φ ) ∗ ] v b ℑ (Φ ) ∗ dx i ] , gener. funct. Φ effective Λ for v b ℑ . Formulas (36) and (37) encode cosmological solitonic hierarchies determined by generating functions. Agenerating source v b ℑ and effective cosmological constant Λ do not involve (in general) any solitonic behaviour.Nonlinear symmetries (35) mix different cosmological solitonic structures of generating functions and anycosmological functional for sources. 19 .3.3 Off-diagonal Levi-Civita cosmological solitonic hierarchies We can solve the equations (34) for zero torsion conditions considering special classes of generating func-tions and sources. For instance, we prescribe a Ψ( τ ) = ˇΨ( τ, x i , t ) for which ( ∂ i ˇΨ) ∗ = ∂ i ( ˇΨ ∗ ) and fix a v b ℑ ( τ, x i , t ) = v b ℑ [ ˇΨ] = v ˇ ℑ ( τ ) , or v b ℑ = const. The nonlinear symmetries (35) transforms into
Λ ˇΨ = ˇΦ | v b ℑ| − Z dt Φ | v b ℑ| ∗ and ˇΦ = − h ( τ, x i , t ) , ˇΨ = Z dt v b ℑ ˇ h ∗ . In the second case, the coefficient h ( τ ) = ˇ h ( τ, x i , t ) can be considered also as generating function when h andthe N-connection coefficients are computed using corresponding formulas an certain nonlinear symmetries andnonholonomic constraints. To generate zero torsion cosmologic solitonic hierarchies, we find some functions ˇ A ( τ ) = ˇ A ( τ, x i , t ) and n ( τ ) = n ( τ, x i ) when the coefficients of N-connection are n k ( τ ) = ˇ n k ( τ ) = ∂ k n ( τ, x i ) and w i ( τ ) = ∂ i ˇ A ( τ ) = ∂ i ( R dt v ˇ ℑ ˇ h ∗ ]) v ˇ ℑ ˇ h ∗ = ∂ i ˇΨˇΨ ∗ = ∂ i [ R dt v ˇ ℑ ( ˇΦ ) ∗ ] v ˇ ℑ ( ˇΦ ) ∗ . In result, the quadratic line elements for new classes of off-diagonal zero torsion locally anisotropic cosmologicsolutions encoding solitonic hierarchies and defined as subclasses of solutions (37), ds = e ψ ( τ,x k ) [( dx ) + ( dx ) ] − ˇ h (cid:2) dy + ( ∂ k n ) dx k (cid:3) + (ˇ h ∗ ) | R dt v ˇ ℑ ˇ h ∗ | ˇ h [ dt + ( ∂ i ˇ A ) dx i ] , or gener. funct. ˇ h , source v ˇ ℑ , or Λ;( h [0]3 − R dt ( ˇΨ ) ∗ v ˇ ℑ ) (cid:2) dy + ( ∂ k n ) dx k (cid:3) + ( ˇΨ ) ∗ v ˇ ℑ ) ( h [0]3 − R dt (ˇΨ2) ∗ v ˇ ℑ ) [ dt + ( ∂ i ˇ A ) dx i ] or gener. funct. ˇΨ , source v ˇ ℑ ;( h [0]3 − ˇΦ ) (cid:2) dy + ( ∂ k n ) dx k (cid:3) + [(ˇΦ ) ∗ ] | Λ R dt v ˇ ℑ (ˇΦ ) ∗ | ( h [0]3 − ˇΦ24Λ ) [ dt + ( ∂ i ˇ A ) dx i ] , gener. funct. ˇΦ effective Λ for v ˇ ℑ . (38) For any value of flow parameter τ, such cosmological solitonic metrics are generic off-diagonal and define newclasses of solutions which are different, for instance, from the FLRW metric. We may check if the anholonomycoefficients C γαβ = { C bia = ∂ a N bi , C aji = e j N ai − e i N aj } are not zero for solitonic values of N i = ∂ i ˇ A and N k = ∂ k n and conclude if certain metrics are or not generic off-diagonal. We can study certain nonholonomiccosmologic solitonic configurations determined, for instance, by data ( v ˇ ℑ , ˇΨ , h [0]3 , ˇ n k ) , with w i = ∂ i ˇ A → and ∂ k n → , see Appendix B. We consider the coefficient h ( τ ) = h ( τ, x i , t ) in (37) as a generating function. Such a value can bedetermined by a family of solitonic hierarchies, h ( τ ) = h [ ι ] with explicit dependence on a time likecoordinate t. We can perform a deformation procedure for constructing a class of off–diagonal solutions withKilling symmetry on ∂ determined by cosmolgical solitonic hierarchies b ℑ [ ι ] = [ h b ℑ [ ι ] , v b ℑ [ ι ]] (26) and aparametric running cosmological constant Λ( τ ) ,ds = e ψ ( τ,x k ) [( dx ) + ( dx ) ] + h ( τ )[ dy + ( n k + 4 n k Z dt ( h ∗ ( τ )) | R dt v b ℑ ( τ )( h ∗ ( τ )) | ( h ( τ )) / ) dx k ] − [ h ∗ ( τ )] | R dt v b ℑ ( τ )( h ∗ ( τ )) | h [ dt + ∂ i ( R dt v b ℑ ( τ ) h ∗ ]) v b ℑ ( τ ) h ∗ ( τ ) dx i ] . Such classes of cosmological solutions involve different types of solitonic hierarchies and, in general, arewith nontrivial nonholonomically induced torsion (it is possible to impose nonholonomic constraints to LC-configurations (38)). 20 able 2: Off-diagonal cosmological flows with solitonic hierarchies
Exact solutions of b R µν ( τ ) = b ℑ µν ( τ ) (27) transformed into a system of nonlinear PDEs (28)-(31)d-metric ansatz withKilling symmetry ∂ ds = g i ( τ )( dx i ) + g a ( τ )( dy a + N ai ( τ ) dx i ) , for g i = e ψ ( τ,x i ) , g a = h a ( τ, x i , t ) , N i = n i ( τ, x i , t ) , N i = w i ( τ, x i , t ) , y = t ; Effective matter sources b ℑ µν ( τ ) = [ h b ℑ ( τ, x i ) δ ij , v b ℑ ( τ, x i , t ) δ ab ]; ∂ q = q • , ∂ q = q ′ , ∂ q = q ⋄ , ∂ q = q ∗ Nonlinear PDEs (33) ψ •• + ψ ′′ = 2 h b ℑ [ ι ]; ̟ ∗ h ∗ = 2 h h v b ℑ [ ι ]; n ∗∗ k + γn ∗ k = 0; βw i − α i = 0; for ̟ = ln | h ∗ / p | h h || ,α i = h ∗ ( ∂ i ̟ ) , β = h ∗ ̟ ∗ ,γ = (cid:0) ln | h | / / | h | (cid:1) ∗ , Generating functions: h [ ι ] , Ψ( τ, x i , t ) = e ̟ , Φ[ ι ]; integration functions: h [0]3 ( τ, x k ) , n k ( τ, x i ) , n k ( τ, x i ) (Ψ ) ∗ = − R dt v b ℑ h ∗ , Φ = − τ ) h , see nonlinear symmetries (35); h ( τ ) = h [0]3 − Φ / τ ) , h ∗ = 0 , Λ( τ ) = 0 = const Off-diag. solutions, d–metricN-connec. g i ( τ ) = e ψ ( τ,x k ) as a solution of 2-d Poisson eqs. ψ •• + ψ ′′ = 2 h b ℑ ( τ ); h ( τ ) = h [0]3 − R dt (Ψ ) ∗ / v b ℑ = h [0]3 − Φ / τ ); h ( τ ) = − (Ψ ∗ ) / v b ℑ ) h , see (36); n k ( τ ) = n k + n k R dt (Ψ ∗ ) / ( v b ℑ ) | h [0]3 − R dt (Ψ ) ∗ / v b ℑ| / ; w i ( τ ) = ∂ i Ψ / Ψ ∗ = ∂ i Ψ / | (Ψ ) ∗ | . LC-configurations (34) w ∗ i = ( ∂ i − w i ∂ t ) ln p | h ( τ ) | , ( ∂ i − w i ∂ t ) ln p | h ( τ ) | = 0 ,∂ k w i ( τ ) = ∂ i w k ( τ ) , n ∗ i ( τ ) = 0 , ∂ i n k ( τ ) = ∂ k n i ( τ ); Ψ = ˇΨ[ ι ] , ( ∂ i ˇΨ) ∗ = ∂ i ( ˇΨ ∗ ) and v b ℑ ( τ, x i , t ) = v b ℑ [ ˇΨ] = v ˇ ℑ , or v b ℑ = const. N-connections, zero torsion n k ( τ ) = ˇ n k ( τ ) = ∂ k n ( τ, x i ) and w i ( τ ) = ∂ i ˇ A ( τ ) = ∂ i ( R dt ˇ ℑ ˇ h ∗ ]) / ˇ ℑ ˇ h ∗ ; ∂ i ˇΨ / ˇΨ ∗ ; ∂ i ( R dt ˇ ℑ (ˇΦ ) ∗ ) / (ˇΦ) ∗ ˇ ℑ ; . polarization functions ˚g → b g =[ g α = η α ˚ g α , η ai ˚ N ai ] ds = η [ ι ]˚ g ( x i )[ dx ] + η [ ι ]˚ g ( x i )[ dx ] + η [ ι ]˚ g ( x i )[ dy + η i [ ι ] ˚ N i ( x i ) dx i ] + η [ ι ]˚ g ( x i )[ dt + η i [ ι ] ˚ N i ( x k ) dx i ] , Prime metric for a cosm.sol. [˚ g i ( x i ) , ˚ g a = ˚ h a ( x i ); ˚ N k = ˚ n k ( x i )] , ˚ N k = ˚ w k ( x i ) , diagonalizable by frame/ coordinate transforms.Example of a prime metric ˚ g = ˚ a ( ς ) , ˚ g = ˚ a ( ς ) , ˚ h = ˚ a ( ς ) , ˚ h = − , ς = ς ( t ) a FLRW or Biachi type solution ; Solutions for polarization funct. η i ( τ ) = e ψ ( τ,x k ) / ˚ g i ; η ˚ h = − | η ˚ h | / ) ⋄ ] | R dt v b ℑ [( η ˚ h )] ∗ | ; η ( τ ) = η ( τ, x i , t ) = η [ ι ] as a generating function ; η k ( τ ) ˚ N k = n k + 16 n k R dt (cid:16) [( η ˚ h ) − / ] ∗ (cid:17) | R dt v b ℑ [( η ˚ h )] ∗ | ; η i ( τ ) ˚ N i = ∂ i R dt v b ℑ ( η ˚ h ) ∗ v b ℑ ( η ˚ h ) ∗ Polariz. funct. with zero torsion η i ( τ ) = e ψ ( τ,x k ) / ˚ g i ; η = ˇ η ( τ, x i , t ) as a generating function ; η ( τ ) = − | η ˚ h | / ) ∗ ] ˚ g | R dt v ˇ ℑ [(ˇ η ˚ h )] ∗ | ; η k ( τ ) = ∂ k n ˚ n k , η i ( τ ) = ∂ i ˇ A ˚ w k , In this section, we consider applications of the anholonomic frame deformation method (AFDM, outlinedin Tables 1 and 2) for constructing in explicit form exact and parametric off-diagonal cosmological solutionsdescribing solitonic geometric flows.
There will be studied solutions of modified Einstein equations (27) transformed into systems of nonlinearPDEs with decoupling (33).
Let us introduce such conventions: We shall write that int b ℑ ( τ ) = intv b ℑ [ ι ] (i.e. put a left label "0") fora geometric flow source v b ℑ ( τ ) if it contains a term in eff b Υ µν ( τ ) (26) defined as a source functional ona solitonic hierarchy [ ι ] . If it will be written intv b ℑ ( τ ) without a left label "0", we shall consider that such21 term corresponds to a general (effective) eff b Υ µν ( τ ) (not prescribing any solitonic configurations) encodingcontributions from a distortion tensor b Z (2). An effective source term flv b ℑ determined by geometric flows(with left label "fl") of the d-metric, ∂ τ g α ′ ( τ ) , in (26) is contained. It is of cosmological solitonic character ifthe d-metric coefficients are also cosmological solitons. We can consider cosmological solitonic hierarchies forRicci soliton configurations, i.e. nonholonomic Einstein systems, with flv b ℑ = 0 . For this class of solutions, we consider a source (26) with a left label a is used for "additive functionals" av b ℑ ( τ ) = av b ℑ [ ι ] = av b ℑ ( τ, x i , t ) = flv b ℑ [ ι ] + intv b ℑ [ ι ] + intv b ℑ [ ι ] . (39)In such source functionals, it is considered that we prescribe an effective cosmological solitonic hierarchy formatter fields even, in general, such gravitational and matter field interactions can be of non–solitonic type.The second equation (33) with cosmological source v b ℑ [ ι ] = av b ℑ [ ι ] can be integrated on time like coordinate y = t. This allows us to construct off-diagonal cosmological metrics and generalized connections encodingsolitonic hierarchies determined by a generating function h ( τ, x i , t ) with Killing symmetry on ∂ , by effectivesources a b ℑ [ ι ] = ( ah b ℑ [ ι ] , av b ℑ [ ι ]) and an effective cosmological constant a Λ( τ ) = fl Λ( τ ) + m Λ( τ ) + int Λ( τ ) . (40)This constant is related to av b ℑ [ ι ] (39) via nonlinear symmetry transforms (35).Following the AFDM summarized in Table 2, we construct such a class of quadratic line elements forgeneric off-diagonal cosmological solutions determined by effective sources encoding solitonic hierarchies, ds = e ψ [ ι ] [( dx ) + ( dx ) ] − [ h ∗ ( τ )] | R dt av b ℑ [ ι ] h ∗ ( τ ) | h ( τ ) " dt + ∂ i ( R dt av b ℑ [ ι ] h ∗ ( τ )) av b ℑ [ ι ] h ∗ ( τ ) dx i + h ( τ ) " dy + n k ( τ ) + 4 n k ( τ ) Z dt [ h ∗ ( τ )] | R dt av b ℑ [ ι ] h ∗ ( τ ) | [ h ( τ )] / ! dx k . (41)Such solutions can be nonholonomically constrained in order to extract LC-configurations. The formulas (41)can be re-defined equivalently in terms of generating functions Ψ( τ, x i , t ) or Φ( τ, x i , t ) which can be of anon–solitonic character. Another class of cosmological NESs can be generated as (off-) diagonal cosmological solutions using gen-erating functionals encoding cosmological solitonic hierarchies Φ( τ ) = Φ[ i ] . Such functionals are subjected tononlinear symmetries of type (35) and general effective sources v b ℑ ( τ ) which can be of non-solitonic character.The second equation into (33) transforms into ̟ ∗ ( τ )[ Φ[ i ] , Λ( τ )] h ∗ ( τ )[Φ[ i ] , Λ( τ )] = 2 h ( τ )[Φ[ i ] , Λ( τ )] h ( τ )[Φ[ i ] , Λ( τ )] v b ℑ ( τ ) . This equation can be solved together with other equations (28)-(31) following the AFDM, see Table 2.The solutions for such cosmological configurations determined by general nonlinear functionals for gener-ating functions can be written in all forms (37). We present here the quadratic line element correspondingonly to solutions of third type parametrization ds = e ψ ( τ,x k ) [( dx ) + ( dx ) ] + ( h [0]3 ( τ, x k ) − (Φ[ i ]) τ ) ) (42) [ dy + ( n k ( τ, x k ) + n k ( τ, x k ) Z dt [(Φ[ i ]) ] ∗ | Λ( τ ) R dt v b ℑ ( τ ) [(Φ[ i ]) ] ∗ | | h [0]3 ( τ, x k ) − (Φ[ i ]) τ ) | − / ) dx k ] − (Φ[ i ]) [(Φ[ i ]) ] ∗ | Λ( τ ) R dt v b ℑ ( τ )[(Φ[ i ]) ] ∗ | ( h [0]3 − (Φ[ i ]) τ ) ) [ dt + ∂ i (cid:16)R dt v b ℑ ( τ )[(Φ[ i ]) ] ∗ (cid:17) v b ℑ ( τ ) [(Φ[ i ]) ] ∗ dx i ] . ds = e ψ ( τ ) [( dx ) + ( dx ) ] + ( h [0]3 ( τ, x k ) − ( ˇΦ[ i ]) τ ) ) h dy + ( ∂ k n ( τ )) dx k i − ( ˇΦ[ i ]) [( ˇΦ[ i ]) ] ∗ | Λ( τ ) R dt v b ℑ ( τ )[( ˇΦ[ i ]) ] ∗ | ( h [0]3 − (ˇΦ[ i ]) τ ) ) [ dt + ( ∂ i ˇ A ( τ )) dx i ] , (43)where ˇ A ( τ ) and n ( τ ) are also generating functions. Dualizing such solutions and their symmetries, we cangenerate stationary configurations. Let us analyze some classes of cosmological solitonic flow solutions with small parametric deformations for awell-known cosmological metric in GR (for instance, a FLRW or Bianchi one as in Appendix B) ˚g = [˚ g i , ˚ g a , ˚ N jb ] (21) when ∂ ˚ g = ˚ g ∗ = 0 . We formulate a geometric formalism for small generic off–diagonal parametricdeformations of ˚g into certain target cosmological solitonic metrics of type g (20) when ds = η i ( ε, τ )˚ g i ( dx i ) + η a ( ε, τ )˚ g a ( e a ) , (44) e = dy + n η i ( ε, τ )˚ n i dx i , e = dt + w η i ( ε, τ )˚ w i dx i . The coefficients [ g α = η α ˚ g α , w η i ˚ w i , n η i n i ] in these formulas depend on a small parameter ε, ≤ ε ≪ , onevolution parameter τ and on coordinates x i and t. We suppose that a family of (44) define a solution ofcosmological solitonic flow equations described by a system of nonlinear PDEs with decoupling (33). The ε -deformations are parameterised in the form η i ( ε, τ ) = 1 + ευ i ( τ, x k ) , η a = 1 + ευ a ( τ, x k , t ) for the coefficients of d-metrics ; (45) w η i ( ε, τ ) = 1 + ε w υ i ( τ, x k , t ) , n η i ( τ, x k , t ) = 1 + ε n υ i ( τ, x k , t ) for the coefficients of N-connection , where a generating function can be given by g ( τ ) = η ( τ )˚ g = η ( τ, x i , t )˚ g ( τ, x i , t ) = [1 + ευ ( τ, x i , t )]˚ g , for υ = υ ( τ, x i , t ) . The deformations of h -components of a prime cosmological d-metric are ε g i = ˚ g i (1 + ευ i ) = e ψ ( τ,x k ) for asolution of the 2-d Laplace equation in (33). For parameterizations ψ ( τ ) = ψ ( τ, x k ) + ε ψ ( τ, x k ) and h b ℑ ( τ )( τ ) = h b ℑ ( τ, x k ) + ε h b ℑ ( τ, x k ) , we compute the deformation polarization functions in the form υ i = e ψ ψ/ ˚ g i h b ℑ . The horizontal generatingand source functions are solutions of ψ •• + ψ ′′ = h b ℑ and ψ •• + ψ ′′ = h b ℑ . Using ε -decompositions(45) and similar formulas for v -components, we compute ε -decomposition of the target cosmological solitonicd-metric and N-connection coefficients ˚ g i η i ( τ ) = e ψ ( τ,x k ) as a solution of 2-d Poisson equations ε g i ( τ ) = [1 + εe ψ ψ/ ˚ g i h b ℑ ]˚ g i , also constructed as a solution of 2-d Poisson equations for ψ ˚ g η ( τ ) = − | η ( τ )˚ g | / ) ∗ ] | R dt v b ℑ ( τ )[ η ( τ )˚ g ] ∗ | i.e. ε g ( τ ) = [1 + ε υ ]˚ g for υ ( τ, x i , t ) = 2 ( υ ˚ g ) ∗ ˚ g ∗ − R dt v b ℑ ( τ )( υ ˚ g ) ∗ R dt v b ℑ ( τ )˚ g ∗ . For such formulas, the system of coordinates is chosen in such a form that there are satisfied the condition (˚ g ∗ ) = ˚ g | R dt v b ℑ ( τ )˚ g ∗ | , which allow to find ˚ g for any prescribed values ˚ g and v b ℑ ( τ ) . ε -deformations of N-connection coefficients are computed η k ( τ )˚ n k = n k ( τ ) + 16 n k ( τ ) Z dt (cid:0) [( η ( τ )˚ g ) − / ] ∗ (cid:1) | R dt v b ℑ ( τ )( η ( τ )˚ g ) ∗ | i.e. ε n i ( τ ) = [1 + ε n υ i ( τ )]˚ n k = 0 for n υ i ( τ, x i , t ) = 0 , if the integration functions are chosen n k ( τ ) = 0 and n k ( τ ) = 0 , and η i ( τ )˚ w i = ∂ i R dt v b ℑ ( τ )[ η ( τ ) ˚ g ] ∗ v b ℑ ( τ ) [ η ( τ ) ˚ g ] ∗ i.e. ε w i ( τ ) = [1 + ε w υ i ( τ )]˚ w i for w υ i ( τ, x i , t ) = ∂ i R dt v b ℑ ( τ )( υ ˚ g ) ∗ ∂ i R dt v b ℑ ( τ )˚ g ∗ − ( υ ˚ g ) ∗ ˚ g ∗ , when a prime ˚ w i = ∂ i R dt v b ℑ ( τ )˚ g ∗ / v b ℑ ( τ )˚ g ∗ is well defined for some prescribed v b ℑ ( τ ) and ˚ g ∗ . Finally, we conclude that ε –deformed quadratic line elements for cosmological flow deformations can bewritten in a general form ds εt = ε g αβ ( τ, x k , t ) du α du β = ε g i ( τ, x k )[( dx ) + ( dx ) ] + ε g ( τ, x k , t )( dy ) + ε h ( τ, x k , t ) [ dt + ε w i ( τ, x k , t ) dx i ] . We can impose additional nonholonomic constraints (38) in order to extract LC–configurations for ε –deformations with zero torsion. Various classes of generic off-diagonal cosmological solitonic solutions can be constructed and parameter-ized in terms of η –polarization functions introduced in formulas (20) and applying the AFDM summarized inTables 1 and 2. A primary cosmological d-metric can be parameterized in a necessary form as in Appendix (B)when ˚g =[˚ g i ( x i , t ) , ˚ g a = ˚ h a ( x i , t ); ˚ N k = ˚ n k ( x i , t ) , ˚ N k = ˚ w k ( x i , t )] (21) which for a FLRW configuration can bediagonalized by frame/ coordinate transforms. A cosmological solitonic target metric g can be generated bynonholonomic η –deformations, ˚g → g ( τ )=[ g i ( τ, x k ) = η i ( τ )˚ g i , g b ( τ, x k , t ) = η b ( τ )˚ g b , N ai ( τ, x k , t ) = η ai ( τ ) ˚ N ai ] , and constrained to the conditions to define exact and parametric solutions of the system of nonlinear PDEswith decoupling (33). A corresponding quadratic line element for g can be parameterized in a form (20), ds = η i ( τ, x i , t )˚ g i [ dx i ] + η a ( τ, x i , t )˚ g a [ dy a + η ak ( τ, x i , t ) ˚ N ak dx k ] , (46)with summation on repeating contracted low-up indices. The polarizaton values η α ( τ ) and η ai ( τ ) are deter-mined by geometric and cosmological solitonic flows and nonlinear interactions. We prescribe that the effective v –source is determined by a solitonic hierarchy b Υ ( τ, x i , t ) = v b ℑ [ ι ] (26)and compute the coefficients for a target d-metric (46) following formulas summarized in Table 2, η i ( τ ) = e ψ ( τ,x k ) ˚ g i ; η ( τ ) = η ( τ, x i , t ) as a generating function ; η ( τ ) = − | η ( τ )˚ h | / ) ∗ ] ˚ h | R dt v b ℑ [ ι ]( η ( τ )˚ h ) ∗ | ; η k ( τ ) = n k ˚ n k + 16 n k ˚ n k Z dt (cid:16) [( η ( τ )˚ h ) − / ] ∗ (cid:17) | R dt v b ℑ [ ι ]( η ( τ )˚ h ) ∗ | ; η i ( τ ) = ∂ i R dt v b ℑ [ ι ]( η ( τ )˚ h ) ∗ ˚ w i v b ℑ [ ι ] ( η ( τ )˚ h ) ∗ , (47)with integration functions n k ( τ, x i ) and n k ( τ, x i ) .
24n (47), the gravitational polarization η ( x i , t ) is taken as a (non) singular generating function subjectedto nonlinear symmetries of type (35) which can be written in the form Φ = − τ ) h ( τ ) = − η ( τ, x i , t )˚ h ( τ, x i , t ) and (Ψ ) ∗ = − Z dt v b ℑ [ ι ][ η ( τ, x i , t )˚ h ( τ, x i , t )] ∗ . In this section, the values Φ , h and η may not encode soltionic hierarchies but Ψ and other coefficients ofsuch target cosmological d-metric are solitonic ones if they are computed using v b ℑ [ ι ] . We can constrainthe coefficients (47) to a subclass of data generating target LC-configurations when the d-metrics satisfy theconstraints (38) for zero torsion. The nonlinear functionals for the soliton v-source and (effective) cosmologicalconstant can be changed into additive functionals v b ℑ → av b ℑ and Λ → a Λ as av b ℑ [ ι ] (39) and a Λ (40). Solutions with geometric flow and cosmological η –polarizations (44) can be constructed with coefficientsof the d-metrics determined by nonlinear generating functionals Φ[ i ] , or prescribed additive functionals a Φ[ i ] corresponding to (39). This includes terms with integration functions h [0]3 ( τ, x i ) for h [ i ] with explicit depen-dence on a time like variable t. Such configurations can be generated also by some prescribed cosmologicaldata v b ℑ ( τ, x i , t ) and Λ( τ ) , which are not obligatory of solitonic nature. We can compute correspondingnonlinear functionals η ( τ, x i , t ) (we omit here similar formulas for additive functionals a η ( τ, x i , t )) usingnonlinear symmetries (35) and related polarization functions, η [ i ] = − Φ [ i ] / τ )˚ h ( x i , t ) , [Ψ ( τ )] ∗ = − Z dt v b ℑ ( τ, x i , t ) h ∗ ( τ ) = − Z dt v b ℑ ( τ, x i , t )[ η [ i ]˚ h ( x i , t )] ∗ . We apply these formulas for the AFDM outlined in Table 2 and compute the coefficients of a cosmologicalsolitonic d-metric of type (46), η i ( τ ) = e ψ ( τ,x k ) ˚ g i ; η ( τ ) = η ( τ, x i , t ) = η [ i ] as a generating function ; η = − | η [ i ]˚ h | / ) ∗ ] ˚ h | R dt v b ℑ ( τ ) η [ i ]˚ h ) ∗ | ; (48) η k ( τ ) = n k ( τ )˚ n k + 16 n k ( τ )˚ n k Z dt (cid:16) [( η [ i ]˚ h ) − / ] ∗ (cid:17) | R dt v b ℑ ( τ )( η [ i ]˚ h ) ∗ | ; η i ( τ ) = ∂ i R dt v b ℑ ( τ )( η [ i ] ˚ h ) ∗ ˚ w i v b ℑ ( τ ) ( η [ i ] ˚ h ) ∗ , for integrating functions n k τ, x i ) and n k τ, x i ) . Using (48), target cosmological solitonic off-diagonal metrics with zero torsion which solve (34) can begenerated by polarization functions subjected to additional nonholonomic constraints, η i ( τ ) = e ψ ( τ,x k ) ˚ g i ; η ( τ ) = ˇ η ( τ, x i , t ) = ˇ η [ i ] as a generating function ; η ( τ ) = − | ˇ η [ i ]˚ h | / ) ∗ ] ˚ h | R dt v ˇ ℑ ( τ )(ˇ η [ i ]˚ h ) ∗ | ; η k ( τ ) = ∂ k n ( τ )˚ n k , η i ( τ ) = ∂ i ˇ A ( τ )˚ w k , for an integrating function n ( τ, x i ) and a generating function ˇ A ( τ, x i , t ) . The solutions constructed in this subsection describe certain nonholonomically deformed cosmologicalgeometric flow configurations self-consistently imbedded into a solitonic gravitational evolution media whichcan model nonholonomic dark energy and dark energy configurations. A compatibility with observationaldata can be chosen for respective integration functions and constants and corresponding classes of solitonichierarchies. 25 .2.3 Cosmological deformations by solitonic sources & solitonic generating functions
General classes of cosmological solutons and nonholonomic deformations can be constructed with nonlin-ear solitonic functionals both for generating functions and generating sources. Nonlinear superpositions ofsolutions of type (47) and (48) can be performed if the coefficients of d-metric are computed η i ( τ ) = e ψ ( τ,x k ) ˚ g i ; η ( τ ) = η ( τ, x i , t ) = η [ ι ] as a generating function ; η ( τ ) = − | η [ ι ]˚ h | / ) ∗ ] ˚ h | R dt v b ℑ [ ι ]( η [ ι ]˚ h ) ∗ | ; η k ( τ ) = n k ( τ )˚ n k + 16 n k ( τ )˚ n k Z dt (cid:16) [( η [ ι ]˚ h ) − / ] ∗ (cid:17) | R dt v b ℑ [ ι ]( η [ ι ] ˚ h ) ∗ | ; η i ( τ ) = ∂ i R dt v b ℑ [ ι ]( η [ ι ] ˚ h ) ∗ ˚ w i v b ℑ [ ι ] ( η [ ι ] ˚ h ) ∗ , (49) where n k ( τ, x k ) and n k ( τ, x k ) are integration functions. In formulas (49), we consider two differentprescribed a nonlinear generating functional, Φ[ ι ] , and a nonlinear functional for source, v b ℑ [ ι ] . and runningconstant Λ( τ ) related via nonlinear symmetries of type (35). This allows us to compute a correspondingnonlinear functional η ( τ, x i , t ) = η [ ι, ι, ... ] and a polarization function, η ( τ ) = − Φ [ ι ] / τ )˚ h ( x i , t ) , ( Ψ ( τ )) ∗ = − Z dt v b ℑ [ ι ] h ∗ [ ι ] = − Z dt v b ℑ [ ι ][ η ( τ )˚ h ( x i , t )] ∗ . (50)Imposing additional constraints (38) for a zero torsion, LC-cosmological solitonic metrics with geometric flowsare generated.In result, the quadratic line element corresponding to such classes of cosmological solutions (49) can bewritten for generating data (Φ[ ι ] , Λ( τ )) : ds = e ψ ( ι ) [( dx ) + ( dx ) ] + ( h [0]3 ( τ, x k ) − (Φ[ ι ]) τ ) )[ dy + ( n k ( τ, x k ) + n k ( τ, x k ) Z dt [(Φ[ ι ]) ] ∗ | Λ( τ ) R dt v b ℑ [ ι ] [(Φ[ ι ]) ] ∗ | | h [0]3 ( τ, x k ) − (Φ[ ι ]) τ ) | − / ) dx k ] − (Φ[ ι ]) [(Φ[ ι ]) ] ∗ | Λ( τ ) R dt v b ℑ [ ι ][(Φ[ ι ]) ] ∗ | ( h [0]3 − (Φ[ ι ]) τ ) ) [ dt + ∂ i (cid:16)R dt v b ℑ [ ι ][(Φ[ ι ]) ] ∗ (cid:17) v b ℑ [ ι ] [(Φ[ ι ]) ] ∗ dx i ] (51)The data for a primary cosmological solution can be extracted using nonlinear symmetries (50), when ˚ g i = e ψ ( τ,x k ) /η i ( τ ) and ˚ h ( x i , t ) = − Φ [ ι ] / τ ) η ( τ ) are considered for certain values which for a τ are prescribed in some forms that the integration functions h [0]4 ( τ, x k ) , n k ( τ, x k ) and n k ( τ, x k ) encode aprime d-metric ˚g = [˚ g i , ˚ g a , ˚ N jb ] (21). Such cosmological scenarios (51) describe geometric evolution for τ > τ self-consistently imbedded into solitonic gravitational (dark energy) backgrounds and solitonic dark and/orstandard matter. We study how effective sources for geometric flows with cosmological solitonic hierarchies flows result ingeneric off–diagonal deformations and generalizations of a FLRW metric. Such nonholonomic deformationsof geometric objects define new classes of exact solutions of systems of nonlinear PDEs (33). To applythe AFDM is necessary to define some nonholonomic variables which allow decoupling and integration ofcorresponding systems of equations describing N-adapted nonholonomic deformations of a prime d-metric ˚g =[˚ g i , ˚ g a = ˚ h a ; ˚ N k = ˚ n k , ˚ N k = ˚ w k , ] (21), see formulas for parameterizations of cosmological d-metrics inAppendix B. We cite [45] as a standard monograph on GR with necessary details on geometry of cosmologicalspaces and [62, 61, 20, 60, 23] for examples of nonholonomic deformations of BH and cosmological solutionsin geometric flows and gravity theories. 26 .3.1 Nonholonomic evolution of FLRW metrics with induced (or zero) torsion We consider a primary cosmological d-metric parameterized as in Appendix (B) when ˚g =[˚ g i ( x i , t ) , ˚ g a =˚ h a ( x i , t ); ˚ N k = ˚ n k ( x i , t ) , ˚ N k = ˚ w k ( x i , t ) , ] (21) with ˚ g ∗ = 0 , which for a FLRW configuration can be diagonal-ized by frame/ coordinate transforms. This allows us to construct nonholonomic cosmological deformationsfollowing the geometric formalism outlined in section 4.1.3 and Table 2.For general η –deformations (44) and constraints n i = 0 , the solitonic flow modifications of the FLRWmetric are computed ds = e ψ ( τ,x k ) [( dx ) + ( dx ) ] + η [ ι ]˚ h ( e ) − | η [ ι ]˚ h | / ) ∗ ] | R dt v b ℑ [ ι ][ η [ ι ]˚ h ] ∗ | ˚ h ( e ) , e = dy , e = dt + ∂ i R dt v b ℑ [ ι ][ η [ ι ]˚ h ] ∗ v b ℑ [ ι ] [ η [ ι ]˚ h ] ∗ dx i ′ , (52)where η ( τ ) = η ( τ, x k , t ) = η [ ι ] is a generating function and v b ℑ ( τ ) = v b ℑ [ ι ] is a flow generating source asin (49) and ψ ( τ, x k ) is a solution of a 2-d Poisson equation. Let us elaborate on models of geometric cosmological flows for nonholonomic distributions describing ε -deformations (45) for for ˚ g ∗ = 0 in target metrics of type (44). The corresponding quadratic line elementsare ds = [1 + εe ψ ψ ˚ g i h b ℑ ]˚ g i ( dx i ) + [1 + ευ ]˚ g ( e ) + " ε (2 [ υ ˚ g ] ∗ ˚ g ∗ − R dt v b ℑ ([ υ ˚ g ]) ∗ R dy v b ℑ ˚ g ∗ ) ˚ g ( e ) , e = dx , e = dt + [1 + ε ( ∂ i R dt v b ℑ ( υ ˚ g ) ∗ ∂ i R dt v b ℑ ˚ g ∗ − ( υ ˚ g ) ∗ ˚ g ∗ )] ˚ w i dx i . (53)In these formulas, ψ ( τ ) = ψ ( τ, x k ) and ψ ( τ ) = ψ ( τ, x k ) are solutions of 2-d Poisson equationswith a generating h-source h b ℑ ( τ ) = h b ℑ ( τ, x k ) = h b ℑ ( τ, x k ) + ε h b ℑ ( τ, x k ) as described in section 4.1.3; v b ℑ ( τ ) = v b ℑ [ ι ] is a generating v-source with ε -decomposition where υ = υ ( τ ) = υ ( τ, x k , t ) = υ [ ι ] is agenerating function. The cosmological flow solution (53) is for a N-adapted system of references and spacecoordinates [ x i , y ] for which the condition (˚ g ∗ ) = ˚ g | R dt v b ℑ ˚ g ∗ | allows us to compute well-defined coefficients ˚ g and ˚ w i = ∂ i [ dt v b ℑ ˚ g ∗ ] / v b ℑ ˚ g ∗ when there are prescribed certain values v b ℑ and ˚ g ∗ = 0 . We can fix theconditions n k ( τ ) = 0 and n k ( τ ) = 0 for which N i = n i = 0 but even in such cases a non-zero coefficient N i = w i ( ε, τ, x k , t ) results in nontrivial nonholonomic torsion and anholonomy coefficients. To extract LC-configurations we can impose on υ ( τ ) and sources additional zero torsion constraints (38).Considering for nonlinear symmetries of type (50) the formula η ( τ ) = − Φ [ ι ] / τ )˚ g for (53), weconclude that as a generating function (including solitonic hierarchies) can be used the value ευ [ ι ] = − (cid:0) [ ι ] / τ )˚ g (cid:1) or Φ[ ι ] ≃ p | Λ( τ )˚ g | (1 − ε υ [ ι ]) . (54)Other types of geometric flow and cosmologic solitonic hierarchies can be encoded into generated sources h b ℑ ( τ, x k ) = h b ℑ ( τ, x k ) + ε h b ℑ ( τ, x k ) , v b ℑ ( τ ) = v b ℑ [ ι ] and (Λ( τ ) , υ [ ι ] , h b ℑ + ε h b ℑ , v b ℑ [ ι ]) . We provide explicit examples how G. Perelman’s W-entropy and related thermodynamic values can becomputed for cosmological solitonic solutions under geometric flow evolution. To simplify formulas we fix27ertain values for normalization and integration functions corresponding to a constant normalizing function, b f ( τ ) = b f = const = 0 , in (11). In result, the F- and W-functionals (5) are written b F = 18 π Z τ − q | g [Φ( τ, x i , t )] | δ u [ h Λ( τ ) + Λ( τ )] , (55) c W = 14 π Z τ − q | g [Φ( τ, x i , t )] | δ u ( τ [ h Λ( τ ) + Λ( τ )] − , where q | g [Φ( τ, x i , t )] | = q | q q q ( q N ) | = 2 e ψ ( τ,x k ) (cid:12)(cid:12) Φ( τ, x i , t ) (cid:12)(cid:12) s | [Φ ( τ, x i , t )] ∗ || Λ( τ ) R dt v b ℑ ( τ, x i , t )[Φ ( τ, x i , t )] ∗ | is computed for d-metrics parameterized in the form (3) with q ( τ ) = q ( τ ) = e ψ ( τ,x k ) , q ( τ ) = − Φ ( τ, x i , t )4Λ( τ ) , [ q N ( τ )] = h ( τ, x k , t ) = − ( τ, x i , t )] ∗ | R dt v b ℑ ( τ, x i , t )[Φ ( τ, x i , t )] ∗ | for h [0]4 = 0 . The N-adapted differential δ u = dx dx e e = dx dx [ dy + n i ( τ ) dx i ][ dt + w i ( τ ) dx i ] is forN-connection coefficients with fixed integration functions n k ( τ ) = 0 and n k ( τ ) = 0 , when N ai = [ n i ( τ ) = 0 , w i ( τ ) = ∂ i (cid:16)R dt v b ℑ ( τ, x i , t )[Φ ( τ, x i , t )] ∗ (cid:17) v b ℑ ( τ, x i , t )[Φ ( τ, x i , t )] ∗ ] . The statistical thermodynamic values can be computed using the thermodynamic generating function (13)corresponding to c W (55) (for simplicity, with fixed normalization) b Z [ g ( τ )] = 14 π Z τ − d V ( τ ) . (56)The effective integration volume functional d V ( τ ) = d V ( ψ ( τ, x k ) , Φ( τ, x i , t ) , v b ℑ ( τ, x i , t ) , Λ( τ )) is determinedby data ( ψ ( τ ) , Φ( τ ) , v b ℑ ( τ ) , Λ( τ )) and computed d V ( τ ) = e ψ ( τ ) | Φ( τ ) | s | [Φ ( τ )] ∗ || Λ( τ ) R dt v b ℑ ( τ )[Φ ( τ )] ∗ | dx dx dy dt + ∂ i (cid:16)R dy v b ℑ ( τ )[Φ ( τ )] ∗ (cid:17) v b ℑ ( τ ) [Φ ( τ )] ∗ dx i . (57)These formulas allow us to compute thermodynamic values for cosmological solitonic geometric flows, b E ( τ ) = − π Z (cid:18) [ h Λ( τ ) + Λ( τ )] − τ (cid:19) d V ( τ ) , b S ( τ ) = − π τ Z ( τ [ h Λ( τ ) + Λ( τ )] − d V ( τ ) . (58) We can consider Perelman and/or Carathéodory thermodynamic values are computed for a 3+1 spitting(3) determined by a cosmological solitonic d-metric (51) (for LC-configurations, we can consider (34)), when q = q [ h i ] = e ψ [ h i ] , q [ i ] = − (Φ[ i ]) τ ) , [ q N ( τ )] = h [ i ] = − i ]) ] ∗ | R dt v b ℑ [ i ][(Φ[ i ]) ] ∗ | and N ai = [ n i ( τ ) = 0 , w i ( τ ) = ∂ i (cid:16)R dt v b ℑ [ i ][Φ [ i ]] ∗ (cid:17) v b ℑ [ i ][Φ [ i ]] ∗ ] . Having defined such values in a convenient system of reference/coordinates, we can consider changing to any system ofreference. d V [ h i, i, i ] = e ψ [ h i ] | Φ[ i ] | s | [Φ [ i ]] ∗ || Λ( τ ) R dt v b ℑ [ i ][Φ [ i ]] ∗ | dx dx dy [ dt + ∂ i (cid:16)R dt v b ℑ [ i ][Φ [ i ]] ∗ (cid:17) v b ℑ [ i ] [Φ [ i ]] ∗ dx i ] . This volume element allows us to define the thermodynamic generating function (56) and compute respectivethermodynamic values (58) for geometric flows of such solitonic hierarchies, b Z [ h i, i, i ] = 14 π Z τ − d V [ h i, i, i ] and b E [ h i, i, i ] = − π Z (cid:18) [ h Λ( τ ) + Λ( τ )] − τ (cid:19) d V [ h i, i, i ] , b S [ h i, i, i ] = − π Z ( τ [ h Λ( τ ) + Λ( τ )] − τ − d V [ h i, i, i ] . The d-metrics for such parametric solutions are described by quadratic elements (53) and generatingfunctions Φ[ ι, ˚ h ] ≃ q | Λ( τ )˚ h | (1 − ε υ [ ι ]) (54) and a primary cosmological metric. The ε –decompositionfor the respective effective volume form is d V [ h i, ευ [ ι ] , i, ˚ h ] = 2 e ψ [ h i ] (cid:12)(cid:12)(cid:12) (1 − ε υ [ ι ]) (cid:12)(cid:12)(cid:12) vuut (cid:12)(cid:12)(cid:12) ˚ h | υ [ ι ] | ∗ (cid:12)(cid:12)(cid:12) | R dt v b ℑ [ i ] | υ [ ι ] | ∗ | dx dx dy [ dt + ∂ i (cid:16)R dt v b ℑ [ i ] | υ [ ι ] | ∗ (cid:17) v b ℑ [ i ] | υ [ ι ] | ∗ dx i ] , which allows to compute corresponding thermodynamic generating function (56) and canonical energy andentropy (58) for geometric flow cosmological solitonic flow parametric ε –deformations.Finally, we emphasize that it is not possible to define and compute the Bekenstein–Hawking entropy forlocally anisotropic cosmological solutions constructed in this section. The axiomatic side of thermodynamics due to Constantin Carathéodory [24, 25] and, further, the axiomatictreatment of physics were of constant and deep interest both to mathematicians and physicists (including D.Hilbert, W. Pauli, M. Born, and, in a critical sense, M. Planck) [63, 64, 65, 66]. Relativistic generalizationsof Grigory Perelman geometric flow thermodynamics [10] allows us to include in the scheme and find geomet-ric connections to mathematical physics, (modified) gravity theories, cosmology and (quantum) informationtheory [16, 17, 18, 19, 22, 23].In this article, we developed Carathéodory’s axiomatic approach to foundations of thermodynamics andstatistical physics (considering Pfaff forms but also certain work on measure theory) and demonstrated alsothat his methods are useful for research and applications in modern gravity and cosmology theories. AlthoughC. Carathéodory and G. Perelman geometric thermodynamic construction played a strategic role towardsfinding solutions of most important and difficult problems in geometry and physics, their contributions havenot yet properly appreciated by many physicists and mathematicians. This is probably due to the multi- andinter-disciplinary character of their works when further applications request both a "deep physical intuition"and "advanced mathematical education and very sophisticate geometric methods". In our works, we try toestablish a bridge between different communities of researchers.Let us speculate on certain further perspectives on elaborating unified geometric methods to thermody-namics of geometric flows, gravity & cosmology, quantum information etc.:29. Carathéodory’s works on measure theory [67, 68], see further developments in [69] and (on symbolicshifts and ergodic theory) [51, 70], seem to be useful in modeling information sources and processing[71, 72] and (recent applications) quantum information theory [21, 22, 23].2. The extended spectral decompositions are applicable for various conservative systems, complex systemsand their macroscopic descriptions, with new possibilities for probabilistic prediction and control [73,74, 47, 48, 52, 53, 54], in the theory of locally anisotropic kinetic processes and diffusion [52, 53, 54], seealso applications in noncommutative geometric flow theory and physics [16].3. P. Finsler elaborated his geometry as a postgraduate of C. Carathéodory. Such Finsler-Lagrange-Hamilton theories and their modifications of Einstein gravity, geometric flows theories, cosmology andastrophysicsa have been recently axiomatized for nonholonomic Lorentz manifolds and (co) tangentbundles, see recent reviews in [75, 76]. This paper should be considered as a cosmological partner ofthe works [20, 77] on Finsler and other types modified black hole configurations and their generalizedPerelman thermodynamics.
Acknowledgments:
This research develops former programs partially supported by IDEI, PN-II-ID-PCE-2011-3-0256, CERN and DAAD and extended to collaborations at California State University at Fresno,the USA, and Yu. Fedkovych Chernivtsi National University, Ukraine. Author S. Vacaru is grateful to Prof.Dr. P. Stavrinos for former support and collaboration. He thanks Prof. Dr. I. Antoniou for providing veryimportant references on Carathéodory research in mathematics and physics.
A Pfaffian differential equations
Let us provide a brief introduction into the theory of Pfaff forms and thermodynamics, see details andreferences in [37, 38, 39]. A Pfaff differential form is δφ = P I X I dz I , where I runs integer values (forsimplicity, we consider I = 1 , ) and δf is differential 1-form but may be not a differential of a real valuedfunction φ ( z I ) of real variables z I , where ∂ I := ∂/∂z I . An equation δφ = 0 (A.1)is called a non-exact Pfaff equation. If δφ = dφ = ( ∂ I φ ) dz I is an exact differential of a function φ ( z I ) , i.e. wehave an exact Pfaff equation, it is possible to integrate (A.1) along a path C connecting two points z I [1] and z I [2] (when φ is path–independent) and express the solution in the form φ = Z C dφ = φ ( z I [2] ) − φ ( z I [1] ) = const. The H. A. Schwarz criterium is the necessary and sufficient condition to detect a total differential equation ∂ I X J = ∂ J X I , for I = J, i.e. ∂ φ∂x ∂x = ∂ φ∂x ∂x . (A.2)In many cases, a non-exact Pfaffian with ∂ I X J = ∂ J X I can be transformed into an exact one by the aidof an integating factor K ( z I ) , when the coefficients of P I KX I dz I satisfy the Schwarz condition ∂ I ( KX J ) = ∂ J ( KX I ) , for I = J. (A.3)In such a case, the equation Kδφ = d ( Kφ ) = 0 (A.4)can be integrated in an explicit form which allows us to find φ for any prescribed K satisfying (A.3).30n a more general context, if we are not able to transform (A.1) into a (A.4), we can additionally add to δ ( Kφ ) = X I KX I dz I = d ( Kφ ) a differential of a new function B ( z I ) , dB = ( ∂ I B ) dz I and search for such K and B when ∂ I ( KX J + B ) = ∂ J ( KX I + B ) , for I = J and δ ( Kφ ) + dB = d ( Kφ + B ) . In such a case, we can integrate d ( Kφ + B ) = 0 (A.5)for any suitable K and B and find φ in nonexplicit form from a so-called nonholonomic (non-integrable)function F ( φ, z I ) = const. Usually, in thermodynamics we deal with equations of type (A.1) into a (A.4), buton nonholonomic manifolds, equations of type (A.5) are involved.
B Parameterizatons for families of cosmological d-metrics
We consider basic notations for quadratic line elements describing geometric flow evolutions and nonholo-nomic deformations of prime metrics into target cosmological ones.
B.1 Target d-metrics with geometric evolution of polarization functions
Families of target quadratic line elements can be represented in off-diagonal form, g αβ = [ g i , h a , n i , w i ] , and/or using η -polarization functions, ds ( τ ) = g i ( τ, x k )[ dx i ] + h ( τ, x k , t )[ dy + n i ( τ, x k , t ) dx i ] + h ( τ, x k , t )[ dt + w i ( τ, x k , t ) dx i ] (B.1) = η i ( τ, x k , t )˚ g i ( x k , t )[ dx i ] + η ( τ, x k , t )˚ h ( x k , t )[ dy + η i ( τ, x k , t ) ˚ N i ( x k , t ) dx i ] + η ( τ, x k , t )˚ h ( x k , t )[ dt + η i ( τ, x k , t ) ˚ N i ( x k , t ) dx i ] = η i ( τ )˚ g i [ dx i ] + η ( τ )˚ h [ dy + η k ( τ ) ˚ N k dx k ] + η ( τ )˚ h [ dt + η k ( τ ) ˚ N k dx k ] , (B.2)where τ is a temperature like geometric evolution parameter and, for simplicity, we consider that prime metricsdo not depend on such a parameter. There will be stated dependencies of type η a ( τ ) = η a ( τ, x k , t ) if suchnot notations do not result in ambiguities. We consider a coordinate transform to a new time like coordinate y = t → ς when t = t ( x i , ς ) ,dt = ∂ i tdx i + ( ∂t/∂ς ) dς ; dς = ( ∂t/∂ς ) − ( dt − ∂ i tdx i ) , i.e. ( ∂t/∂ς ) dς = ( dt − ∂ i tdx i ) , and rewrite the target d-metric using the new time variable ς . For instance, the 4th term in (B.2) is computed η ( τ )˚ h [ dt + η k ( τ ) ˚ N k dx k ] = η ( τ )˚ h [ ∂ k tdx k + ( ∂t/∂ς ) dς + η k ( τ ) ˚ N k dx k ] = η ( τ )˚ h [( ∂ k t ) dx k + ( ∂t/∂ς ) dς + η k ( τ ) ˚ N k dx k ] = η ( τ )˚ h [( ∂t/∂ς ) dς + ( ∂ k t + η k ( τ ) ˚ N k ) dx k ] = ˚ h [ η ( τ )( ∂t/∂ς ) dς + η ( τ )( ∂ k t + η k ( τ ) ˚ N k ) dx k ] = ˚ h [ η ( τ )( ∂t/∂ς ) dς + η ( τ )( ∂ k t/ ˚ N k + η k ( τ )) ˚ N k dx k ] If η ∂t/∂ς = 1 , when ∂t/∂ς = ( η ) − is introduced for dt = ∂ i tdx i + ( ∂t/∂ς ) dς, we obtain dt = ( ∂ i t ) dx i + ( η ) − dς for ˇ η k = η ( ∂ k t + η k ˚ N k ) . In result, a new time coordinate ς can be found from ∂t/∂ς = ( η ) − which results in dς = η ( x k , t ) dt ; ς = Z η ( x k , t ) dt + ς ( x k ) . Such coordinates with flow parameter τ and time like ς are useful for computations of geometric evolutionand nonholonomic deformations of the FLRW metrics.31 .2 Off-diagonal and diagonal parameterizations of prime d-metrics Let us consider a target line quadratic element for an off-diagonal cosmological solution written in theform (B.2). We can introduce an effective target locally anisotropic cosmological scaling factor ˇ a ( τ, x k , ς ) := η ( τ, x k , ς )˚ a ( x i , ς ) with gravitational polarization η ( τ, x k , ς ) and prime cosmological scaling factor ˚ a ( τ, x i , ς ) , which allows to consider limits ˚ a ( τ, x i , ς ) → ˚ a ( ς ) with typical FLRW configurations. This can be performedfollowing formulas ds = η ( τ ) { η i ( τ ) η ( τ )˚ g i [ dx i ] + ˚ h [ dy + η k ˚ N k dx k ] } + ˚ h [ dτ + ˇ η k ˚ N k dx k ] (B.3) = ˇ a ( τ, x k , ς ) { ˇ η i ( τ, x k , ς )˚ g i [ dx i ] + ˚ h [ dy + ˇ η k ( τ, x k , ς ) ˚ N k dx k ] } + ˚ h [ dς + ˇ η k ( τ, x k , ς ) ˚ N k dx k ] = η ( τ, x k , ς )˚ a ( τ, x i , ς ) { ˇ η i ( τ, x k , ς )˚ g i [ dx i ] + ˚ h [ dy + ˇ η k ( τ, x k , ς ) ˚ N k dx k ] } + ˚ h [ dς + ˇ η k ( τ, x k , ς ) ˚ N k dx k ] , where ˇ a ( τ, x k , ς ) := η ( τ, x k , t ( x i , ς )) = η ( τ, x k , t ( x i , ς ))˚ a ( x k , t ( x i , ς )) = η ( τ, x k , ς )˚ a ( x i , ς );ˇ η i ( τ, x k , ς ) := η i ( τ, x k , t ( x i , ς )) η ( τ, x k , t ( x i , ς )) ; ˇ η k ( τ, x k , ς ) := η k ( τ, x k , t ( x i , ς ));ˇ η k ( τ, x k , ς ) := η { τ, , ∂ k t ( x i , ς )[ ˚ N k ( x i , t ( x i , ς ))] − + η k ( τ, x i , t ( x i , ς )) } ˚ N k ( x i , t ( x i , ς )) . Considering a prime d-metric as a flat FLRW metric written in local coordinates u = { u α ( x i , y , ς ) = ( x ( x i , y , ς ) , x ( x i , y , ς ) , y ( x i , y , ς ) , y ( x i , y , ς )) } , a d-metric (B.1) can be written incurved coordinate form ˚ a ( u ) , with local coordinated u α using a prime cosmological scaling factor ˚ a ( ς ) ,d ˚ s = ˚ a ( u ) { ˚ g i ( u )[ dx i ] + ˚ h ( u )[ dy + ˚ N k ( u ) dx k ] } + ˚ h ( u )[ dy + ˚ N k ( u ) dx k ] → ˚ a ( ς )[ dx ˇ i ] − dς , for u α → ( x i , y , ς ) , ˚ g i → , ˚ h → , ˚ h → − , ˚ N ak ( u ) → and ˚ a ( u ) → ˚ a ( ς ) . By definition, a quasi FLRW configuration is stated by a diagonalized solution for a d-metric is of type (B.3)when the integration functions and coordinates result in ˇ η ak ( τ, x k , ς ) = 0 ,ds = η ( τ, x k , ς )˚ a { ˇ η i ( τ, x k , ς )˚ g i [ dx i ] + ˚ h [ dy ] } + ˚ h [ dς ] . (B.4)Small nonholonomic deformations of such d-metrics can be parameterized ˇ η i τ ) ≃ ε ˇ χ i ( τ, x k , ς ) (see belowformulas relevant to (B.9)) by the polarization of the target cosmological factor, η ( τ, x k , ς ) can be arbitraryone and not a value of εχ ( τ, x k , ς ) with a small parameter ε. We can consider a resulting scaling factor a ( τ, x k , ς ) = η ( τ, x k , ς )˚ a ( x k , ς ) , with possible further re-parametrizations or limits to a ( τ, ς ) = η ( τ, ς )˚ a ( ς ) encoding possible nonlinear off-diagonal and parametric interactions determined by systems of nonlinear PDEs. B.3 Approximations for flows of target d-metrics
To study nonlinear properties of cosmological models is convenient to consider different types of param-eterizations and approximations for nonholonomic deformations of a prime metric to a target d-metric (B.3)being under geometric flow evolution. For our purposes, there are important six classes of exact, or para-metric, solutions which can be generated by a respective subclass of generating functions and/or generatingsources and, for certain cases, making some diagonal approximations, or by introducing small ε -parameters.1. We can chose mutual re–parametrization of generating functions (Ψ , Υ) ⇐⇒ (Φ , Λ = const ) andintegrating functions when the coefficients of a family of target d-metric b g αβ ( τ, ς ) depend only a timelike coordinate ς, when η ( τ, x k , ς ) → e η ( τ, ς ) and a ( τ, x k , ς ) → e a ( τ, ς ) = e η ( τ, ς )˚ a ( ς ) . Respective familiesof linear quadratic elements (B.3) can be represented in the form ds ( τ ) = η ( τ, ς )˚ a ( ς ) { ˇ η i ( τ, ς )˚ g i [ dx i ] + ˚ h [ dy + ˇ η k ( τ, ς ) ˚ N k dx k ] } + ˚ h [ dς + ˇ η k ( τ, ς ) ˚ N k dx k ] . (B.5)32ith respect to coordinate bases, such families of cosmological solutions can be generic off-diagonal andcould be chosen in some forms describing nonholonomic deformations of Bianchi cosmological models.2. For FLRW prime configurations, we can consider families of generation functions and integration func-tions which result in zero values of the target N-connection coefficients under geometric flow evolutonsand/or consider limits ˚ N ak → . For such cases, we can transform families (B.5) into families of diagonalmetrics ds ( τ ) = η ( τ, ς )˚ a ( ς ) { ˇ η i ( τ, ς )˚ g i [ dx i ] + ˚ h ( dy ) } + ˚ h ( dτ ) (B.6)modeling locally anisotropic interactions with a "memory" of nonholonomic/ off-diagonal structures.3. Flow evolution with small parametric nonholonomic deformations of a prime metric into families oftarget off-diagonal cosmological solutions (B.3) can be approximated ˇ η i ( τ, x k , ς ) ≃ ε i ˇ χ i ( τ, x k , ς ) , η ( τ, x k , ς ) ≃ ε χ ( τ, x k , ς ) , ˇ η ak ( τ, x k , ς ) ≃ ε ak ˇ χ ak ( τ, x k , ς ) , where small parameters ε i , ε , ε ak satisfy conditions of type ≤ | ε i | , | ε | , | ε ak | ≪ and, for instance, χ ( τ, x k , ς ) is taken as a generating function. Such approximations restrict the class of generating func-tions subjected to nonlinear symmetries and may impose certain relations between such ε -constants and χ -functions. Corresponding quadratic line elements can be parameterized ds ( τ ) = [1 + ε χ ( τ, x k , ς )]˚ a ( x i , ς ) { [1 + ε i ˇ χ i ( τ, x k , ς )]˚ g i [ dx i ] + (B.7) ˚ h [ dy + (1 + ε k ˇ χ k ( τ, x k , ς )) ˚ N k dx k ] } + ˚ h [ dς + (1 + ε k ˇ χ k ( τ, x k , ς )) ˚ N k dx k ] . Such τ -families of off-diagonal solutions define cosmological metrics with certain small independentfluctuations, for instance, a FLRW embedded self-consistently into a locally anisotropic backgroundunder geometric flow evolution.4. We can consider also families of off-diagonal cosmological solutions with small parameters ε i , ε , ε ak when the generating functions and d-metric and N-connection coefficients do not depend on space likecoordinates, which is typical for a number of cosmological models. For such approximations, the familyof quadratic line element (B.7) transforms into ds ( τ ) = [1 + ε χ ( τ, ς )]˚ a ( ς ) { [1 + ε i ˇ χ i ( τ, ς )]˚ g i [ dx i ] + (B.8) ˚ h [ dy + (1 + ε k ˇ χ k ( τ, ς )) ˚ N k dx k ] } + ˚ h [ dς + (1 + ε k ˇ χ k ( τ, ς )) ˚ N k dx k ] .
5. There are off-diagonal deformations, for instance, of a FLRW metric into a family of locally anisotropiccosmological solutions which can be constructed using only one small parameter ε = ε i = ε = ε ak , andwhen the formulas (B.8) transform into ds ( τ ) = [1 + εχ ( τ, x k , ς )]˚ a ( x i , ς ) { [1 + ε ˇ χ i ( τ, x k , ς )]˚ g i [ dx i ] + (B.9) ˚ h [ dy + (1 + ε ˇ χ k ( τ, x k , ς )) ˚ N k dx k ] } + ˚ h [ dς + (1 + ε ˇ χ k ( τ, x k , ς )) ˚ N k dx k ] . Such flows with ε -deformations can be generated by corresponding small ε –deformations of flows gener-ating functions.6. We can impose on families (B.9) the condition that the ε –deformations depend only on evoluiton tem-perature like parameter and a time like coordinate. This results in d-metrics ds ( τ ) = [1 + εχ ( τ, ς )]˚ a ( ς ) { [1 + ε ˇ χ i ( τ, ς )]˚ g i [ dx i ] +˚ h [ dy + (1 + ε ˇ χ k ( τ, ς )) ˚ N k dx k ] } + ˚ h [ dς + (1 + ε ˇ χ k ( τ, ς )) ˚ N k dx k ] which can be considered as some ansatz used, for instance, for describing geometric evolution of quantumfluctuations of FLRW metrics. 33n various classes of cosmological models with families of solutions with parametric ε -decompositions canbe performed in a self-consistent form by omitting quadratic and higher order terms after a class of locallyanisotropic solutions have been found for some general data ( η α , η ai ) . They are more general than approximatesolutions found, for instance, for classical and quantum fluctuations of standard FLRW metrics and mayinvolve flow evolution parameters of cosmological constants and generating functions and sources. For certainsubclasses of generic off-diagonal solutions, we can consider that ε i , ε a , ε ai ∼ ε, when only one small parameteris considered for all coefficients of nonholonomic deformations. References [1] A. Sommerfeld, Thermodynamics and Statistical Mechanics (Academic, New York, 1955)[2] M. Zemansky, Heat and Thermodynamics, 5th edn. (McGraw Hill, London, 1968)[3] A. Pippard, The Elements of Classical Thermodynamics (Cambridge University Press, London, 1997)[4] H. Callen, Thermodynamics (John Wiley, New York, 1960).[5] P. Landsberg, Thermodynamics (Interscience, New York, 1961).[6] J. D. Bekenstein, Black holes and the second law, Nuovo Cimento Letters 4 (1972) 737-740[7] J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333-2346[8] J. M. Bardeen, B. Carter and S. W. Hawking, The four laws of black hole mechanics, Commun. Math.Phys. 31 (1973) 161[9] S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199-220[10] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159[11] D. Friedan, Nonlinear models in εε