#  Dark energy and dark matter configurations for wormholes and solitionic
hierarchies of nonmetric Ricci flows and $F(R,T,Q,T_{m})$ gravity

Laurenţiu Bubuianu
SRTV - Studioul TVR Iaşi and University Appolonia, 2 Muzicii street, Iaşi,
700399, Romania email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Sergiu
I. Vacaru emails<EMAIL_ADDRESS><EMAIL_ADDRESS>;
Address for correspondence in 2024 as a visiting fellow at CAS LMU, Munich,
Germany and YF CNU, Chernitvsi, Ukraine:  Vokzalna street, 37-3, Chernivtsi,
Ukraine, 58008 Department of Physics, California State University at Fresno,
Fresno, CA 93740, USA;  Institute of Applied-Physics and Computer Sciences,
Yu. Fedkovych University, Chernivtsi, 58012, Ukraine; CAS Center for Advanced
Studies, Ludwig-Maximilians-Universität, Seestrasse 13, München, 80802,
Germany Elşen Veli Veliev
Department of Physics, Kocaeli University, 41380, Izmit, Turkey  email:
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Assel Zhamysheva
Department of General and Theoretical Physics, Eurasian National University,
Astana, 010000, Kazakhstan email<EMAIL_ADDRESS>
(accepted by EPJC: February 6, 2024)

###### Abstract

We extend the anholonomic frame and connection deformation method, AFCDM, for
constructing exact and parametric solutions in general relativity, GR, to
geometric flow models and modified gravity theories, MGTs, with nontrivial
torsion and nonmetricity fields. Following abstract geometric or variational
methods, we can derive corresponding systems of nonmetric gravitational and
matter field equations which consist of very sophisticate systems of coupled
nonlinear PDEs. Using nonholonomic frames with dyadic spacetime splitting and
applying the AFCDM, we prove that such systems of PDEs can be decoupled and
integrated in general forms for generic off-diagonal metric structures and
generalized affine connections. We generate new classes of quasi-stationary
solutions (which do not depend on time like coordinates) and study the
physical properties of some physically important examples. Such exact or
parametric solutions are determined by nonmetric solitonic distributions
and/or ellipsoidal deformations of wormhole hole configurations. It is not
possible to describe the thermodynamic properties of such solutions in the
framework of the Bekenstein-Hawking paradigm because such metrics do not
involve, in general, certain horizons, duality, or holographic configurations.
Nevertheless, we can always elaborate on associated Grigori Perelman
thermodynamic models elaborated for nonmetric geometric flows. In explicit
form, applying the AFCDM, we construct and study the physical implications of
new classes of traversable wormhole solutions describing solitonic deformation
and dissipation of non-Riemannian geometric objects. Such models with
nontrivial gravitational off-diagonal vacuum are important for elaborating
models of dark energy and dark matter involving wormhole configurations and
solitonic- type structure formation.

Keywords: exact and parametric solutions; modified gravity; nonholonomic
geometric flows; dark energy; dark matter; wormholes

###### Contents

  1. 1 Introduction, motivations and objectives
  2. 2 Metric noncompatible geometric flows and MGTs
    1. 2.1 Geometric preliminaries on metric-affine spaces and nonholonomic deformations
      1. 2.1.1 N-adapted metric-affine structures with nonholonomic (2+2) splitting
      2. 2.1.2 Geometric objects adapted to a N-connection structure and nonmetricity
    2. 2.2 Relativistic geometric flows encoding nonmetricity fields
      1. 2.2.1 $Q$-modified Perelman’s F- and W-functionals in canonical nonholonomic variables
      2. 2.2.2 Hamilton-Friedan geometric flow equations with nonmetricity
    3. 2.3 Nonmetric Ricci solitons and modified Einstein equations
      1. 2.3.1 Connecting nonholonomic solitons to nonmetric modified gravitational equations
      2. 2.3.2 Generating sources for $\tau$-running quasi-stationary effective matter fields and nonmetricity deformations
  3. 3 Off-diagonal quasi-stationary solutions encoding nonmetricity
    1. 3.1 Effective nonmetric and $\tau$-running Einstein equations
    2. 3.2 Nonmetric solitonic hierarchies
      1. 3.2.1 $\tau$-running quasi-stationary generating functions with solitonic hierarchies
      2. 3.2.2 Nonmetric quasi-stationary generating sources with solitonic hierarchies
      3. 3.2.3 Interacting $\tau$-running solitonic hierarchies for generating functions with respective solitonic hierarchies for nonmetric sources
    3. 3.3 Nonmetric wormhole solutions and their solitonic deformations
      1. 3.3.1 Prime metrics for 4-d metric compatible wormhole configurations
      2. 3.3.2 Nonholonomic quasi-stationary gravitational polarizations of wormholes
      3. 3.3.3 Off-diagonal quasi-stationary solitonic deformations of wormhole d-metrics
      4. 3.3.4 Small parametric off-diagonal quasi-stationary deformations of wormhole d-metrics
      5. 3.3.5 On traversable nonmetric wormholes
  4. 4 Perelman thermodynamics for nonmetric quasi-stationary configurations
    1. 4.1 Statistical thermodynamic variables for $Q$-deformed relativistic geometric flows
    2. 4.2 Thermodynamic variables for nonmetric deformations of wormhole solutions
  5. 5 Conclusions and open questions
  6. A Decoupling and integrability of nonmetric quasi-stationary geometric flow equations
    1. A.1 Decoupling of nonlinear PDEs with nonmetricity fields
      1. A.1.1 The coefficients of quasi-stationary ansatz and canonical Ricci d-tensor
      2. A.1.2 Decoupling of nonmetric geometric flow equations
    2. A.2 Off-diagonal solutions for nonmetric quasi-stationary configurations
      1. A.2.1 Generating functions and sources for nonmetric quasi-stationary off-diagonal solutions
      2. A.2.2 Nonlinear symmetries of nonmetric quasi-stationary off-diagonal solutions
      3. A.2.3 Using some d-metric coefficients as generating functions and nonmetricity
      4. A.2.4 Gravitational nonmetric polarizations
      5. A.2.5 Generating solutions with small parametric decompositions encoding nonmetricity
      6. A.2.6 Extracting Levi-Civita configurations
  7. B Generating nonmetric solitonic hierarchies via solitonic metrics and effective sources
    1. B.1 Preliminaries on geometric models and solitons
    2. B.2 Examples of solitonic space like stationary distributions and nonlinear waves
      1. B.2.1 Quasi-stationary solitonic distributions
      2. B.2.2 Generating nonlinear solitonic waves on temperature like parameter
      3. B.2.3 Ansatz for quasi-stationary geometric flows and solitonic hierarchies

##  1 Introduction, motivations and objectives

The standard approach to the gravity theory and general relativity, GR (i.e.
Einstein’s gravity theory) is formulated in the framework of pseudo-Riemannian
geometry, see [1, 2, 3, 4] as typical monographs and reviews of physically
important exact solutions. In GR, a four dimensional, 4-d, curved spacetime
$V$ is modelled as a Lorentz manifold endowed with metric structure,
$g=\\{g_{\alpha\beta}\\},$ when the Levi-Civita, LC-connection $\nabla=\\{\
^{\nabla}\Gamma_{\ \beta\gamma}^{\alpha}\\}$ is uniquely determined by
coefficients $g_{\alpha\beta}$ following two conditions: 1) zero nonmetricity
(i. e. metric compatibility), $Q:=\nabla g=0$ and 2) zero torsion, $\
{}^{\nabla}T=0.$111In our approach, we use the signature $(+++-)$, when
coordinate indices $\alpha,\beta,....$ run values 1,2,3,4; in general, indices
may be abstract ones; we use also certain left/up labels for
geometric/physical objects; local coordinates are labelled
$u=\\{u^{\alpha}\\}$ and the Einstein summation rule on up-low indices is
used. All necessary notation conventions and definitions of geometric/physical
objects are defined in the next sections and appendices. We assume that
readers are familiar with standard results and methods outlined in the
mentioned monographs and understand basic ideas and motivations for how to
extend the geometric constructions to modern modified gravity theories, MGTs,
and accelerating cosmology. The Einstein relativity theory has a remarkable
success and deep influence both in physics and mathematics. Nevertheless,
various alternatives and modifications of GR were elaborated when instead of
standard Lorentz manifolds there are considered metric-affine spaces
determined by some general metric and (independent) affine/ linear connection
structures, $(g,D=\\{\Gamma_{\ \beta\gamma}^{\alpha}\\}).$ Such non-Riemannian
spaces can be characterized by nonzero torsion, $T=\\{T_{\
\beta\gamma}^{\alpha}\\},$ and/or nontrivial nonmetricity,
$Q:=Dg=\\{Q_{\alpha\beta\gamma}:=D_{\alpha}g_{\beta\gamma}\\},$ fields. We
cite [5] as an early review of metric-affine gravity theories. For
applications in modern cosmology, various generalizations of such geometric
and gravity models are formulated as modified gravity theories, MGTs, when the
gravitational and matter field Lagrangians in GR are changed into some
functionals $F(R,T,Q,\ ^{tr}T),$ where $R$ is the Ricci scalar for $D,$ and $\
{}^{tr}T$ is the trace of the energy-momentum tensor for matter [6,
7].222Readers may find in the cited works many references with chronological
historical remarks, criticism, and applications in modern cosmology and
astrophysics. In this paper, we do not provide a comprehensive review on
nonmetricity and MGTs but concentrate on elaborating new geometric methods for
constructing exact and parametric solutions in nonmetric geometric flow and
gravity models. In our works, we write $F(R,T,...)$ instead of $f(,R,T,...)$
used in papers by other authors.

Generalizations of the Einstein equations with geometric distortions of linear
connections, $\nabla\rightarrow D$, can be written in some effective forms:

$E_{\mu\nu}=\varkappa[T_{\mu\nu}^{[br]}+T_{\mu\nu}^{[DM]}(\phi,\psi,...)+T_{\mu\nu}^{[DE]}(\phi,\psi,...)+T_{\mu\nu}^{[geom]}(g,R,T,Q,\mathcal{L}^{[m]},T^{[m]},\square
R,\square T,...)+...].$ (1)

In such formulas, $\varkappa$ is defined by the gravitational constant and the
energy-momentum tensors contain respective labels $[br],$ for barionic matter
(which can be written also $T_{\mu\nu}^{[m]});$ [DM] is for dark matter models
with some scalar, $\phi,$ spinor, $\psi,$ and other type fields. The label
[DE] is used for respective dark energy terms. The tensor
$T_{\mu\nu}^{[geom]}$ is a functional of different geometric and physical
values including classical and quasi-classical and/or extra dimension
contributions, string terms contributions, distortions of mater fields
Lagrange densities, $\mathcal{L}^{[m]},$ and corresponding traces of energy-
momentum tensors, $T^{[m]}$. In some MGTs, there are considered nonlocal terms
of type $\square R$ and/or $\square T,$ where $\square$ is a corresponding
variant of d’Alambert (wave operator) for respective curved spacetime etc. The
modified Einstein tensor $E_{\mu\nu}$ and respective (effective) barionic
matter terms $T_{\mu\nu}\approx T_{\mu\nu}^{[br]}+...$ can be derived from a
corresponding action

$S=\int[\varkappa^{-1}F(R,...)+\mathcal{L}^{[m]}+...]\sqrt{|g|}d^{4}u.$

Typically, such formulas define a four-dimensional, 4-d, spacetime model which
is distorted in the symbolic form $D=\nabla+Z[g,T,Q,...],$ where $Z$ is the
distortion tensor. Physical motivations and details on such MGTs and dark
gravity/ matter / energy formalism are presented in [6, 7, 8] and references
therein ( there are studied also certain applications in modern acceleration
cosmology and DM and DE physics).

Modified Einstein equations of type (1) consist of very sophisticate systems
of coupled nonlinear partial differential equations, PDEs. It is very
difficult to find exact/ parametric solutions for such dynamical equations
using standard methods elaborated in GR [4], when for some higher symmetry and
diagonal ansatz form metrics, nonlinear PDEs transform into certain nonlinear
systems of ordinary differential equations, ODEs. For instance, it is not
clear how to construct black hole, BH, like solutions for nontrivial
$Q$-terms, for generic off-diagonal $g_{\alpha\beta};$ and how to define
extensions of the Einstein-Dirac equations. Such problems were discussed in
detail in [9, 10], (in a general form for Finsler modifications of gravity
theories). Similar problems exist for metric-affine distortions of physical
models elaborated on Lorentz manifolds. The DM and DE coupling theories and
various MGTs involve constructions with nonminimal coupling between geometry
and matter. The equations (1) lead to the "nonconservation" of matter energy-
momentum tensor which makes more sophisticate the physical interpretation of
such models and solutions of dynamical or evolution equations. Nevertheless,
we can elaborate on a nonholonomic deformation formalism with adapted
distortions $\nabla\rightarrow D,$ when "nonconservation" is related to
certain (equivalently, anholonomic, i.e. nonintegrable) constraints like in
nonholonomic mechanics. By introducing integration constants, then solving the
constraint equations and redefining the effective Lagrangians, we can
formulate some types of modified conservation laws.

The main goal of this work is to prove that modified Einstein equations of
type (1) can be decoupled and integrated in some general forms for $Q\neq 0$.
We shall provide explicit examples for generic off-diagonal solutions defining
$Q$-deformations of gravitational solitonic hierarchies and wormhole
configurations. Such methods for generating exact/ parametric solutions of
physically important systems of nonlinear PDEs are not contained in standard
monographs on GR and various MGTs reviews [1, 2, 3, 4, 5, 6, 7, 8]. During the
last 30 years, it was elaborated the anholonomic frame and connection
deformation method, AFCDM, for constructing exact and parametric off-diagonal
solutions of physically important systems of nonlinear PDEs. We cite [9, 10,
11, 12] for a review of main results and methods, examples and applications.
Here we note that the AFCDM involves nonholonomic distributions of geometric
objects and nonholonomic frames, with respective types 2+2+2+…, 3+1 and
(3+1)+(2+2), 3+2+2+… (correspondingly, for 4-d, and extra dimension
spacetimes). Then, it is important to construct an auxiliary connection
$\widehat{\mathbf{D}}=\nabla+\widehat{\mathbf{Z}}$ when equations of type (1)
with redefined linear connections, $\nabla\rightarrow\widehat{\mathbf{D}}$ can
be decoupled and integrated in certain general forms. In the next section, we
shall provide the necessary definitions and explicit formulas.

Our nonholonomic geometric approach to generating solutions in gravity
theories should not be confused with the well-known Cartan moving frame
method, Neuman-Penrose formalism or other nontrivial string torsion
generalizations and/or equivalent constructions involving various types of
tetradic, dyadic, and Arnowit-Deser-Wheeler, ADM, formalisms. Main geometric
and analytic constructions related to GR are summarized in [4]). The main
difference of the AFCDM from other ones is that it involves deformations both
of the frame and linear connection structures which are adapted to certain
canonical nonholonomic distributions. In such nonholonomic adapted variables,
physically important systems of nonlinear PDEs can be decoupled and integrated
in certain general forms, when the technique of constructing solutions is not
restricted only to special diagonal ansatz transforming systems of nonlinear
PDEs into systems of nonlinear ODEs. For generic off-diagonal ansatz, we can
prescribe some special symmetries with Killing vectors and/or Lie algebra
structure, for spherical, cylindrical, toroid or other type configurations.
Such solutions also posses certain nonlinear symmetries. We can prescribe
necessary symmetries of solutions, and compute certain deformations (for
instance, by nonmetricity fields) for some physically important classes of
solutions. Usually, it is possible to extract LC configurations if there are
imposed additional nonholonomic constraints on the nonlinear and linear
connection structures, and respective generating and integration functions.
For some models, the off-diagonal terms of metrics and the nontrivial
nonholonomic structure may encode nonmetricity contributions even for
constraints to LC models.

This paper is a generalization to nonmetric geometric flows providing a
metric-affine development of the methods reviewed in [10, 11, 12]. Such
geometric constructions can be performed for nonassociative and noncommutative
geometric flow and gravity theories [13], when the constructions are performed
for star product R-flux deformations in metric compatible forms but including
nonsymmetric metrics. In this work, we consider associative and commutative
metric-affine structures with symmetric metrics when the nonholonomic
deformations and nonmetricity fields are related both to nonmetricity-induced
torsion fields and canonically induced torsions. For this work, we state three
general objectives:

The first objective (Obj1, in section 2) is to provide an introduction to
theories of geometric flow and gravity including nonmetricity. The 4-d
metric–affine geometry is formulated in an N-connection adapted form with
(dyadic) nonholonomic (2+2)-splitting. We define $Q$-modified (for Riemannian
metrics introduced by G. Perelman [15]) F- and W-functionals and sketch how
they can be derived from respective Hamilton-Friedan geometric flow equations
with nonmetricity. Nonmetric Ricci solitons and related modified Einstein
equations are derived as self-similar nonholonomic geometric flow
configurations for a fixed flow parameter.

The second objective, Obj2, stated for section 3, is to construct and analyse
solitonic deformations of exact/parametric quasi-stationary geometric flow
solutions encoding nonmetricity fields, which for fixed flow parameters and
LC-configurations define solutions of modified Einstein equations (1) and
their nonholonomic deformations. Such systems of nonlinear PDEs are extended
in certain forms encoding nonmetric geometric flow data. We provide necessary
examples of solitonic distributions defined in quasi-stationary geometric form
and study models with locally anisotropic wormhole solutions encoding
nonmetricity. Nonmetric quasi-stationary deformations of 4-d wormhole metrics
in GR are constructed for general and small parametric off-diagonal and
nonmetric gravitational polarizations.

The third objective, Obj3, is stated for section 4. It consists of a study of
$Q$-modified Grigori Perelman thermodynamics and its applications for quasi-
stationary configurations. We show how to define and compute (using
integration functions and nonlinear symmetries to some flow-running
cosmological constants) respective volume elements. This allows us to compute
thermodynamic variables with running cosmological constants and nonmetricity.
For nonmetric modified wormhole configurations and their solitonic
deformations, we show that, in principle, we can construct two thermodynamic
models: a ’l’a Perelman’ and/ or do not follow the Bekenstein-Hawking approach
because the last one is applicable only for solutions with conventional
horizons, holographic models and similar. Modified Perelman thermodynamic
models can be formulated for all classes of nonholonomic geometric flow
theories, including theories with $Q$-deformations.

In this article and a series of further partner works on nonmetric geometric
flow and gravity theories we follow the Hypothesis: _Metric-affine geometric
flow models can be exploited as alternatives for describing DE and DM effects
and elaborating new physical theories. Such approaches can be elaborated in
self-consistent and solvable forms using nonholonomic variables with
conventional 2(3)+2+… splitting which allows to decouple and integrate of
physically important systems of nonlinear PDEs for such theories. The
solutions with conventional $\tau$ -running effective cosmological constants
physics can be used for modelling DE physics when nonlinear symmetries relate
such configurations to models of DM physics. Generic off-diagonal metrics are
determined by respective generating functions and effective matter sources
which encode nonmetric $Q$-deformations and describe DE and DM off-diagonal
interactions or geometric evolution scenarios. Corresponding systems of
$Q$–deformed geometric flow/ gravitational and (effective) matter field
equations admit exact and parametric solutions describing certain quasi-
stationary (BH, wormhole etc.) configurations and (locally anisotropic and
inhomogeneous) cosmological scenarios. For well-defined nonholonomic geometric
constraints, such models can be defined almost equivalently in canonical
metric compatible backgrounds with effective N-connection structure, when
generalized conservation laws and nonlinear symmetries are well-defined. In
such cases, we can formulate self-consistent nonmetric modifications of
classical gravity and quantum gravity theories, quantum mechanical models and
quantum field theories, which can be unified as thermodynamic information
theories in the framework of respective geometric and quantum information
models._

In Appendix A, we revise in a nonmetric quasi-stationary form all formulas
which are necessary for general decoupling and integration of $Q$-modified
nonholonomic Ricci flow/ soliton equations. Details and proofs for general
metric compatible canonical d-connections are provided in [10, 11, 12, 13]. In
this work, abstract and N-adapted coefficient formulas are re-defined by
$Q$-deformations and $Q$-generating sources. A summary of basic concepts and
formulas for generating solitonic hierarchies via d-metrics and nonmetricity
effective sources is presented in Appendix B.

##  2 Metric noncompatible geometric flows and MGTs

This section contains an introduction to the geometry of four dimensional,
4-d, metric-affine spaces with nontrivial torsion and nonmetricity fields. The
approach is formulated in canonical nonholonomic variables with
(2+2)-splitting defined by a nonlinear connection, N-connection, structure
stating dyadic frame decompositions. We consider N-adapted distortions of
linear connections and fundamental geometric objects. The constructions are
performed in such forms that physically important systems of nonlinear PDEs
(such as nonmetric geometric flow evolution and modified Einstein equations)
can be decoupled and integrated in certain general off-diagonal forms.
Necessary concepts and additional technical formulas for the nonmetric
anholonomic frame and connection deformation method, AFCDM, are outlined in
appendix A. Such a nonholonomic geometric formalism was elaborated for (co)
tangent bundles, see reviews of results and methods in [10, 12]. In this
section, we develop the approach in a form which allows to construct exact and
parametric solutions with nonmetricity for gravity theories of type in [5, 6,
7, 8]. The Grigori Perelman functionals [15] (see [16] for recent developments
related to GR, MGTs and quantum information flow theories) are modified in
nonmetric forms. For self-similar configurations (i.e. for nonholonomic Ricci
solitons) such models encode the action functionals for gravity theories with
nontrivial $Q$-fields [6].

###  2.1 Geometric preliminaries on metric-affine spaces and nonholonomic
deformations

In this work, the background geometric arena consists from a Lorentz spacetime
manifold $V$ enabled with standard geometric data $(V,\mathbf{g,}\nabla).$
Such a (primary) spacetime is defined as a 4-d pseudo-Riemannian manifold of
necessary smooth/ differentiability class, when the symmetric metric tensor
$\mathbf{g}$ is of signature $(+++-)$ and can be written in the form

$\mathbf{g}=g_{\alpha^{\prime}\beta^{\prime}}(u)e^{\alpha^{\prime}}\otimes
e^{\beta^{\prime}},$ (2)

using the tensor product $\otimes$ of general co-frames $e^{\alpha^{\prime}}$,
which are dual to frame bases $e_{\alpha^{\prime}}.$ In general form, the
geometric and physical constructions are performed for metric-affine spaces
(target ones) determined by geometric data $(V,\mathbf{g,}D),$ when
$\nabla\rightarrow D$ and, in general, the nonmetricity field is nontrivial,
$Q:=D\mathbf{g}\neq 0.$ To elaborate theories of geometric flows [15, 16] one
considers families of metrics
$\mathbf{g}(\tau)=\\{g_{\alpha\beta}(\tau,u)\\},$ where $\tau$ is a
temperature like parameter considered for an interval $0\leq\tau\leq\tau_{1},$
or $\tau=\tau_{0}$ for a fixed value. Frame vectors can be prescribed to
depend, or not, on $\tau$-parameter, i.e. $e_{\alpha^{\prime}}(\tau),$ or
$e_{\alpha^{\prime}}.$ For simplicity, we shall write only
$g_{\alpha\beta}(\tau)$ instead of $g_{\alpha\beta}(\tau,u)$ if that will not
result in ambiguities.

####  2.1.1 N-adapted metric-affine structures with nonholonomic (2+2)
splitting

We introduce a nonlinear connection, N-connection, structure as a Whitney sum:

$\mathbf{N}:\ TV=hV\oplus vV,$ (3)

which is globally defined on $V$ and its tangent bundle $TV.$ A $\mathbf{N}$
defines a conventional horizontal and vertical splitting ( h- and
v–decomposition) into respective 2-d and 2-d subspaces, $hV$ and $vV.$ In
local coordinate form, a N-connection is defined by a set of coefficients
$N_{i}^{a}(u)$ when $\mathbf{N}=N_{i}^{a}(x,y)dx^{i}\otimes\partial/\partial
y^{a}.$333We can always define local coordinates
$u=\\{u^{\alpha}=(x^{i},y^{a})\\}$ involving a conventional $2+2$ splitting
into h-coordinates, $x=(x^{i}),$ and v-coordinates, $y=(y^{a}),$ for indices
$j,k,...=1,2$ and $a,b,c,...=3,4,$ when $\alpha,\beta,...=1,2,3,4.$ Using
partial derivatives, local coordinate basis and a co-base are computed
respectively as $e_{\alpha}=\partial_{\alpha}=\partial/\partial u^{\beta}$ and
$e^{\beta}=du^{\beta}.$ Transforms to arbitrary frames (tetrads / vierbeinds)
are defined as $e_{\alpha^{\prime}}=e_{\
\alpha^{\prime}}^{\alpha}(u)e_{\alpha}$ and $e^{\alpha^{\prime}}=e_{\alpha\
}^{\ \alpha^{\prime}}(u)e^{\alpha}.$ Usually, such (co) bases are
orthonormalized by the conditions $e_{\alpha\ }^{\ \alpha^{\prime}}e_{\
\alpha^{\prime}}^{\beta}=\delta_{\alpha}^{\beta},$ where
$\delta_{\alpha}^{\beta}$ is the Kronecker symbol. On Lorentz manifolds, a
N-connection (3) states a nonholonomic distribution defining a fibred 2+2
structure. We use the term nonholonomic Lorentz / pseudo-Riemannian manifold
when a conventional h-v-splitting is considered. Typically, ”boldface” symbols
are used to emphasize that certain spaces or geometric objects are enabled
(adapted) with (to) a N-connection structure.

N–elongated/adapted local bases, $\mathbf{e}_{\nu},$ and co-bases
(N–differentials), $\mathbf{e}^{\mu},$ are defined

$\displaystyle\mathbf{e}_{\nu}$ $\displaystyle=$
$\displaystyle(\mathbf{e}_{i},e_{a})=(\mathbf{e}_{i}=\partial/\partial x^{i}-\
N_{i}^{a}(u)\partial/\partial y^{a},\ e_{a}=\partial_{a}=\partial/\partial
y^{a}),\mbox{ and }$ (4) $\displaystyle\mathbf{e}^{\mu}$ $\displaystyle=$
$\displaystyle(e^{i},\mathbf{e}^{a})=(e^{i}=dx^{i},\ \mathbf{e}^{a}=dy^{a}+\
N_{i}^{a}(u)dx^{i}),$ (5)

to be linear on $N_{i}^{a}.$ The term nonholonomic (equivalently, anholonomic)
is used because, for instance, a N-elongated basis (4) satisfies certain
nonholonomy relations

$[\mathbf{e}_{\alpha},\mathbf{e}_{\beta}]=\mathbf{e}_{\alpha}\mathbf{e}_{\beta}-\mathbf{e}_{\beta}\mathbf{e}_{\alpha}=W_{\alpha\beta}^{\gamma}\mathbf{e}_{\gamma},$
(6)

with nontrivial anholonomy coefficients

$W_{ia}^{b}=\partial_{a}N_{i}^{b},W_{ji}^{a}=\Omega_{ij}^{a}=\mathbf{e}_{j}\left(N_{i}^{a}\right)-\mathbf{e}_{i}(N_{j}^{a}).$
(7)

In these formulas, $\Omega_{ij}^{a}$ define the coefficients of N-connection
curvature. If all $W_{ia}^{b}$ (7) are zero for a $\mathbf{e}_{\alpha},$ such
a N-adapted base is holonomic and we can write it as a partial derivative
$\partial_{\alpha}$ with $N_{i}^{a}=0.$ In curved local coordinates, the
coefficients $N_{j}^{a}$ may be nontrivial even all
$W_{\alpha\beta}^{\gamma}=0$ and we may chose a holonomic base.444For
instance, we define and write a d–vector in N-adapted form as
$\mathbf{X}=(hX,vX)$. The geometric objects on a nonholonomic manifold
$\mathbf{V}$ enabled with a N-connection structure $\mathbf{N}$ are called
distinguished (i.e d-objects, d-vectors, d-tensors etc) if they are adapted to
the N–connection structure via corresponding decompositions with respect to
frames of type (4) and (5).

The geometric objects on a nonholonomic manifold $\mathbf{V}$ enabled with a
N-connection structure $\mathbf{N}$ (and on extensions to tangent,
$T\mathbf{V,}$ and cotangent, $T^{\ast}\mathbf{V}$, bundles; and their tensor
products, for instance, $T\mathbf{V\otimes}T^{\ast}\mathbf{V}$) are called
distinguished (in brief, d-objects, d-vectors, d-tensors etc) if they are
adapted to the N–connection structure via corresponding decompositions with
respect to frames of type (4) and (5). For instance, we write a d–vector as
$\mathbf{X}=(hX,vX)$.

Any spacetime metric $\mathbf{g}=(hg,vg)$ (2) can be represented equivalently
as a d–metric555Introducing coefficients of (5) into (8) and regrouping with
respect to the coordinate dual basis, we obtain the formulas for the
coefficients in (9),
$\underline{g}_{\alpha\beta}=\left[\begin{array}[]{cc}g_{ij}+N_{i}^{a}N_{j}^{b}g_{ab}&N_{j}^{e}g_{ae}\\\
N_{i}^{e}g_{be}&g_{ab}\end{array}\right].$ A metric
$\mathbf{g}=\\{\underline{g}_{\alpha\beta}\\}$ is generic off–diagonal if the
anholonomy coefficients $W_{\alpha\beta}^{\gamma}$ are not identical to zero.
For 4-d spacetimes, such a matrix can’t be diagonalized via coordinate
transforms., when

$\displaystyle\ \mathbf{g}$ $\displaystyle=$ $\displaystyle\ g_{ij}(x,y)\
e^{i}\otimes e^{j}+\ g_{ab}(x,y)\ \mathbf{e}^{a}\otimes\mathbf{e}^{b},\mbox{
in N-adapted form with }hg=\\{\ g_{ij}\\},vg=\\{g_{ab}\\};$ (8)
$\displaystyle=$
$\displaystyle\underline{g}_{\alpha\beta}(u)du^{\alpha}\otimes du^{\beta}.$
(9)

A d–connection $\mathbf{D}=(hD,vD)$ is defined as a linear connection
preserving under parallelism the N–connection splitting (3). In N-adapted
coefficient form with respect to frames (4) and (5), we can write
decompositions of $\mathbf{D}$ in terms of h- and v-indices,

$\mathbf{D}=\\{\mathbf{\Gamma}_{\
\alpha\beta}^{\gamma}=(L_{jk}^{i},\acute{L}_{bk}^{a};\acute{C}_{jc}^{i},C_{bc}^{a})\\},\mbox{
where }hD=(L_{jk}^{i},\acute{L}_{bk}^{a})\mbox{ and
}vD=(\acute{C}_{jc}^{i},C_{bc}^{a}).$ (10)

We define a nonholonomic metric-affine space by geometric data
$(\mathbf{V,N,g,D}).$

####  2.1.2 Geometric objects adapted to a N-connection structure and
nonmetricity

The fundamental geometric d-objects of nonholonomic metric-affine space are
defined:

$\displaystyle\mathcal{T}(\mathbf{X,Y})$ $\displaystyle:=$
$\displaystyle\mathbf{D}_{\mathbf{X}}\mathbf{Y}-\mathbf{D}_{\mathbf{Y}}\mathbf{X}-[\mathbf{X,Y}],\mbox{
torsion d-tensor, d-torsion};$ (11) $\displaystyle\mathcal{R}(\mathbf{X,Y})$
$\displaystyle:=$
$\displaystyle\mathbf{D}_{\mathbf{X}}\mathbf{D}_{\mathbf{Y}}-\mathbf{D}_{\mathbf{Y}}\mathbf{D}_{\mathbf{X}}-\mathbf{D}_{\mathbf{[X,Y]}},\mbox{
curvature d-tensor, d-curvature};$ $\displaystyle\mathcal{Q}(\mathbf{X})$
$\displaystyle:=$ $\displaystyle\mathbf{D}_{\mathbf{X}}\mathbf{g,}\mbox{\
nonmetricity d-fiels, d-nonmetricity}.$

The N-adapted coefficient formulas involving (8), (9) and (10) are provided
and computed in [10, 11, 12]. Here we only present, respectively, their 4-d
N-adapted coefficient representations,

$\mathcal{R}=\mathbf{\\{R}_{\
\beta\gamma\delta}^{\alpha}\\},\mathcal{T}=\\{\mathbf{T}_{\
\alpha\beta}^{\gamma}\\},\mathcal{Q}=\mathbf{\\{Q}_{\gamma\alpha\beta}\\}.$

In geometric flow and gravity theories, there are also another important
geometric d-objects:

$\displaystyle\mathbf{R}ic$ $\displaystyle:=$ $\displaystyle\\{\mathbf{R}_{\
\beta\gamma}:=\mathbf{R}_{\ \beta\gamma\alpha}^{\alpha}\\},\mbox{ the Ricci
d-tensor };$ $\displaystyle\mathbf{R}sc$ $\displaystyle:=$
$\displaystyle\mathbf{g}^{\alpha\beta}\mathbf{R}_{\ \alpha\beta},\mbox{ the
scalar curvature },$

where we uses the inverse d-tensor $\\{\mathbf{g}^{\alpha\beta}\\}$ of a
d-metric (8).

Using a d-metric $\mathbf{g}$ (8), we can define two important linear
connection structures:

$(\mathbf{g,N})\rightarrow\left\\{\begin{array}[]{cc}\mathbf{\nabla:}&\mathbf{\nabla
g}=0;\ _{\nabla}\mathcal{T}=0,\ \mbox{\ LC--connection };\\\
\widehat{\mathbf{D}}:&\widehat{\mathbf{Q}}=0;\
h\widehat{\mathcal{T}}=0,v\widehat{\mathcal{T}}=0,\
hv\widehat{\mathcal{T}}\neq 0,\mbox{ the canonical
d-connection}.\end{array}\right.$ (12)

In this paper, "hat" labels are used for geometric d-objects defined by a
$\widehat{\mathbf{D}}.$ Such an auxiliary d-connection defines a canonical
distortion relation

$\widehat{\mathbf{D}}[\mathbf{g}]=\nabla[\mathbf{g}]+\widehat{\mathbf{Z}}[\mathbf{g}],$
(13)

when the canonical distortion d-tensor, $\widehat{\mathbf{Z}}$, and
$\nabla[\mathbf{g}]$ are determined by the same metric structure
$\mathbf{g.}$666A canonical d-connection (12) is defined by N-adapted
coefficients $\widehat{\mathbf{D}}=\\{\widehat{\mathbf{\Gamma}}_{\
\alpha\beta}^{\gamma}=(\widehat{L}_{jk}^{i},\widehat{L}_{bk}^{a},\widehat{C}_{jc}^{i},\widehat{C}_{bc}^{a})\\},$
for  $\displaystyle\widehat{L}_{jk}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{ir}(\mathbf{e}_{k}g_{jr}+\mathbf{e}_{j}g_{kr}-\mathbf{e}_{r}g_{jk}),\widehat{L}_{bk}^{a}=e_{b}(N_{k}^{a})+\frac{1}{2}g^{ac}(\mathbf{e}_{k}g_{bc}-g_{dc}\
e_{b}N_{k}^{d}-g_{db}\ e_{c}N_{k}^{d}),$ $\displaystyle\widehat{C}_{jc}^{i}$
$\displaystyle=$ $\displaystyle\frac{1}{2}g^{ik}e_{c}g_{jk},\
\widehat{C}_{bc}^{a}=\frac{1}{2}g^{ad}(e_{c}g_{bd}+e_{b}g_{cd}-e_{d}g_{bc}).$

The coefficients of the canonical fundamental geometric d-objects (11) are
labeled by "hat" symbols, for instance,
$\widehat{\mathcal{R}}=\\{\widehat{\mathbf{R}}_{\
\beta\gamma\delta}^{\alpha}\\}.$ Similar fundamental geometric objects can be
defined and computed for $\nabla,$ for instance, $\
{}_{\nabla}\mathcal{R}=\\{\ _{\nabla}R_{\ \beta\gamma\delta}^{\alpha}\\}$ (in
such cases, boldface indices are not used). Considering the canonical
distortion relation for linear connections (13), we can compute respective
canonical distortions of fundamental geometric d-objects (11). Such formulas
relate, for instance, two different curvature tensors, $\
{}_{\nabla}\mathcal{R}=\\{\ _{\nabla}R_{\ \beta\gamma\delta}^{\alpha}\\}$ and
$\widehat{\mathcal{R}}=\\{\widehat{\mathbf{R}}_{\
\beta\gamma\delta}^{\alpha}\\}$ etc.

An arbitrary d-connection $\mathbf{D=\\{\mathbf{\Gamma}}_{\
\alpha\beta}^{\gamma}\\}$ with nontrivial nonmetric d-tensor
$\mathbf{Q}_{\gamma\alpha\beta}$ can be expressed via distortion d-tensors
with respect to $\nabla=\\{\breve{\Gamma}_{\ \alpha\beta}^{\gamma}\\}$ or
$\widehat{\mathbf{D}}=\\{\widehat{\mathbf{\mathbf{\Gamma}}}_{\
\alpha\beta}^{\gamma}\\},$

$\mathbf{\mathbf{\Gamma}}_{\ \alpha\beta}^{\gamma}=\breve{\Gamma}_{\
\alpha\beta}^{\gamma}+K_{\ \alpha\beta}^{\gamma}+\ ^{q}Z_{\
\alpha\beta}^{\gamma}\mbox{ and/or }\mathbf{\mathbf{\Gamma}}_{\
\alpha\beta}^{\gamma}=\widehat{\mathbf{\mathbf{\Gamma}}}_{\
\alpha\beta}^{\gamma}+\widehat{\mathbf{K}}_{\ \alpha\beta}^{\gamma}+\
^{q}\widehat{\mathbf{Z}}_{\ \alpha\beta}^{\gamma}.$ (14)

In these formulas, for instance, for $(T,Q)$-deformations of LC-
configurations, we use and construct such distortion tensors:

$K_{\alpha\beta\gamma}=\frac{1}{2}(T_{\alpha\beta\gamma}+T_{\beta\gamma\alpha}-T_{\gamma\alpha\beta}),\
S_{\gamma\ }^{\ \alpha\beta}=\frac{1}{2}(K_{\ \ \
\gamma}^{\alpha\beta}+\delta_{\gamma}^{\alpha}T_{\ \ \
\tau}^{\tau\beta}-\delta_{\gamma}^{\beta}T_{\ \ \ \tau}^{\tau\alpha}),\
^{q}Z_{\
\alpha\beta\gamma}=\frac{1}{2}(Q_{\alpha\beta\gamma}-Q_{\beta\gamma\alpha}-Q_{\gamma\beta\alpha}),$

where $Q_{\alpha\beta\gamma}:=D_{\alpha}g_{\beta\gamma}$ and $T_{\
\beta\gamma}^{\alpha}$ is computed for any $D=\\{\Gamma_{\
\alpha\beta}^{\gamma}\\}$ which can an arbitrary affine connection (with
coefficients in coordinate or arbitrary frames). The torsion d-tensor
$\mathbf{T}_{\ \beta\gamma}^{\alpha}$ can be computed for an arbitrary
d-connection $\mathbf{D}=\\{{\Gamma}_{\ \alpha\beta}^{\gamma}\\},$ using
N-adapted bases. Such objects are defined with respect to coordinate or
N-adapted frames and used for introducing three scalar values considered in
the Weyl–Cartan geometry:

$Rsc[D]=R=g^{\beta\gamma}\ R_{\beta\gamma},\ ^{s}T=\ S_{\gamma\ }^{\
\alpha\beta}T_{\ \alpha\beta}^{\gamma},Q=\ ^{q}Z_{\ \beta\alpha}^{\alpha}\
{}^{q}Z_{\quad\mu}^{\beta\mu}-\ ^{q}Z_{\ \beta\mu}^{\alpha}\
{}^{q}Z_{\quad\alpha}^{\beta\mu}.$

For the nonholonomic Weyl-Cartan geometry, corresponding scalar values are
defined for $\mathbf{D}=\\{{\Gamma}_{\ \alpha\beta}^{\gamma}\\}$ and
$\mathbf{Q}_{\alpha\beta\gamma}:=\mathbf{D}_{\alpha}\mathbf{g}_{\beta\gamma}$
as distortions (13) of $\widehat{\mathbf{D}}=\\{\widehat{\Gamma}_{\
\alpha\beta}^{\gamma}\\}$ and can be written

$\mathbf{R}sc[\mathbf{D}]=\ ^{s}R=\mathbf{g}^{\beta\gamma}\
\mathbf{R}_{\beta\gamma},\ ^{s}\mathbf{T}=\ \mathbf{S}_{\gamma\ }^{\
\alpha\beta}\mathbf{T}_{\ \alpha\beta}^{\gamma},\ ^{q}\mathbf{Q}=\
^{q}\mathbf{Z}_{\ \beta\alpha}^{\alpha}\
{}^{q}\mathbf{Z}_{\quad\mu}^{\beta\mu}-\ ^{q}\mathbf{Z}_{\ \beta\mu}^{\alpha}\
{}^{q}\mathbf{Z}_{\quad\alpha}^{\beta\mu}.$

MGTs with torsion and nonmetricity are modelled for actions of type

$S(g,D,\phi)=\int\sqrt{|g|}d^{4}u[\ ^{g}\mathcal{L}+\ ^{m}\mathcal{L}],$ (15)

where the gravitational Lagrange density is a functional $\
{}^{g}\mathcal{L}=F(R,\ ^{s}T,Q,T^{[m]})$ and $\ {}^{m}\mathcal{L}[\phi]$ is
the Lagrange density for conventional matter fields $\phi.$ Such a model is
studied in [6] but, in this work, we follow a different system of notations
considered, for instance, in [11] (for instance, we write $F(...)$ instead of
$f(...)$). The gravitational and matter field equations of type (1) derived
variationally from an action (15) consist sophisticate coupled systems of
nonlinear PDEs. It is very difficult to find exact/parametric solutions in
such MGTs even, for instance, certain cosmological and DE and DM models were
studied in [6, 7, 8].

In this work, we shall consider MGTs of type (15) written in canonical
d-variables $(\mathbf{g,D}=\widehat{\mathbf{D}}+\widehat{\mathbf{K}}+\
^{q}\widehat{\mathbf{Z}}),$ see distortions (14), and following a N-adapted
variational calculus for actions of type

$S(g,\mathbf{D},\phi)=\int\sqrt{|\mathbf{g}|}\delta^{4}u[\
^{g}\widehat{\mathcal{L}}\mathcal{+}\ ^{e}\widehat{\mathcal{L}}+\
^{m}\widehat{\mathcal{L}}],$ (16)

where $\delta^{4}u$ is the volume element defined with N-elongated
differentials (5), $\ {}^{g}\widehat{\mathcal{L}}=F(\widehat{\mathbf{R}}sc,\
^{s}\widehat{T},T^{[m]})$ is computed as $\ {}^{g}\mathcal{L}$ from (15) when
$\nabla\rightarrow\widehat{\mathbf{D}},$ for
$\widehat{\mathbf{Q}}_{\alpha\beta\gamma}=0$; $\
{}^{e}\widehat{\mathcal{L}}(\mathbf{Q}_{\alpha\beta\gamma},\widehat{\mathbf{D}},\phi)$
includes distortions of geometric d-objects and Lagrangians deformations for
$\widehat{\mathbf{D}}\rightarrow\mathbf{D}$ is nontrivial
$\mathbf{Q}_{\alpha\beta\gamma};$ and $\
{}^{m}\widehat{\mathcal{L}}(\mathbf{g}_{\alpha\beta},\phi,\widehat{\mathbf{D}})$
if $\ {}^{m}\mathcal{L}(g_{\alpha\beta},\phi,\nabla),$ or (we can consider
some simplified models) when $\
{}^{m}\widehat{\mathcal{L}}(\mathbf{g}_{\alpha\beta},\phi)=\
^{m}\mathcal{L}(\mathbf{g}_{\alpha\beta},\phi).$

In a series of works [11, 12, 13, 16], we proved that geometric flow and
gravitational field equations in MGTs with $\mathbf{D}=\widehat{\mathbf{D}}$
or $\mathbf{D\rightarrow}\widehat{\mathbf{D}}\rightarrow\nabla$ can be
decoupled and integrated in certain general off-diagonal forms using the
AFCDM. The general goal of this article is to show how those methods can be
generalized for nontrivial $\mathbf{Q}_{\alpha\beta\gamma}(\tau)$ and applied
for research of the relativistic Ricci flows of nonholonomic metric-affine
structures, or corresponding nonholonomic Ricci soliton equations for any
$\tau_{0}.$

###  2.2 Relativistic geometric flows encoding nonmetricity fields

The theory of Ricci flows has a high scientific impact in modern mathematics
and physics after Grigori Perelman proved [15] the famous Poincaré–Thurston
conjecture, see original works [19] and [17, 18] and monographs [20, 21, 22]
for reviews of mathematical results and methods. A crucial step in elaborating
such theories consisted in definition of the so-called $\mathcal{F}$\- and
$\mathcal{W}$-functionals from which the geometric flow equations (called also
as R. Hamilton or Hamilton-Friedan equations) can be proved in variational
form. It is not clear how mathematically can be formulated and proved
relativistic variants of such conjectures and generalizations for nonmetric/
supersymmetric/ nonassociative / noncommutative / Finsler like geometries.
Nevertheless, generalizations of $\mathcal{F}$\- and $\mathcal{W}$-functionals
allow to prove modified versions of geometric flow equations and solve such
systems of nonlinear PDEs using the AFCDM. The results of a series of recent
papers [23, 16] demonstrate that Perelman like information thermodynamics may
play an important role in the theory of quantum geometric and information,
QGIF, flows. In this work, we show how such constructions can be performed for
nonmetric geometric flows for families $\tau$-evolving metric-affine data
$(\mathbf{g}(\tau),\mathbf{D}(\tau))$ and Lagrange densities $\
{}^{g}\widehat{\mathcal{L}}(\tau)+\ ^{e}\mathcal{L}(\tau)+\
^{m}\widehat{\mathcal{L}}(\tau).$

####  2.2.1 $Q$-modified Perelman’s F- and W-functionals in canonical
nonholonomic variables

The modified Perelman’s functionals for nonmetric geometric flows are
postulated in the form

$\displaystyle\mathcal{F}(\tau)$ $\displaystyle=$
$\displaystyle\int_{t_{1}}^{t_{2}}\int_{\Xi_{t}}e^{-f(\tau)}\sqrt{|\mathbf{g}(\tau)|}\delta^{4}u[F(\widehat{\mathbf{R}}sc(\tau),\
^{s}\widehat{T}(\tau),T^{[m]}(\tau))+\ ^{e}\mathcal{L}(\tau)+\
^{m}\widehat{\mathcal{L}}(\tau)+|\mathbf{D}(\tau)f(\tau)|^{2}],$ (17)
$\displaystyle\mathcal{W}(\tau)$ $\displaystyle=$
$\displaystyle\int_{t_{1}}^{t_{2}}\int_{\Xi_{t}}\left(4\pi\tau\right)^{-2}e^{-f(\tau)}\sqrt{|\mathbf{g}(\tau)|}\delta^{4}u[\tau(F(\widehat{\mathbf{R}}sc(\tau),\
^{s}\widehat{T}(\tau),T^{[m]}(\tau))+$ $\ {}^{e}\mathcal{L}(\tau)+\
^{m}\widehat{\mathcal{L}}(\tau)+|\mathbf{D}(\tau)f(\tau)|^{2}+f(\tau)-4],$

where the condition
$\int_{t_{1}}^{t_{2}}\int_{\Xi_{t}}\left(4\pi\tau\right)^{-2}e^{-f(\tau)}\sqrt{|\mathbf{g}|}d^{4}u=1$
is imposed on the normalizing function $f(\tau)=f(\tau,u).$ The difference
from the original F- and W-functionals [15] introduced for 3-d Riemannian
$\tau$-flows $(g(\tau),\nabla(\tau))$ is that in this work we study geometric
flows of canonical geometric data $(\mathbf{g}(\tau),$
$\mathbf{N}(\tau),\mathbf{D}(\tau))$ for $Q$-deformations of nonholonomic
Lorentz manifolds.

We can compute relativistic effective functionals (17) and (2.2.1) for any 3+1
splitting with 3-d closed hypersurface fibrations $\widehat{\Xi}_{t}$ and
considering nonholonomic canonical d-connections and respective geometric
variables. In general, it is possible to work with any class of normalizing
functions $f(\tau)$. Such a function can be fixed by some constant values or
some parametrization conditions simplifying corresponding systems of nonlinear
PDEs. Such $f(\tau)$ define respective integration measures which may be
important, or not, for elaborating topological and/or geometric models. The
$\mathcal{W}$-functional possess the properties of "minus" entropy. This can
be stated by choosing corresponding nonholonomic configurations along some
causal curves taking values $\mathcal{W}(\tau)$ on $\widehat{\Xi}_{t}.$ Using
N-adapted variations, we can derive nonmetric geometric flow evolution
equations which can be solved using the AFCDM for metrics with pseudo-
Euclidean signature even analogs of Poincaré–Thurston conjecture have not been
formulated and proven in modern mathematics.

####  2.2.2 Hamilton-Friedan geometric flow equations with nonmetricity

There are two possibilities to derive geometric flow equations from
functionals $\mathcal{F}(\tau)$ (17) and $\mathcal{W}(\tau)$ (2.2.1). In the
first case, we can use $\mathbf{D}(\tau)$ instead of $\nabla(\tau)$ and
reproduce in N-adapted form all covariant differential and integral calculus
from [15, 20, 21, 22]. This would consist proofs on some hundred of pages.

We can follow geometric abstract principles [1] when all geometric and
physically important objects and fundamental physical equations are derived by
corresponding generalizations of Riemannian geometry to certain nonholonomic
metric-affine geometries with
$\nabla(\tau)\rightarrow\mathbf{D}(\tau)=\widehat{\mathbf{D}}(\tau)+\widehat{\mathbf{K}}(\tau)+\
^{q}\widehat{\mathbf{Z}}(\tau)$ respective generalizations of Ricci, torsion,
and energy-momentum d-tensors. Such an abstract geometric calculus allows to
prove for some primary data
$(\mathbf{g}=\\{\mathbf{g}_{\mu\nu}=[g_{ij},g_{ab}]\\},\mathbf{N}=\\{N_{i}^{a}\\},\mathbf{D},\
^{tot}\widehat{\mathcal{L}}=\ ^{e}\mathcal{L}+\ ^{m}\widehat{\mathcal{L}}),$
see definitions related to formulas (16) the nonholonomic geometric flow
evolution equations:

$\displaystyle\partial_{\tau}g_{ij}(\tau)$ $\displaystyle=$
$\displaystyle-2[\widehat{\mathbf{R}}_{ij}(\tau)-\
^{tot}\widehat{\Upsilon}_{ij}(\tau)];\
\partial_{\tau}g_{ab}(\tau)=-2[\widehat{\mathbf{R}}_{ab}(\tau)-\
^{tot}\widehat{\Upsilon}_{ab}(\tau)];$ (19)
$\displaystyle\widehat{\mathbf{R}}_{ia}(\tau)$ $\displaystyle=$
$\displaystyle\widehat{\mathbf{R}}_{ai}(\tau)=0;\widehat{\mathbf{R}}_{ij}(\tau)=\widehat{\mathbf{R}}_{ji}(\tau);\widehat{\mathbf{R}}_{ab}(\tau)=\widehat{\mathbf{R}}_{ba}(\tau);\
$ $\displaystyle\partial_{\tau}\widehat{f}(\tau)$ $\displaystyle=$
$\displaystyle-\widehat{\square}(\tau)[\widehat{f}(\tau)]+\left|\widehat{\mathbf{D}}(\tau)[\widehat{f}(\tau)]\right|^{2}-\
^{s}\widehat{R}(\tau)+\ ^{tot}\widehat{\Upsilon}_{\ \alpha}^{\alpha}(\tau).$

In these formulas, there are used such geometric d-objects and N-adapted
operators:
$\widehat{\square}(\tau)=\widehat{\mathbf{D}}^{\alpha}(\tau)\widehat{\mathbf{D}}_{\alpha}(\tau)$
when the conditions $\widehat{\mathbf{R}}_{ia}=\widehat{\mathbf{R}}_{ai}=0$
for the Ricci tensor
$\widehat{R}ic[\widehat{\mathbf{D}}]=\\{\widehat{\mathbf{R}}_{\alpha\beta}=[\widehat{R}_{ij},\widehat{R}_{ia},\widehat{R}_{ai},\widehat{R}_{ab,}]\\}$
are necessary if we want to keep the metric $\mathbf{g}(\tau)$ to be symmetric
under nonholonomic Ricci flow evolution. Such constraints are not obligatory,
for instance, in nonassociative geometric flow theory with nonsymmetric
metrics [13].

The definition of $\ {}^{tot}\widehat{\Upsilon}_{ab}(\tau)$ from (19) will be
discussed in section (2.3) for $\tau=\tau_{0}.$ Here we note that such
equations describe nonmetric geometric flow evolution of d-metrics
$\mathbf{g}_{\mu\nu}(\tau)$ described in nonholonomic canonical variables.
Alternatively, such equations can be introduced as relativistic
generalizations and nonholonomic canonical deformations of the R. Hamilton
[19] and D. Friedan [17, 18] Ricci flow equations for $\nabla(\tau).$

The normalizing function $f(\tau)$ can be re-defined in such a way that it
compensates certain $Q$-deforms, or other type nonholonomic distortions, when
$\widehat{f}(\tau)\rightarrow f(\tau)$ for

$\partial_{\tau}f=-\square f+\left|\mathbf{D}f\right|^{2}-\ ^{s}R+\
^{tot}\Upsilon_{\ \alpha}^{\alpha}.$

Such an equation involves nonlinear partial differential operators and usually
it is not possible to solve it in an explicit form and define the evolution of
topological configurations determined, for instance, by nontrivial nonmetric
structures. Nevertheless, we can fix a variant of $\widehat{f}(\tau)$ which
together with some off-diagonal ansatz for metrics we can solve the
nonholonomic system of nonlinear PDEs (19) in certain general/ parametric
forms and then to re-define the constructions in for arbitrary systems of
reference, other types of distortions of connections and normalizing
functions. The formulas for nonholonomic frame/coordinate/ normalizing
transforms could be found in certain series/ recurrent form when the solutions
of geometric flow equations are generated in explicit form.

###  2.3 Nonmetric Ricci solitons and modified Einstein equations

A nonholonomic and nonmetric Ricci soliton configuration is a self-similar one
for the geometric flow equations (19). For Riemannian metrics, such
configurations homothetically strink, remain steady or expand under geometric
flow evolution, see details in [15, 19, 20, 21, 22], and can be respectively
studied for a fixed point $\tau=\tau_{0}$. Considering relativistic and
torsion and nonmetricity modified nonlinear systems with
$\partial_{\tau}\mathbf{g}_{\mu\nu}=0$ and for a specific choice of the
normalizing geometric flow function $f,$ the equations (19) transform into
nonholonomic Ricci soliton equations encoding $Q$-distortions into effective
sources. Such systems of nonlinear PDEs are equivalent to modified Einstein
equations in nonholonomic metric-affine gravity for corresponding definitions
of effective sources $\ {}^{tot}\widehat{\Upsilon}_{\alpha\beta}(\tau_{0})=[\
^{tot}\widehat{\Upsilon}_{ij},\ ^{tot}\widehat{\Upsilon}_{ab}],$

$\displaystyle\widehat{\mathbf{R}}_{ij}$ $\displaystyle=$ $\
{}^{tot}\widehat{\Upsilon}_{ij},\ \widehat{\mathbf{R}}_{ab}=\
^{tot}\widehat{\Upsilon}_{ab},$ (20) $\displaystyle\widehat{\mathbf{R}}_{ia}$
$\displaystyle=$
$\displaystyle\widehat{\mathbf{R}}_{ai}=0;\widehat{\mathbf{R}}_{ij}=\widehat{\mathbf{R}}_{ji};\widehat{\mathbf{R}}_{ab}=\widehat{\mathbf{R}}_{ba}.$

The effective sources in (20) are can be parameterized as for the effective
Lagrangians (16)

$\ {}^{tot}\widehat{\mathbf{\Upsilon}}_{\mu\nu}:=\varkappa\left(\
{}^{tot}\widehat{\mathbf{T}}_{\mu\nu}-\frac{1}{2}\mathbf{g}_{\mu\nu}\
^{tot}\widehat{\mathbf{T}}\right)=(\frac{\partial F}{\partial\ ^{s}R})^{-1}\
{}^{m}\widehat{\mathbf{\Upsilon}}_{\mu\nu}+\
^{F}\widehat{\mathbf{\Upsilon}}_{\mu\nu}+\
^{e}\widehat{\mathbf{\Upsilon}}_{\mu\nu},$ (21)

where $\varkappa$ is determined in standard form by the Newton gravitational
constant $G,$

$\
{}^{m}\widehat{\mathbf{T}}_{\mu\nu}:=-\frac{2}{\sqrt{|\mathbf{g}|}}\frac{\delta(\sqrt{|\mathbf{g}|}\
^{m}\widehat{\mathcal{L}}\mathcal{)}}{\delta\mathbf{g}^{\mu\nu}}=2\frac{\delta\
^{m}\widehat{\mathcal{L}}}{\delta\mathbf{g}^{\mu\nu}}+\mathbf{g}_{\mu\nu}\
^{m}\widehat{\mathcal{L}}.$

For the full system relating nonholonomic Ricci solitons to modified gravity,
the effective energy-momentum d-tensor is computed for

$\ {}^{F}\widehat{\mathbf{T}}_{\beta\gamma}=[\frac{1}{2}(F-\frac{\partial
F}{\partial\
^{s}R})\mathbf{g}_{\beta\gamma}-(\mathbf{g}_{\beta\gamma}\widehat{\mathbf{D}}_{\alpha}\widehat{\mathbf{D}}^{\alpha}-\widehat{\mathbf{D}}_{\beta}\widehat{\mathbf{D}}_{\gamma})\frac{\partial
F}{\partial\ ^{s}R}](\frac{\partial F}{\partial\ ^{s}R})^{-1},$

when

$\ {}^{tot}\widehat{\mathbf{T}}_{\mu\nu}=(\frac{\partial F}{\partial\
^{s}R})^{-1}\ {}^{m}\widehat{\mathbf{T}}_{\mu\nu}+\
^{F}\widehat{\mathbf{T}}_{\mu\nu}+\ ^{e}\widehat{\mathbf{T}}_{\mu\nu}.$ (22)

Choosing $F(\ ^{s}R)=\ ^{s}R$ and the Levi-Civita connection
$\mathbf{D}\rightarrow\nabla,$ we can relate above formulas to GR.

####  2.3.1 Connecting nonholonomic solitons to nonmetric modified
gravitational equations

The gravitational field equations in Weyl-Cartan MGT can be constructed by
considering variations of the action on a metric-affine manifold determined by
geometric objects using such values constructed for the affine connection
$\mathbf{D}$ expressed as a canonical distortion (14) from
$\widehat{\mathbf{D}}$ involving $\ {}^{e}\widehat{\mathbf{T}}_{\mu\nu}$ as an
effective source containing nontrivial contributions from
$\mathbf{Q}_{\alpha\beta\gamma}:$

$\displaystyle\widehat{\mathbf{H}}_{\lambda}^{\ \mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{2}{\sqrt{|\mathbf{g}|}}\frac{\delta(\sqrt{|\mathbf{g}|}\
\delta\
^{m}\widehat{\mathcal{L}}\mathcal{)}}{\delta\widehat{\mathbf{\mathbf{\Gamma}}}_{\
\mu\nu}^{\alpha}}\mbox{ the canonical hyper-momentum d-tensor },$ and
$\displaystyle\mathbf{A}_{\mu\alpha\nu}=\frac{\partial F}{\partial
Q}(\mathbf{g}_{\mu\nu}\ ^{q}\mathbf{Z}_{\ \
\alpha\nu}^{\alpha}+\mathbf{g}_{\alpha\nu}\
^{q}\mathbf{Z}_{\mu\beta}^{\quad\beta}-\ ^{q}\mathbf{Z}_{\ \mu\alpha\nu}-\
^{q}\mathbf{Z}_{\ \nu\alpha\mu}).$

Such values can be constructed in terms of
$\widehat{\mathbf{\mathbf{\Gamma}}}_{\ \mu\nu}^{\alpha}$ and/or
$\mathbf{\mathbf{\Gamma}}_{\ \mu\nu}^{\alpha}.$ In coordinate bases and in non
N-adapted form, such results are presented, for instance by formulas (23) -
(26) in [6].

In this work, we consider a model of nonholonomic nonmetric Ricci solitons
with a Weyl d-vector $\mathbf{q}_{\alpha},$ when
$\mathbf{Q}_{\alpha\beta\gamma}=\mathbf{q}_{\alpha}\mathbf{g}_{\beta\gamma}$
and nontrivial d-torsion
$\mathbf{T}_{\mu\nu\alpha}=\mathbf{A}_{\nu}\mathbf{g}_{\mu\alpha}-\mathbf{A}_{\alpha}\mathbf{g}_{\mu\nu},$
for $\mathbf{A}_{\mu}=q\mathbf{q}_{\mu},q=const.$ For such approximations
considering $\frac{\partial F}{\partial T}=1/2q\frac{\partial F}{\partial Q},$
the variational N-adapted gravitational field equations with
$\widehat{\mathbf{D}}$ can be written in the form:

$\displaystyle\widehat{\mathbf{D}}_{\beta}\frac{\partial F}{\partial\
^{s}R}+[\frac{\partial F}{\partial Q}+(1-q)\frac{\partial F}{\partial\
^{s}R}]\mathbf{q}_{\beta}$ $\displaystyle=$ $\displaystyle 0;$ (23)
$\displaystyle\widehat{\mathbf{R}}_{\ \beta\gamma}$ $\displaystyle=$ $\
{}^{q}\widehat{\Upsilon}_{\alpha\beta}(\tau_{0})$ (24)

for effective $Q$-source

$\ {}^{q}\widehat{\Upsilon}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(\frac{\partial F}{\partial\
^{s}R})^{-1}[\frac{F}{2}\mathbf{g}_{\alpha\beta}+\frac{\partial F}{\partial\
^{m}T}(\widehat{\mathbf{T}}_{\alpha\beta}-\mathbf{g}_{\alpha\beta}\
^{m}\widehat{\mathcal{L}})-\mathbf{g}_{\alpha\beta}\widehat{\square}\frac{\partial
F}{\partial\
^{s}R}-\widehat{\mathbf{D}}_{\alpha}\widehat{\mathbf{D}}_{\beta}\frac{\partial
F}{\partial\ ^{s}R}$
$\displaystyle-(q-\frac{1}{2}+\frac{3}{q})(\widehat{\mathbf{D}}_{\alpha}\mathbf{q}_{\beta}-\widehat{\mathbf{D}}_{\beta}\mathbf{q}_{\alpha})(\frac{\partial
F}{\partial\ ^{s}R})-\frac{1}{2}\widehat{\mathbf{T}}_{\alpha\beta}]$
$\displaystyle+(2+\frac{1}{2}-\frac{3}{q})\mathbf{q}_{\alpha}\mathbf{q}_{\beta}-3(\frac{1}{2}-\frac{1}{q})(\widehat{\mathbf{D}}_{\alpha}\mathbf{q}_{\beta}-\widehat{\mathbf{D}}_{\beta}\mathbf{q}_{\alpha})+\frac{1}{2}(1+q)(\mathbf{q}_{\tau}\mathbf{q}^{\tau})~{}\mathbf{g}_{\alpha\beta}.$

We defined the system of constraints and nonlinear PDEs (23)-(2.3.1) in a form
that for $\widehat{\mathbf{D}}\rightarrow\nabla$ it transforms into respective
equations (37) and (38) in [6]. For such nonholonomic Ricci soliton equations,
we can decouple and integrate in certain general forms the modified Einstein
equations (24) if the $Q$-source (2.3.1) is generated by two effective sources
(see below). It is not possible to decouple such equations for generic off-
diagonal $\mathbf{g}_{\beta\gamma}$ if it is considered only the LC-connection
$\nabla$ and/or general nonmetricity fields.

####  2.3.2 Generating sources for $\tau$-running quasi-stationary effective
matter fields and nonmetricity deformations

In this work, we shall construct and study physical implications of quasi-
stationary solutions of nonmetric geometric flow equations (19) when the
metric (2) (in equivalent form, the d-metric (8)) is determined by N-adapted
coefficients $\mathbf{g}(\tau)=[g_{i}(\tau),g_{a}(\tau),N_{i}^{a}(\tau)],$
when such coefficients do not depend on variable $y^{4}=t$ and can be
parameterized in the form

$g_{i}(\tau)=e^{\psi{(\tau,x^{j})}},g_{a}(\tau)=h_{a}(\tau,x^{k},y^{3}),\
N_{i}^{3}=w_{i}(\tau,x^{k},y^{3}),\,\,\,\,N_{i}^{4}=n_{i}(\tau,x^{k},y^{3}).$
(26)

Let us consider effective sources $\ {}^{tot}\widehat{\Upsilon}_{ab}(\tau)$
from (19) which via N–adapted frames can be parameterized in the form

$\mathbf{\Upsilon}_{\ \nu}^{\mu}(\tau)=\mathbf{e}_{\
\mu}^{\mu^{\prime}}(\tau)\mathbf{e}_{\nu}^{\ \nu^{\prime}}(\tau)[~\
{}^{tot}\mathbf{\Upsilon}_{\mu^{\prime}\nu^{\prime}}(\tau)-\frac{1}{2}~{}\partial_{\tau}\mathbf{g}_{\mu^{\prime}\nu^{\prime}}(\tau)]=[~\
{}_{h}^{tot}\Upsilon(\tau,{x}^{k})\delta_{j}^{i},~\
{}^{tot}\Upsilon(\tau,x^{k},y^{3})\delta_{b}^{a}].$ (27)

In these formulas, there are considered $\tau$-families of vierbein transforms
$\mathbf{e}_{\ \mu^{\prime}}^{\mu}(\tau)=\mathbf{e}_{\
\mu^{\prime}}^{\mu}(\tau,u^{\gamma})$ and their dual $\mathbf{e}_{\nu}^{\
\nu^{\prime}}(\tau,u^{\gamma})$, when $\mathbf{e}_{\ }^{\mu}=\mathbf{e}_{\
\mu^{\prime}}^{\mu}du^{\mu^{\prime}}.$ The values $[\
_{h}^{tot}\Upsilon(\tau,{x}^{k}),\ ^{tot}\Upsilon(\tau,x^{k},y^{3})]$ can be
fixed as generating functions for (effective) matter sources imposing
nonholonomic frame constraints on quasi-stationary distributions of
(effective) matter fields. In particular, we can change $\
{}^{tot}\Upsilon_{\mu^{\prime}\nu^{\prime}}(\tau)\rightarrow\
^{q}\widehat{\Upsilon}_{\alpha\beta}(\tau)$ for modeling nonholonomic flow
evolution of a $Q$-source (2.3.1), when the generating sources are written in
the form

$\ {}^{q}\mathbf{\Upsilon}_{\ \nu}^{\mu}(\tau)=[~\
{}_{h}^{q}\Upsilon(\tau,{x}^{k})\delta_{j}^{i},~\
{}^{q}\Upsilon(\tau,x^{k},y^{3})\delta_{b}^{a}].$ (28)

In Appendix A, we show how nonlinear systems of PDEs (19) can be decoupled in
general forms for any quasi-stationary ansatz (26) and any variant of
generating sources (27) or (28). For certain classes of nonholonomic
constraints and small parametric deformations, we are able to change symbolic
data $[\ _{h}^{q}\Upsilon,\ ^{q}\Upsilon]$ into some approximations of (2.3.1)
and study in explicit form for contributions from any $\frac{\partial
F}{\partial\ ^{s}R},\frac{\partial F}{\partial\ ^{m}T}$ and/or $q$-term. In
general, we can consider that $[\ _{h}^{tot}\Upsilon,~\ {}^{tot}\Upsilon]$ or
$[\ _{h}^{q}\Upsilon,\ ^{q}\Upsilon]$ impose certain nonholonomic constraints
on respective geometric evolution / dynamical field generating sources which
allow to generate solutions with nontrivial canonical d-torsion
$\widehat{\mathbf{T}}_{\ \alpha\beta}^{\gamma}(\tau,x^{k},y^{3}).$ Such
nonholnomic values can be eliminated by additional nonholonomic constraints
$\widehat{\mathbf{D}}[\mathbf{g}]\rightarrow\nabla[\mathbf{g}]$ even the
d-torsion
$\mathbf{T}_{\mu\nu\alpha}=\mathbf{A}_{\nu}\mathbf{g}_{\mu\alpha}-\mathbf{A}_{\alpha}\mathbf{g}_{\mu\nu},$
for $\mathbf{A}_{\mu}=q\mathbf{q}_{\mu},q=const,$ can be nonzero because of
nonmetricity.

##  3 Off-diagonal quasi-stationary solutions encoding nonmetricity

In this section, we construct and analyze physical properties of two classes
of respective nonholonomic geometric flow and Ricci soliton equations encoding
quasi-stationary nonmetricity effects. First, we consider examples of
nonmetric solitionic hierarchies. Then, we generate wormhole solutions
determined by nonmetric fields and study generic off-diagonal deformations,
ellipsoidal deformations and embedding into nonmetric backgrounds determined
by solitonic hierarchies with general or small parametric polarizations. In
Appendix B, we provide necessary formulas for bi-Hamilton structures and
solitonic hierarchies.

###  3.1 Effective nonmetric and $\tau$-running Einstein equations

Let us consider a system of nonlinear PDEs (19), for nonmetric Ricci flows, or
(20), for nonmetric Ricci solitons, with generating sources of nonmetric type
$[\ _{h}^{q}\Upsilon(\tau),\ ^{q}\Upsilon(\tau)]$ (28). Such effective sources
can be substituted by formulas of type $[\ _{h}^{tot}\Upsilon(\tau),\
^{tot}\Upsilon(\tau)]$ (27) involving additional effective sources
$\frac{1}{2}\partial_{\tau}\mathbf{g}_{\mu\nu}(\tau).$ Such conditions involve
a more special class of nonholonomic constraints on the geometric evolution
and dynamics of effective sources which allows to decouple the nonlinear
systems of PDEs in general form. For elaborating evolution scenarios in
explicit forms, we can consider product parameterizations of type
$\mathbf{g}_{\mu\nu}(\tau,x^{k},y^{3})=\ ^{1}\mathbf{g}_{\mu\nu}(\tau)\times\
^{2}\mathbf{g}_{\mu\nu}(x^{k},y^{3})$. In this work, we shall write the
general form only parametric solutions in terms of generating functions and
generating sources without discussing particular details on how we may apply
methods with separation of variables. In abstract geometric form, any quasi-
stationary d-metric generated by a $\ {}^{q}\mathbf{\Upsilon}_{\
\nu}^{\mu}(\tau)$ and respective generating functions and nonlinear symmetries
to effective cosmological constants, see details in appendix A.2.2, can be
transformed by frame and connection deformations into a more general
$\mathbf{\Upsilon}_{\ \nu}^{\mu}(\tau).$ Different types of nonholonomic
deformations determined by a $\ {}^{q}\mathbf{\Upsilon}_{\ \nu}^{\mu}$ or a
general type $\mathbf{\Upsilon}_{\ \nu}^{\mu}$, and their physical properties
and implications in physical theories can be studied using Perelman
thermodynamic variables as we shall consider in section 4.

With respect to N-adapted frames (4) and (5), we can write the modified
Einstein equations (20) in $\tau$-parametric form for nonmetric sources (28)
and using the canonical d-connection $\widehat{\mathbf{D}},$

$\displaystyle\widehat{\mathbf{R}}_{\ \ \beta}^{\alpha}(\tau)$
$\displaystyle=$ $\ {}^{q}\mathbf{\Upsilon}_{\ \ \beta}^{\alpha}(\tau),$ (29)
$\displaystyle\widehat{\mathbf{T}}_{\ \alpha\beta}^{\gamma}(\tau)$
$\displaystyle=$ $\displaystyle 0,\mbox{ for }\nabla.$ (30)

The equations (30) do not involve zero conditions for another types of torsion
which may be present in theory, for instance, of type
$\mathbf{T}_{\mu\nu\alpha}=\mathbf{A}_{\nu}\mathbf{g}_{\mu\alpha}-\mathbf{A}_{\alpha}\mathbf{g}_{\mu\nu}.$
Such a torsion is induced by a nontrivial nonmetricity d-vector
$\mathbf{A}_{\nu}$ for a d-metric $\mathbf{g}_{\mu\alpha}$ defined as a
solution of (29).

We note that, in general, for above considered systems of nonmetric
gravitational and matter fields,

$\widehat{\mathbf{D}}^{\beta}\widehat{\mathbf{E}}_{\ \
\beta}^{\alpha}=\widehat{\mathbf{D}}(\widehat{\mathbf{R}}_{\ \
\beta}^{\alpha}-\frac{1}{2}\ ^{s}\widehat{R}\delta_{\ \ \beta}^{\alpha})\neq
0\mbox{ and }\widehat{\mathbf{D}}^{\beta}\widehat{\mathbf{\Upsilon}}_{\ \
\beta}^{\alpha}\neq 0,$

which is typical for nonholonomic systems. In some similar forms, such
nonholonomic configurations are modeled in nonholonomic mechanics when the
conservation laws are not formulated in a standard form. For mechanical
systems, there are introduced the so-called Lagrange multiples associated to
certain classes of nonholonomic constraints. Solving the constraint equations,
it is possible to re-define the variables. Such nonholonomic variables allow
us to introduce new effective Lagrangians and, finally, to define conservation
laws in certain standard form if $\mathbf{Q}_{\alpha\beta\gamma}=0.$ In
explicit general forms, such constructions can be performed only for some
"toy" models. Using distortions of connections, we can rewrite (29) in terms
of $\nabla,$ when $\nabla^{\beta}E_{\ \ \beta}^{\alpha}=\nabla^{\beta}T_{\ \
\beta}^{\alpha}=0$ for $\mathbf{Q}_{\alpha\beta\gamma}\rightarrow 0.$

In Appendix B, we show how using the AFCDM the equations (29) and (30) can be
decoupled and integrated in general quasi-stationary forms for certain
prescribed nonmetric effective sources (28).

###  3.2 Nonmetric solitonic hierarchies

Nonholonomic geometric flow models with solitonic hierarchies, in metric
compatible form, are studied in sections 4 and 5 of [16] for constructing
theories of geometric information flows of nonholonomic Einstein systems.
Similar geometric models can be elaborated for solutions of $\tau$-modified
Einstein equations (29), when the effective source $\
{}^{q}\mathbf{\Upsilon}_{\ \ \beta}^{\alpha}(\tau)$ is determined by
nonmetricity fields as we considered in previous subsection. The physical
interpretation of such off-diagonal solutions involving solitonic wave/
distributions and describing geometric evolution flow processes is different.
In this work, the nonmetricity is encoded into effective generating sources.
We present a brief summary on the theory of quasi-stationary solitonic
hierarchies and their nonmetric $\tau$-evolution in Appendix B. The goal of
this subsection is to provide explicit formulas for general off-diagonal
solutions defining nonmetric geometric flow equations determined by solitonic
distributions for generating functions and/or generating sources written in
solitonic functional form, for instance, as $[\
_{1}^{q}\Upsilon[\mathbf{\wp}],\ _{2}^{q}\Upsilon[\mathbf{\wp}]]$, see
formulas (B.15).

####  3.2.1 $\tau$-running quasi-stationary generating functions with
solitonic hierarchies

We show how $\tau$-evolution of quasi-stationary solitonic configurations can
be defined by respective classes of generating functions when the nonmetricity
generating source $\ {}^{q}\mathbf{\Upsilon}_{\ \ \beta}^{\alpha}(\tau)$ is an
arbitrary one (i.e. it is not obligatory of solitonic nature). For instance,
we consider that $\Phi[\mathbf{\wp}]$ is any functional on a solitonic
hierarchy $\wp(\tau,x^{i},y^{3})$ as we stated for d-metrics
$\mathbf{g}(\tau)=(g_{i}[\mathbf{\wp}],g_{a}[\mathbf{\wp}])$ (B.14). Using
nonlinear symmetries (A.14), we can consider as a generating function any
coefficient

$h_{4}[\tau,x^{i},\mathbf{\wp}]=h_{4}^{[0]}(\tau,x^{i})-\Phi^{2}[\mathbf{\wp}]/4\
_{2}^{q}\Lambda(\tau),\mbox{ for }h_{4}^{\ast}(\tau)\neq 0.$

We can also express

$\ (\Psi^{2})^{\ast}=\ \left|\
{}_{2}^{q}\Upsilon(\tau)\right|(h_{4})^{\ast}\mbox{ and
}\Psi^{2}[\tau,x^{i},\mathbf{\wp}]=\left|\ \
{}_{2}^{q}\Upsilon(\tau)\right|h_{4}-\int dy^{3}\left|\
{}_{2}^{q}\Upsilon(\tau)\right|^{\ast}h_{4}$

and use such a $Q$-deformed (by a generating source $\
{}_{2}^{q}\Upsilon(\tau)$) solitonic configuration $\Psi(\tau)$ as a new
generating function. For simplicity, the integration function
$h_{4}^{[0]}(\tau,x^{i})$ can be approximated to some $\tau$-running values or
constants when the generating functions are some functionals of type
$h_{4}[\mathbf{\wp}],\Psi[\mathbf{\wp}]$ and $\Phi[\mathbf{\wp}].$

We can express above quadratic element in three equivalent forms:

$\displaystyle ds^{2}(\tau)$ $\displaystyle=$ $\displaystyle e^{\
\psi[\mathbf{\wp}]}[(dx^{1})^{2}+(dx^{2})^{2}]+$ (43)
$\displaystyle\left\\{\begin{array}[]{c}\begin{array}[]{c}-\frac{[h_{4}^{\ast}]^{2}}{|\int
dy^{3}[\ _{2}^{q}\mathbf{\Upsilon}(\tau)h_{4}]^{\ast}|\
h_{4}}[dy^{3}+\frac{\partial_{i}[\int dy^{3}(\ \
_{2}^{q}\mathbf{\Upsilon}(\tau))\ h_{4}^{\ast}]}{\ \
{}_{2}^{q}\mathbf{\Upsilon}(\tau)\ h_{4}^{\ast}}dx^{i}]-\\\ h_{4}[dt+(\
_{1}n_{k}(\tau)+\ _{2}n_{k}(\tau)\int dy^{3}\frac{[h_{4}^{\ast}]^{2}}{|\int
dy^{3}[\ \ _{2}^{q}\mathbf{\Upsilon}(\tau)h_{4}]^{\ast}|\ [h_{4}]^{5/2}}\
)dx^{k}];\\\ \mbox{ or }\end{array}\\\
\begin{array}[]{c}-\frac{[\Psi^{\ast}]^{2}}{4(\
_{2}^{q}\mathbf{\Upsilon})^{2}\\{h_{4}^{[0]}-\int dy^{3}[\Psi^{2}]^{\ast}/4(\
\ _{2}^{q}\mathbf{\Upsilon})\\}}[dy^{3}+\frac{\partial_{i}\ \Psi}{\
\partial_{3}\Psi}dx^{i}]-(h_{4}^{[0]}(\tau)-\int
dy^{3}\frac{[\Psi^{2}]^{\ast}}{4(\ _{2}^{q}\mathbf{\Upsilon})})\\\
[dt+(_{1}n_{k}(\tau)+\ _{2}n_{k}(\tau)\int
dy^{3}\frac{[(\Psi)^{2}]^{\ast}}{4(\
_{2}^{q}\mathbf{\Upsilon}(\tau))^{2}|h_{4}^{[0]}-\int
dy^{3}[\Psi^{2}]^{\ast}/4(\
_{2}^{q}\mathbf{\Upsilon}(\tau))|^{5/2}})dx^{k}];\\\ \mbox{ or }\end{array}\\\
\begin{array}[]{c}-\frac{\Phi^{2}[\Phi^{\ast}]^{2}}{|\
_{2}^{q}\Lambda(\tau)\int dy^{3}\
_{2}{}^{q}\mathbf{\Upsilon}(\tau)[\Phi^{2}]^{\ast}|[g_{4}^{[0]}(\tau)-\Phi^{2}/4\
_{2}^{q}\Lambda(\tau)]}[dy^{3}+\frac{\partial_{i}\ \int dy^{3}\ \
_{2}{}^{q}\mathbf{\Upsilon}(\tau)\ [\Phi^{2}]^{\ast}}{\ \
{}_{2}^{q}\mathbf{\Upsilon}(\tau)\
[\Phi^{2}]^{\ast}}dx^{i}]-(h_{4}^{[0]}-\frac{\Phi^{2}}{4\
_{2}^{q}\Lambda(\tau)})\\\ [dt+(_{1}n_{k}(\tau)+\ _{2}n_{k}(\tau)\int
dy^{3}\frac{\Phi^{2}[\Phi^{\ast}]^{2}}{|\ _{2}^{q}\Lambda(\tau)\int dy^{3}\ \
\
_{2}{}^{q}\mathbf{\Upsilon}(\tau)[\Phi^{2}]^{\ast}|[g_{4}^{[0]}(\tau)-\Phi^{2}/4\
_{2}^{q}\Lambda(\tau)]^{5/2}})dx^{k}].\end{array}\end{array}\right.$

In these formulas, there are used respective generating data: $\\{{\small
h}_{4}{\small[\wp],\ }_{2}^{q}{\small\Upsilon(\tau),\
}_{2}^{q}{\small\Lambda(\tau)}\\}$ from (A.2.3); $\\{{\small\Psi[\wp],\
}_{2}^{q}{\small\Upsilon(\tau)}\\}$ from (A.2.1); and $\\{{\small\Phi[\wp],\
}_{2}^{q}{\small\Lambda(\tau),\ }_{2}^{q}{\small\Upsilon(\tau)}\\}$ from
(A.2.2).

The solutions (43) can be re-defined for $\eta(\tau)$-polarizations in a form
(A.2.4) (for instance, with functionals $\eta_{4}[\mathbf{\wp}]$) or
considering $\kappa$-parametric deformations to $\tau$–families of quasi-
stationary d-metrics with $\chi$-generating functions for solutions of type
(A.18) (with functionals $\chi_{4}[\mathbf{\wp}]$). The prime metrics can be
taken of any nature (being or not solutions of some modified gravitational
equations, or some quasi-periodic/ solitonic configurations) and than
subjected to nonmetric quasi-stationary solitonic deformations via generating/
gravitational polarization functions. In such cases, the target solutions will
define a mixture of solitonic distributions under $\tau$-evolution and
respective prescribed geometric data for primary d-metrics and effective
nonmetric generating sources.

####  3.2.2 Nonmetric quasi-stationary generating sources with solitonic
hierarchies

We can generate a different class of generic off-diagonal solitonic solutions
of nonmetric geometric flow equations (29) when the generating source is of
type $\ {}^{q}\mathbf{\Upsilon}_{\ \nu}^{\mu}(\tau)=[\
_{h}^{q}\Upsilon[\mathbf{\wp}],\ ^{q}\Upsilon[\mathbf{\wp}]]$ (B.15) but the
d-metric coefficients of $\mathbf{g}(\tau)=(g_{i}(\tau),g_{a}(\tau))$ are
computed via corresponding nonholonomic deformations of nonmetric sources. The
respective quadratic elements are written in the form,

$\displaystyle ds^{2}(\tau)$ $\displaystyle=$ $\displaystyle e^{\ \psi[~\
{}_{h}^{q}\Upsilon[\mathbf{\wp}]]}[(dx^{1})^{2}+(dx^{2})^{2}]+$ (56)
$\displaystyle\left\\{\begin{array}[]{c}\begin{array}[]{c}-\frac{[h_{4}^{\ast}(\tau)]^{2}}{|\int
dy^{3}[\ ^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ h_{4}(\tau)]^{\ast}|\
h_{4}(\tau)}[dy^{3}+\frac{\partial_{i}[\int dy^{3}(\ \
^{q}\mathbf{\Upsilon}[\mathbf{\wp}])\ h_{4}^{\ast}(\tau)]}{\ \
{}^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ \ h_{4}^{\ast}(\tau)}dx^{i}]-\\\
h_{4}(\tau)[dt+(\ _{1}n_{k}(\tau)+\ _{2}n_{k}(\tau)\int
dy^{3}\frac{[h_{4}^{\ast}(\tau)]^{2}}{|\int dy^{3}[\
^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ h_{4}(\tau)]^{\ast}|\
[h_{4}(\tau)]^{5/2}}\ )dx^{k}];\\\ \mbox{ or }\end{array}\\\
\begin{array}[]{c}-\frac{[\Psi^{\ast}(\tau)]^{2}}{4(\
_{2}^{q}\mathbf{\Upsilon})^{2}\\{h_{4}^{[0]}(\tau)-\int
dy^{3}[\Psi^{2}(\tau)]^{\ast}/4\ \ ^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\
\\}}[dy^{3}+\frac{\partial_{i}\ \Psi}{\
\partial_{3}\Psi}dx^{i}]-(h_{4}^{[0]}(\tau)-\int
dy^{3}\frac{[\Psi^{2}(\tau)]^{\ast}}{4\ ^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\
})\\\ [dt+(_{1}n_{k}(\tau)+\ _{2}n_{k}(\tau)\int
dy^{3}\frac{[(\Psi(\tau))^{2}]^{\ast}}{4(\ \
^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ )^{2}|h_{4}^{[0]}(\tau)-\int
dy^{3}[\Psi^{2}(\tau)]^{\ast}/4(\ ^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\
)|^{5/2}})dx^{k}];\\\ \mbox{ or }\end{array}\\\
\begin{array}[]{c}-\frac{\Phi^{2}(\tau)[\Phi^{\ast}(\tau)]^{2}}{|\
_{2}^{q}\Lambda(\tau)\int dy^{3}\ \ {}^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\
[\Phi^{2}(\tau)]^{\ast}|[g_{4}^{[0]}(\tau)-\Phi^{2}(\tau)/4\
_{2}^{q}\Lambda(\tau)]}[dy^{3}+\frac{\partial_{i}\ \int dy^{3}\ \
{}^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ \ [\Phi^{2}]^{\ast}}{\ \
{}^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ \ [(\
_{2}\Phi)^{2}]^{\ast}}dx^{i}]-(h_{4}^{[0]}(\tau)\\\ -\frac{\Phi^{2}(\tau)}{4\
_{2}^{q}\Lambda(\tau)})[dt+(_{1}n_{k}(\tau)+\ _{2}n_{k}(\tau)\int
dy^{3}\frac{\Phi^{2}[\Phi^{\ast}]^{2}}{|\ _{2}^{q}\Lambda(\tau)\int dy^{3}\ \
{}^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ \
[\Phi^{2}]^{\ast}|[g_{4}^{[0]}(\tau)-\Phi^{2}/4\
_{2}^{q}\Lambda(\tau)]^{5/2}})dx^{k}].\end{array}\end{array}\right.$

In d-metrics (56), the corresponding generating data for (A.2.3), (A.2.1), and
(A.2.2) are stated in such forms:
$\\{{\small h}_{4}(\tau)={\small h}_{4}(\tau,x^{i},y^{3}){\small,\
^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ ,\ }_{2}^{q}{\small\Lambda(\tau)}\\};$
$\\{{\small\Psi(\tau),\ \ ^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ }\\}$ from
(A.2.1); and $\\{{\small\Phi(\tau),\ }_{2}^{q}{\small\Lambda(\tau),\ \
^{q}\mathbf{\Upsilon}[\mathbf{\wp}]\ }\\}.$

####  3.2.3 Interacting $\tau$-running solitonic hierarchies for generating
functions with respective solitonic hierarchies for nonmetric sources

The generic off-diagonal solutions (43) or (56) can be generalized for
$\tau$-running configurations when a set of solitionic hierarchies $\
{}_{1}{\small\mathbf{\wp}}$ is prescribed for generating functions and another
set $\ {}_{2}{\small\mathbf{\wp}}$ is generated for generating functions of
nonmetric origin. For a prime d-metric $\mathbf{\mathring{g}}$ (A.12) of
arbitrary nature, we generate a target quasi-stationary d-metric double
generating function/source solitonic hierarchies,

$\displaystyle d\widehat{s}^{2}(\tau)$ $\displaystyle=$
$\displaystyle\widehat{g}_{\alpha\beta}(\tau,x^{k},y^{3};\mathring{g}_{\alpha};\psi(\tau),\eta_{4}[\
_{1}\mathbf{\wp}];\ \ ^{q}\mathbf{\Upsilon}[\
_{2}\mathbf{\wp}])du^{\alpha}du^{\beta}=e^{\psi(\tau)}[(dx^{1})^{2}+(dx^{2})^{2}]$
$\displaystyle-\frac{[\partial_{3}(\eta_{4}[\ _{1}\mathbf{\wp}]\
\mathring{g}_{4})]^{2}}{|\int dy^{3}\ _{2}{}^{q}\mathbf{\Upsilon}[\
_{2}\mathbf{\wp}]\partial_{3}(\eta_{4}[\ _{1}\mathbf{\wp}]\
\mathring{g}_{4})|\ \eta_{4}[\
_{1}\mathbf{\wp}]\mathring{g}_{4}}\\{dy^{3}+\frac{\partial_{i}[\int dy^{3}\
_{2}{}^{q}\mathbf{\Upsilon}[\ _{2}\mathbf{\wp}]\ \partial_{3}(\eta_{4}[\
_{1}\mathbf{\wp}]\mathring{g}_{4})]}{\ {}_{2}^{q}\mathbf{\Upsilon}[\
_{2}\mathbf{\wp}]\partial_{3}(\eta_{4}[\
_{1}\mathbf{\wp}]\mathring{g}_{4})}dx^{i}\\}^{2}$ $\displaystyle+\eta_{4}[\
_{1}\mathbf{\wp}]\mathring{g}_{4}\\{dt+[\ _{1}n_{k}(\tau)+\
_{2}n_{k}(\tau)\int dy^{3}\frac{[\partial_{3}(\eta_{4}[\
_{1}\mathbf{\wp}]\mathring{g}_{4})]^{2}}{|\int dy^{3}\ \
_{2}{}^{q}\mathbf{\Upsilon}[\ _{2}\mathbf{\wp}]\partial_{3}(\eta_{4}[\
_{1}\mathbf{\wp}]\mathring{g}_{4})|\ (\eta_{4}[\
_{1}\mathbf{\wp}]\mathring{g}_{4})^{5/2}}]dx^{k}\\}^{2}.$

In such nonlinear quadratic elements, we can prescribe $\mathbf{\mathring{g}}$
to define, for instance, a black hole, BH, solution in GR or a MGT like in
[11, 16]. Nonmetric soltionic deformations of type (3.2.3) do not preserve, in
general, the primary BH character. Nevertheless, for small parametric
deformations like in (A.18), we can generate $\tau$-families of quasi-
stationary d-metrics with $\chi$-generating functions for solutions with
generating functionals $\chi_{4}[\ _{1}\mathbf{\wp}]$ and/or $\
{}_{2}^{q}\mathbf{\Upsilon}[\ _{2}\mathbf{\wp}].$ Various variants with
ellipsoid and solitonic deformations, for instance, for black ellipsoids, BE,
with additional solitonic modifications (of physical constants, or embedding
into certain solitonic gravitational vacuum) can be modelled. We do not study
in this work solutions for BHs, or BEs, solitonic $\tau$-running but
concentrate only on nonmetric wormhole-soliton configurations.

###  3.3 Nonmetric wormhole solutions and their solitonic deformations

The goal of this subsection is to construct analyze physical properties of
wormhole solutions and their solitonic deformations in nonmetric geometric
flow and gravity theories with $\tau$-modified Einstein equations (29). As
prime d-metric configurations, we shell consider certain curve coordinate
transforms of the Morris–Thorne and generalized Ellis–Bronnikov wormholes to
certain trivial off-diagonal configurations which allow to apply the AFCDM. We
cite [25, 26, 27, 28, 33, 31] for details and a recent review of results on
wormhole solutions. Then, we shall construct new classes of quasi-stationary
solutions as target metrics, for certain gravitational polarizations
determined by additional nonmetricity source, and study their possible
traversable nonmetric properties. Necessary technical results are summarized
in appendix A, containing a brief summary of the AFCDM adapted to nonmetric
gravity, and appendix B, for necessary formulas on solitoinic hierarchies.

####  3.3.1 Prime metrics for 4-d metric compatible wormhole configurations

Let us consider a prime d-metric

$d\mathring{s}^{2}=\check{g}_{\alpha}(l,\theta,\varphi)[\mathbf{\check{e}}^{\alpha}(l,\theta,\varphi)]^{2},$
(58)

where the (tortoise) coordinates $u^{\alpha}=(r,\theta,\varphi,t)$ are defined
for $r(l)=(l^{2k}+\ _{0}b^{2k})^{1/2k}$ and the cylindrical angular coordinate
$\phi\in[0,2\pi)$ is called parallel. In such coordinates, $-\infty<l<\infty$
which is different from the standard cylindrical radial coordinate $\rho,$
when $0\leq\rho<\infty.$ We can fix
$\check{g}_{1}=1,\check{g}_{2}=r^{2}(l),\check{g}_{3}=r^{2}(l)\sin^{2}\theta$
and $\check{g}_{4}=-1$ and can consider frame transforms to a parametrization
with trivial N-connection coefficients
$\check{N}_{i}^{a}=\check{N}_{i}^{a}(u^{\alpha}(l,\theta,\varphi,t))$ and
$\check{g}_{\beta}(u^{j}(l,\theta,\varphi),u^{3}(l,\theta,\varphi)),$ when new
coordinates are $u^{1}=x^{1}=l,u^{2}=\theta,$ and $u^{3}=y^{3}=\varphi+\
^{3}B(l,\theta),u^{4}=y^{4}=t+\ ^{4}B(l,\theta),$ for

$\displaystyle\mathbf{\check{e}}^{3}$ $\displaystyle=$ $\displaystyle
d\varphi=du^{3}+\check{N}_{i}^{3}(l,\theta)dx^{i}=du^{3}+\check{N}_{1}^{3}(l,\theta)dl+\check{N}_{2}^{3}(l,\theta)d\theta,$
$\displaystyle\mathbf{\check{e}}^{4}$ $\displaystyle=$ $\displaystyle
dt=du^{4}+\check{N}_{i}^{4}(l,\theta)dx^{i}=du^{4}+\check{N}_{1}^{4}(l,\theta)dl+\check{N}_{2}^{4}(l,\theta)d\theta,$

for $\mathring{N}_{i}^{3}=-\partial\ ^{3}B/\partial x^{i}$ and
$\mathring{N}_{i}^{4}=-\partial\ ^{4}B/\partial x^{i}.$

We consider a prime d-metric (58) which is related via coordinate transforms
to the generic Morris-Thorne wormhole solution [25],

$d\mathring{s}^{2}=(1-\frac{b(r)}{r})^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}-e^{2\Phi(r)}dt^{2},$

where $e^{2\Phi(r)}$ is a red-shift function and $b(r)$ as a shape function
defined in spherically polar coordinates $u^{\alpha}=(r,\theta,\varphi,t).$ We
can also parameterize this metric to get usual Ellis-Bronnikov, EB, wormholes
which are defined for $\Phi(r)=0$ and $b(r)=\ _{0}b^{2}/r$ characterizing a
zero tidal wormhole with $\ {}_{0}b$ the throat radius. A generalized EB is
characterized additionally by even integers $2k$ (with $k=1,2,...$) This
allows us to define a prime metric

$d\mathring{s}^{2}=dl^{2}+r^{2}(l)d\theta^{2}+r^{2}(l)\sin^{2}\theta
d\varphi^{2}-dt^{2},$

when

$dl^{2}=(1-\frac{b(r)}{r})^{-1}dr^{2}\mbox{ and }b(r)=r-r^{3(1-k)}(r^{2k}-\
_{0}b^{2k})^{(2-1/k))}.$

Generic off-diagonal nonholonomic deformations of prime metrics (58) can be
analyzed for effective sources of type (2.3.1) when the $Q$-deformations are
stated to be zero. For such quasi-stationary diagonalizable solutions, the
effective sources are of type $\ {}^{tot}\widehat{\mathbf{\Upsilon}}_{\mu\nu}$
(21) when $\ {}^{m}\widehat{\mathbf{T}}_{\mu\nu}$ in (22) is taken for a
energy-momentum tensor for matter fields. For generating off-diagonal
deformations of wormhole solutions, we shall consider generating sources of
type (27) or (28). Corresponding classes of generating and integration
functions are related via nonlinear symmetries (A.8) or (A.9), when a class of
solutions for $(\ _{2}^{tot}\mathbf{\Upsilon}(\tau)\leftrightarrow\
_{2}^{tot}\Lambda(\tau))$ can be distinguished from another class of solutions
for $(\ _{2}^{q}\mathbf{\Upsilon}(\tau)\leftrightarrow\
_{2}^{q}\Lambda(\tau)).$ In general, such $\tau$-running or nonholonomic Ricci
flow configurations are characterized by different types of thermodynamic
variables as we shall prove in section 4.

####  3.3.2 Nonholonomic quasi-stationary gravitational polarizations of
wormholes

Nonmetric off-diagonal quasi-stationary deformations of wormhole d-metrics
(58) can be generated by introducing
$\left(\check{g}_{\beta},\check{N}_{i}^{a}\right)$ instead of primary
geometric data $\mathbf{\mathring{g}}=[\mathring{g}_{\alpha},\
\mathring{N}_{i}^{a}]$ (A.12) in (A.2.4) using some generating sources $\
{}_{2}^{q}\mathbf{\Upsilon}(\tau)$ and/or ${}_{2}^{q}\Lambda(\tau).$ We
generate new classes of a target $\tau$-family of d-metrics $\mathbf{g}(\tau)$
determined by nonmetric geometric flows and respective $\eta$-polarization
functions, when
$\mathbf{\check{g}}\rightarrow\mathbf{g}(\tau)=[g_{\alpha}(\tau)=\eta_{\alpha}(\tau)\check{g}_{\alpha},N_{i}^{a}(\tau)=\eta_{i}^{a}\
(\tau)\check{g}_{i}^{a}].$ Corresponding quadratic linear elements

$\displaystyle d\widehat{s}^{2}(\tau)$ $\displaystyle=$
$\displaystyle\widehat{g}_{\alpha\beta}(l,\theta,\varphi;\psi,\eta_{4};\
_{2}^{q}\Lambda(\tau)=\ _{2}^{q}\Lambda(\tau),\ \
_{2}^{q}\mathbf{\Upsilon}(\tau),\ \check{g}_{\alpha})du^{\alpha}du^{\beta}$
$\displaystyle=$ $\displaystyle e^{\psi(\tau
l,\theta)}[(dx^{1}(l,\theta))^{2}+(dx^{2}(l,\theta))^{2}]$
$\displaystyle-\frac{[\partial_{\varphi}(\eta_{4}(\tau)\
\check{g}_{4})]^{2}}{|\int d\varphi\ \
_{2}^{q}\mathbf{\Upsilon}(\tau)\partial_{\varphi}(\eta_{4}(\tau)\
\breve{g}_{4})|\ \eta_{4}(\tau)\
\check{g}_{4}}\\{dy^{3}+\frac{\partial_{i}[\int d\varphi\ \
_{2}^{q}\mathbf{\Upsilon}(\tau)\ \partial_{\varphi}(\eta_{4}(\tau)\
\check{g}_{4})]}{\
{}_{2}^{q}\mathbf{\Upsilon}(\tau)\partial_{\varphi}(\eta_{4}(\tau)\
\check{g}_{4})}dx^{i}\\}^{2}$
$\displaystyle+\eta_{4}(\tau)\breve{g}_{4}\\{dt+[\ _{1}n_{k}+\ _{2}n_{k}\int
d\varphi\frac{[\partial_{\varphi}(\eta_{4}(\tau)\ \breve{g}_{4})]^{2}}{|\int
d\varphi\ \ _{2}^{q}\mathbf{\Upsilon}(\tau)\partial_{\varphi}(\eta_{4}(\tau)\
\breve{g}_{4})|\ (\eta_{4}(\tau)\ \breve{g}_{4})^{5/2}}]dx^{k}\\}.$

This class of solutions are determined by respective generating function
$\eta_{4}(\tau)=\eta_{4}(\tau,l,\theta,\varphi)$ and integration functions $\
{}_{1}n_{k}(\tau,l,\theta)$ and $\ {}_{2}n_{k}(\tau,l,\theta).$ The
$\tau$-family of functions $\psi(\tau,l,\theta)$ are defined as a solution of
a respective family of 2-d Poisson equations,
$\partial_{11}^{2}\psi(\tau)+\partial_{22}^{2}\psi(\tau)=2\ \
_{1}^{q}\mathbf{\Upsilon}(\tau,l,\theta)$, when the horizontal generating
effective sources, $\ {}_{1}^{q}\mathbf{\Upsilon}(\tau,l,\theta),$ can be
different from the vertical ones which may depend on a vertical coordinate, $\
{}_{2}^{q}\mathbf{\Upsilon}(\tau,l,\theta,z).$

For families of off-diagonal solutions (3.3.2), we can fix $\
{}_{1}^{q}\Lambda(\tau)=\ _{2}^{q}\Lambda(\tau)$ and analyze quasi-stationary
configurations with running cosmological constants. We suppose that such
effective $\tau$-families of cosmological constants can be expressed in
additive form

$\ {}_{2}^{q}\Lambda(\tau)=\ _{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)=\
_{2}^{m}\Lambda(\tau)+\ _{2}^{F}\Lambda(\tau)+\ _{2}^{e}\Lambda(\tau),$ (60)

where $\ {}_{2}^{tot}\Lambda(\tau)$ model metric compatible configurations and
$\ {}_{2}^{1}\Lambda(\tau)$ describe possible additional nonmetric
contributions. The three terms with left labels $m,F,e$ in these formulas
correspond to (effective) energy-momentum tensors (22). We can consider
nonlinear symmetries of type (A.14) relating $\mathbf{\Upsilon}_{\
\nu}^{\mu}(\tau)\simeq$ $[~\ {}_{h}^{tot}\Upsilon,~\ {}^{tot}\Upsilon]$ (27)
to a $\tau$-family $\ {}_{s}^{tot}\Lambda(\tau),$ for $s=1,2,$ and $\
{}^{q}\mathbf{\Upsilon}_{\ \nu}^{\mu}(\tau)\simeq[\ _{h}^{q}\Upsilon,~\
{}^{q}\Upsilon]$ (28) to another family $\ {}_{2}^{q}\Lambda(\tau)=\
_{2}^{q}\Lambda[\tau,\ _{2}^{tot}\Lambda(\tau)]$ which may be a nonlinear
functional on $\ {}_{2}^{tot}\Lambda(\tau).$ In this work, we elaborate on
models with nontrivial nonmetricity effective sources and additional type
functionals (60) for cosmological constants.

The class of quasi-stationary nonmetric deformed wormholes (3.3.2) can be
written in terms of generating data $\left[\Phi(\tau),\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)\right]$ as in (56),

$\displaystyle d\widehat{s}^{2}(\tau)$ $\displaystyle=$ $\displaystyle e^{\
\psi[~\ {}_{h}^{q}\Upsilon(\tau)]}[(dx^{1})^{2}+(dx^{2})^{2}]-$
$\displaystyle\frac{\Phi^{2}(\tau)[\Phi^{\ast}(\tau)]^{2}}{|\ [\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]\int d\varphi\
^{q}\mathbf{\Upsilon}(\tau)\
[\Phi^{2}(\tau)]^{\ast}|[g_{4}^{[0]}(\tau)-\frac{\Phi^{2}(\tau)}{4\ [\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]}]}$
$\displaystyle[dy^{3}+\frac{\partial_{i}\ \int d\varphi\ \
^{q}\mathbf{\Upsilon}(\tau)\ \ [\Phi^{2}]^{\ast}}{\ \
{}^{q}\mathbf{\Upsilon}(\tau)\ [(\
_{2}\Phi)^{2}]^{\ast}}dx^{i}]-[h_{4}^{[0]}(\tau)-\frac{\Phi^{2}(\tau)}{4\ [\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]}][dt+(_{1}n_{k}(\tau)+$ $\
{}_{2}n_{k}(\tau)\int d\varphi\frac{\Phi^{2}(\tau)[\Phi^{\ast}(\tau)]^{2}}{|\
[\ _{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]\int d\varphi\
^{q}\mathbf{\Upsilon}(\tau)\
[\Phi^{2}(\tau)]^{\ast}|[g_{4}^{[0]}(\tau)-\frac{\Phi^{2}(\tau)}{4\ [\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]}]^{5/2}})dx^{k}].$

In such a form, the data for a prime wormhole metric are "hidden" into
generating functions and the equations for nonlinear symmetries.

The target d-metrics (3.3.2) or (3.3.2) do not describe wormhole
configurations for general classes of generating and integrating data. There
are necessary additional assumptions on polarization functions and integration
functions which allows us to provide certain physical interpretation of such
generic off-diagonal solutions. Typically, we can prescribe some generating
data for solitonic waves, or some small deformations of wormhole
configurations. For instance, to elaborate cosmological scenarios with
acceleration and quasi-periodic structure, or to transform quasi-stationary
d-metrics into locally anisotropic cosmological ones, we have to consider
other types of gravitational polarization and generating data.

####  3.3.3 Off-diagonal quasi-stationary solitonic deformations of wormhole
d-metrics

We can generate a class of d-metrics (3.2.3) when the prime d-metric
$\mathbf{\mathring{g}}$ (A.12) changed into a wormhole one (58) with space
coordinates $(l,\theta,\varphi).$ Such a target d-metric is of type (3.3.2)
when off-diagonal deformations are determined by a generating function
$\eta_{4}(\tau)=\eta_{4}[\wp],$ where $\wp=\wp(\tau,x^{i}),$
$=\wp(\tau,x^{1},y^{3}),$ or $=\wp(\tau,x^{2},y^{3})$ is any solution for
$\tau$-running solitonic configurations (B.13).777We can consider $\wp$ as any
solitonic hierarchy considered in appendix B.1 and re-defined in coordinates
$(\tau,l,\theta,\varphi).$ The quadratic linear element for
$\wp(\tau,x^{1},y^{3})$–solitonic deformations of wormhole metrics is
constructed

$\displaystyle d\widehat{s}^{2}(\tau)$ $\displaystyle=$
$\displaystyle\widehat{g}_{\alpha\beta}(l,\theta,y^{3};\psi,\eta_{4};\
_{2}^{m}\Lambda(\tau)+\ _{2}^{F}\Lambda(\tau)+\ _{2}^{e}\Lambda(\tau),\ \
_{2}^{q}\mathbf{\Upsilon}(\tau),\ \check{g}_{\alpha})du^{\alpha}du^{\beta}$
$\displaystyle=$ $\displaystyle
e^{\psi(\tau,l,\theta)}[(dx^{1}(l,\theta))^{2}+(dx^{2}(l,\theta))^{2}]$
$\displaystyle-\frac{[\partial_{3}(\eta_{4}[\wp]\ \check{g}_{4})]^{2}}{|\int
dy^{3}\ \ _{2}{}^{q}\mathbf{\Upsilon}(\tau)\partial_{3}(\eta_{4}[\wp]\
\breve{g}_{4})|\ \eta_{4}[\wp]\
\check{g}_{4}}\\{dy^{3}+\frac{\partial_{i}[\int dy^{3}\ \
_{2}{}^{q}\mathbf{\Upsilon}(\tau)\ \partial_{3}(\eta_{4}[\wp]\
\check{g}_{4})]}{\
{}_{2}^{q}\mathbf{\Upsilon}(\tau)\partial_{3}(\eta_{4}[\wp]\
\check{g}_{4})}dx^{i}\\}^{2}$
$\displaystyle+\eta_{4}[\wp]\breve{g}_{4}\\{dt+[\ _{1}n_{k}+\ _{2}n_{k}\int
dy^{3}\frac{[\partial_{3}(\eta_{4}[\wp]\ \breve{g}_{4})]^{2}}{|\int dy^{3}\ \
_{2}{}^{q}\mathbf{\Upsilon}(\tau)\partial_{3}(\eta_{4}[\wp]\ \breve{g}_{4})|\
(\eta_{4}[\wp]\ \breve{g}_{4})^{5/2}}]dx^{k}\\}.$

Such solutions describe nonmetric nonholonomic dissipations of a prime
wormhole metric into certain $\tau$-running generic off-diagonal solitonic
distributions. Nonlinear symmetries of type (A.8) or (A.9) involve additive
decompositions of running cosmological constants, $\ {}_{2}^{m}\Lambda(\tau)+\
_{2}^{F}\Lambda(\tau)+\ _{2}^{e}\Lambda(\tau)$, and a nonmetric source $\
{}_{2}^{q}\mathbf{\Upsilon}(\tau)$ which can be of non-solitonic character.

####  3.3.4 Small parametric off-diagonal quasi-stationary deformations of
wormhole d-metrics

We can generate new classes of solutions which preserve wormhole character
under nonmetric geometric flow evolution scenarios of a prime d-metric
$\mathbf{\check{g}}$ (58) if we consider small $\varepsilon$-parametric
deformations of type (A.18). The generating functions can be linearized on
$\varepsilon$ as in (A.17) when in terms of $\chi$-polarization functions, the
quadratic linear elements (3.3.2) can be expressed

$\displaystyle d\ \widehat{s}^{2}(\tau)$ $\displaystyle=$
$\displaystyle\widehat{g}_{\alpha\beta}(l,\theta,\varphi;\psi,\chi_{4};\
_{2}^{q}\Lambda(\tau)=\ _{2}^{q}\Lambda(\tau),\ \
_{2}^{q}\mathbf{\Upsilon}(\tau),\
\check{g}_{\alpha},\check{N}_{i}^{3})du^{\alpha}du^{\beta}$ $\displaystyle=$
$\displaystyle e^{\psi_{0}(l,\theta)}[1+\varepsilon\
^{\psi(l,\theta)}\chi(l,\theta)][(dx^{1}(l,\theta))^{2}+(dx^{2}(l,\theta))^{2}]$
$\displaystyle-\\{\frac{4[\partial_{\varphi}(|\zeta_{4}\
\breve{g}_{4}|^{1/2})]^{2}}{\breve{g}_{3}|\int d\varphi\\{\ \
_{2}^{q}\mathbf{\Upsilon}(\tau)\partial_{\varphi}(\zeta_{4}\
\breve{g}_{4})\\}|}-\varepsilon[\frac{\partial_{\varphi}(\chi_{4}|\zeta_{4}\breve{g}_{4}|^{1/2})}{4\partial_{\varphi}(|\zeta_{4}\
\breve{g}_{4}|^{1/2})}-\frac{\int d\varphi\\{\
_{2}^{q}\mathbf{\Upsilon}(\tau)\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})\chi_{4}]\\}}{\int d\varphi\\{\
_{2}^{q}\mathbf{\Upsilon}(\tau)\partial_{\varphi}(\zeta_{4}\
\breve{g}_{4})\\}}]\\}\ \breve{g}_{3}$
$\displaystyle\\{d\varphi+[\frac{\partial_{i}\ \int d\varphi\ \
_{2}^{q}\mathbf{\Upsilon}(\tau)\
\partial_{\varphi}\zeta_{4}}{(\check{N}_{i}^{3})\
_{2}^{q}\mathbf{\Upsilon}(\tau)\partial_{\varphi}\zeta_{4}}+\varepsilon(\frac{\partial_{i}[\int
d\varphi\ _{2}^{q}\mathbf{\Upsilon}(\tau)\
\partial_{\varphi}(\zeta_{4}\chi_{4})]}{\partial_{i}\ [\int d\varphi\
_{2}^{q}\mathbf{\Upsilon}(\tau)\partial_{\varphi}\zeta_{4}]}-\frac{\partial_{\varphi}(\zeta_{4}\chi_{4})}{\partial_{\varphi}\zeta_{4}})]\check{N}_{i}^{3}dx^{i}\\}^{2}$
$\displaystyle+\zeta_{4}(1+\varepsilon\ \chi_{4})\
\breve{g}_{4}\\{dt+[(\check{N}_{k}^{4})^{-1}[\ _{1}n_{k}+16\ _{2}n_{k}[\int
d\varphi\frac{\left(\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})^{-1/4}]\right)^{2}}{|\int d\varphi\partial_{\varphi}[\ \
_{2}^{q}\mathbf{\Upsilon}(\tau)(\zeta_{4}\ \breve{g}_{4})]|}]$
$\displaystyle+\varepsilon\frac{16\ _{2}n_{k}\int
d\varphi\frac{\left(\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})^{-1/4}]\right)^{2}}{|\int d\varphi\partial_{\varphi}[\ \
_{2}^{q}\mathbf{\Upsilon}(\tau)(\zeta_{4}\
\breve{g}_{4})]|}(\frac{\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})^{-1/4}\chi_{4})]}{2\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})^{-1/4}]}+\frac{\int
d\varphi\partial_{\varphi}[\breve{g}_{4}(\zeta_{4}\chi_{4}\
\breve{g}_{4})]}{\int d\varphi\partial_{\varphi}[\breve{g}_{4}(\zeta_{4}\
\breve{g}_{4})]})}{\ {}_{1}n_{k}+16\ _{2}n_{k}[\int
d\varphi\frac{\left(\partial_{\varphi}[(\zeta_{4}\
\breve{g}_{4})^{-1/4}]\right)^{2}}{|\int
d\varphi\partial_{\varphi}[\breve{g}_{4}(\zeta_{4}\
\breve{g}_{4})]|}]}]\check{N}_{k}^{4}dx^{k}\\}^{2}.$ (62)

We can model elliptic deformations of the wormhole throat as a particular case
of d-metrics of type (62) if we chose a generating function of type

$\chi_{4}(\tau,l,\theta,\varphi)=\underline{\chi}(\tau,l,\theta)\sin(\omega_{0}\varphi+\varphi_{0})$
(63)

as for cylindric configurations with $\varphi$-anisotropic deformations and
$\tau$-running small deformations. Such classes of solutions describe
nonmetric $\tau$-evolution of some ellipsoidal wormholes.

If in (62) we take a vacuum profile with $\zeta_{4}=[\ _{1}\wp]$ i.e. as a
functional of a solitonic distribution $\ {}_{1}\wp=\
_{1}\wp(l,\theta,\varphi)$ as in (B.12) but with correspondingly re-defined
coordinates, the wormhole ellipsoidal configuration is modelled as a
$\tau$-evolution in such nonmetric gravitational vacuum. Instead of
ellipsoidal configurations (63), we can consider small solitonic
$\chi$-deformations. For such classes of generic off-diagonal solutions, we
can consider any $\chi_{4}(\tau,l,\theta,\varphi)$ defined as functional
$\chi_{4}[\wp]$ of a $\wp=\wp(\tau,x^{i}),$ $=\wp(\tau,x^{1},y^{3}),$ or
$=\wp(\tau,x^{2},y^{3}),$ of some solitonic $\tau$-waves (B.13).

The class of nonmetric locally anisotropic wormhole solutions (62) may possess
a multiple solitonic wave or solitionic distribution character even the
generating source $\ {}_{2}^{q}\mathbf{\Upsilon}(\tau)$ can be non-solitonic.
Nonlinear symmetries allow to associate to such configurations certain
effective $\tau$-running cosmological constants of type (60). Considering, for
instance, zero values of $\ {}_{2}^{e}\Lambda(\tau),$ we can model some metric
compatible locally anisotropic wormholes and then to extend the constructions
for nonmetric configurations with nontrivial $\ {}_{2}^{e}\Lambda(\tau).$

####  3.3.5 On traversable nonmetric wormholes

Wormhole solutions in GR and MGTs are considered as hypothetical geometric
structures that link two distinct regions of the same spacetime. References
[25, 26, 27, 28, 29] contain reviews of results and methods of constructing
wormhole solutions. Standard wormhole solutions in GR are not traversable
because for various classes of such solutions it is not possible to send
causal light signals through it throat faster than we can send it through
outside. The first wormhole model of the so-called Einstein-Rosen bridge was
elaborated using as a vacuum solution of gravitational field equations [30].
That solution was derived as a modification of the Schwarzschild BH when the
corresponding wormhole model is not traversable because of singularity of such
solutions. Latter, a static and spherically symmetric wormhole configuration
with a traversable throat at the center was constructed in [25]. Various
classes of wormhole metrics were found in the framework of GR and MGTs [31,
27, 28]. The existence of such solutions demands the presence of some exotic
matter and additional geometric distortions for which the null energy
condition, NEC, is violated in order to achieve a stable and traversable
structure.

Let us remember in brief how traversable wormholes require a violation of the
so-called average null energy condition, ANEC, and how such conditions are
modified in the framework of MGTs with nonmetricity. The ANEC states that the
energy-momentum tensor for matter fields, $T_{\mu\nu},$ for a local quantum
field theory, QFT, along a complete achronal null geodesic, there are
satisfied the conditions $\int T_{\mu\nu}k^{\mu}k^{\nu}d\lambda\geq 0,$ where
$k^{\mu}$ is a tangent d-vector and $\lambda$ is an affine parameter. For
elaborating quantum gravity, QG, models, such conditions have to be considered
for certain effective matter field and distortion of geometric objects. Here
we note that in classical theories the violation of the ANEC is prevented by
the null energy conditions, NEC, $T_{\mu\nu}k^{\mu}k^{\nu}d\lambda\geq 0.$ It
is considered that such conditions must be valid for any physically reasonable
theory at least for classical and semi-classical configurations. Additionally,
there are considered other important criteria on causality, topological
censorship, absence of singularities etc.

Due to problems with transversability, many authors excluded wormholes to be
considered as reliable astrophysical objects. In another turn, a number of
authors concluded that there are possibilities to realize wormholes without
considering exotic matter but modifying GR. In various cosmological models,
there are used solutions with NECs. One supposes that such theories are not
experimentally prohibited in the condition when MGTs are involved and due to
ideas on existence of particles beyond the standard model.

However, for elaborating explicit physical models and applications in modern
cosmology and astrophysics, it is admitted that QM and distortions of
geometric effects may induce negative null energy, leading to violations of
some NECs and/or ANECs. For nonholonomic systems, the variational and
conservation laws are different from those stated for unconstrained ones. For
diagonal traversable systems, to sustain a traversable wormhole there are
introduced certain negative null energy and various nonlocal / nonachronal
constructions. The matter fields are considered as quantum ones, but the
gravitational field is treated classically. In such models, there are solved
some semi-classical Einstein’s equations with an effective source
$<T_{\mu\nu}>$ computed as the expectation value of the stress-energy tensor
in a given quantum state. For certain classes of wormhole solutions, the
1-loop expectation value of the stress-energy tensor satisfy in some spacetime
regions the conditions $\int<T_{\mu\nu}>k^{\mu}k^{\nu}d\lambda<0$. Such
configurations allows us to construct traversable Einstein-Rosen bridges with
certain interesting physical properties.

Quantum effects and wormhole solutions are studied in various MGTs and quantum
information theories. For elaborating quantum computing models, this provides
the possibility to transfer information between the two asymptotic spacetime
boundaries. Such a process can be viewed as a teleportation protocol, see
details and references in [32]. Here we note that wormholes seem to be
traversable for qubits [33, 34, 35].

In this work, we follow an approach to constructing wormhole solutions with
off-diagonal deformations of some prime wormhole metrics in 4-d theories (see
also generalizations for higher dimensions and MGTs [36, 37, 38, 39, 40, 41]).
Such solutions can be constructed using the AFCDM and extended to nonmetric
geometric flow and gravity theories as we have shown in previous subsections.
The generating sources $\ {}^{q}\mathbf{\Upsilon}_{\ \
\beta}^{\alpha}(\tau)\simeq[\ _{h}^{q}\Upsilon(\tau),\ ^{q}\Upsilon(\tau)]$
(28) for $Q$-modified Einstein equations (29) can be prescribed in such a form
that $\int\ ^{q}\mathbf{\Upsilon}_{\mu\nu}k^{\mu}k^{\nu}d\lambda<0$ even using
the matter energy-momentum d-tensor from (22), we have $\
{}^{m}\widehat{\mathbf{T}}_{\mu\nu}k^{\mu}k^{\nu}d\lambda\geq 0.$ This means
that certain types of nonmetric geometric flow and off-diagonal deformations
resulting in certain locally anisotropic wormhole solutions result in
traversable conditions even at the classical level. Such conditions can be
valid even for locally anisotropic wormhole solutions in GR because of
additional generic off-diagonal terms. This is also a result of nonholonomic
modifications of the spacetime structure.

##  4 Perelman thermodynamics for nonmetric quasi-stationary configurations

The parametric solutions constructed in previous section describe nonmetric
geometric flow and solitonic deformations of wormhole metrics. They are
characterized by respective Perelman statistical/ geometric thermodynamic
variables [15]. The Bekenstein-Hawking thermodynamic paradigm is not
applicable to such solutions because, in general, they do not involve certain
hypersurface configurations. We cite [23, 16] for details on relativistic
generalizations and MGTs. For geometric flows and metric and nonmetric gravity
theories, the W-functional (2.2.1) can be treated as a "minus" entropy. The
goal of this section is to show how nonmetric geometric flow thermodynamic
variables can be defined and computed for quasi-stationary off-diagonal
solutions of type (3.3.2).

###  4.1 Statistical thermodynamic variables for $Q$-deformed relativistic
geometric flows

Let us consider nonmetric geometric flow equations (29). They can be derived
in geometric or variational form, following the methods outlined in section
3.1 and 5 of [15],888in this work, we consider 4-d Lorentz manifols which
correspond to dimension $n=4$, which a conventional changing to Riemannian
signature; Perelman’s formulas for deriving geometric flows and thermodynamic
variables can be used for arbitrary dimension $n\geq 3$ and signature but such
formulas can’t be used for formulations/ proofs of analogs of
Poincare–Thorston conjectures in non-Riemannian cases from the W-functional,

$\widehat{\mathcal{W}}(\tau)=\int_{t_{1}}^{t_{2}}\int_{\Xi_{t}}\left(4\pi\tau\right)^{-2}e^{-\widehat{f}(\tau)}\sqrt{|\mathbf{g}(\tau)|}\delta^{4}u[\tau(\widehat{\mathbf{R}}sc(\tau)+|\widehat{\mathbf{D}}(\tau)\widehat{f}(\tau)|^{2}+\widehat{f}(\tau)-4].$
(64)

In this formula, the normalizing function $\widehat{f}(\tau,u)$ and the
parameter $\tau$ are such way re-defined that corresponding geometric flow
equations and normalizing conditions formulas contributions from possible
matter fields, $F$\- and $Q$-distortions of the geometric data
$(g(\tau),\nabla(\tau))$ to canonical ones,
$(\mathbf{g}(\tau),\mathbf{N}(\tau),\widehat{\mathbf{D}}(\tau)).$ We use
different integration measures and nonlinear symmetries for (64) comparing to
$\mathcal{W}(\tau)$ (2.2.1). The effects of nonmetricity will be emphasized
below when there will be used solutions of (29) determined by respective
$\tau$–running generating sources (28) or effective cosmological constants
(60).

On a metric-affine space $\mathcal{M}$ endowed with canonical geometric data
and an additional nonholonomic (3+1) splitting,999such a conventional
splitting is necessary for introducing thermodynamic variables, when the
nonholonomic 2+2 decompositions is important for generating off-diagonal
solutions we introduce the statistical partition function

$\ {}^{q}\widehat{Z}(\tau)=\exp[\int_{\widehat{\Xi}}[-\widehat{f}+2]\
\left(4\pi\tau\right)^{-2}e^{-\widehat{f}}\delta\ ^{q}\mathcal{V}(\tau)],$
(65)

where the volume element is defined and computed as

$\delta\ ^{q}V(\tau)=\sqrt{|\mathbf{g}(\tau)|}dx^{1}dx^{2}\delta y^{3}\delta
y^{4}\ .$ (66)

We use a left label $q$ because nonmetric $Q$-contributions can be encoded in
$\mathbf{g}(\tau).$ Such a label will be omitted in formulas below if that
will not result in ambiguities. Here we note that a a statistical
thermodynamic model can be constructed using a partition function
$Z=\int\exp(-\beta E)d\omega(E)$ for a canonical ensemble at temperature
$\beta^{-1}=\tau$ and when the measure is stated as the density of states
$\omega(E).$ The thermodynamical variables are computed as the average energy,
$\left\langle E\right\rangle:=-\partial\log Z/\partial\beta,$ the entropy
$S:=\beta\left\langle E\right\rangle+\log Z$ and the fluctuation parameter
$\sigma:=\left\langle\left(E-\left\langle
E\right\rangle\right)^{2}\right\rangle=\partial^{2}\log Z/\partial\beta^{2}.$

Using $\widehat{Z}$ (65) and $\widehat{\mathcal{W}}(\tau)$ (64) and following
for canonical variables a variational procedure on a closed region of
$\mathcal{M}$ as in section 5 of [15], we can define and compute respective
thermodynamic variables:

$\ {}^{q}\widehat{\mathcal{E}}\ (\tau)$ $\displaystyle=$
$\displaystyle-\tau^{2}\int_{\widehat{\Xi}}\
\left(4\pi\tau\right)^{-2}\left(\widehat{\mathbf{R}}sc+|\
\widehat{\mathbf{D}}\ \widehat{f}|^{2}-\frac{2}{\tau}\right)e^{-\
\widehat{f}}\ \delta\ ^{q}\mathcal{V}(\tau),$ (67) $\ \
{}^{q}\widehat{S}(\tau)$ $\displaystyle=$
$\displaystyle-\int_{\widehat{\Xi}}\left(4\pi\tau\right)^{-2}\left(\tau(\widehat{\mathbf{R}}sc+|\widehat{\mathbf{D}}\
\widehat{f}|^{2})+\widehat{f}-4\right)e^{-\ \widehat{f}}\delta\
^{q}\mathcal{V}(\tau),$ $\ \ {}^{q}\widehat{\sigma}(\tau)$ $\displaystyle=$
$\displaystyle 2\ \tau^{4}\int_{\widehat{\Xi}}\left(4\pi\tau\right)^{-2}|\
\widehat{\mathbf{R}}_{\alpha\beta}+\widehat{\mathbf{D}}_{\alpha}\
\widehat{\mathbf{D}}_{\beta}\
\widehat{f}-\frac{1}{2\tau}\mathbf{g}_{\alpha\beta}|^{2}e^{-\widehat{f}}\delta\
^{q}\mathcal{V}(\tau).$

We note that such a thermodynamic systems can be associated to solution of the
nonholonomic nonmetric geometric flow equations (19). In particular, we can
fix conventionally the temperature and consider such data for nonmetric Ricci
solitons characterized by $\left[\ {}^{q}\widehat{\mathcal{E}}(\tau_{0}),\
^{q}\widehat{\mathcal{S}}(\tau_{0}),\ ^{q}\widehat{\sigma}(\tau_{0})\right].$
Certain classes of solutions can be not well-defined in the framework of such
a statistical and geometric thermodynamic approach, for instance, if $\
{}^{q}\widehat{\mathcal{S}}(\tau_{0})<0.$ We have to restrict certain classes
of nonholonomic frames/ distributions/distortions in order to generate
physically viable solutions. The nonmetric $Q$-deformations may have different
sign contributions comparing to certain metric compatible classes of solutions
determined by corresponding
$\left[\widehat{\mathcal{E}}(\tau),\widehat{\mathcal{S}}(\tau),\widehat{\sigma}(\tau)\right].$

###  4.2 Thermodynamic variables for nonmetric deformations of wormhole
solutions

We compute in explicit form the variables $\widehat{Z}$ (65), and $\
{}^{q}\widehat{\mathcal{E}}\ (\tau),\ ^{q}\widehat{S}(\tau)$ from (67) for
quasi-stationary off-diagonal solutions (3.3.2).101010We omit more cumbersome
calculations for $\ {}^{q}\widehat{\sigma}(\tau).$ The simplest way is to
consider that

$\widehat{\mathbf{R}}sc=2[\ _{1}^{tot}\Lambda(\tau)+\ _{1}^{1}\Lambda(\tau)+\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]$

choosing such a normalizing function when $\widehat{\mathbf{D}}_{\alpha}\
\widehat{f}=0$ and approximating $\widehat{f}\approx 0.$ Such conditions can
be considered for a frame/coordinate system and then the results can be
redefined for arbitrary bases and normalizing functions. Correspondingly, we
obtain

$\ {}^{q}\widehat{Z}(\tau)$ $\displaystyle=$
$\displaystyle\exp[\int_{\widehat{\Xi}}\frac{1}{8\left(\pi\tau\right)^{2}}\
\delta\ ^{q}\mathcal{V}(\tau)],$ (68) $\ {}^{q}\widehat{\mathcal{E}}\ (\tau)$
$\displaystyle=$ $\displaystyle-\tau^{2}\int_{\widehat{\Xi}}\
\frac{1}{8\left(\pi\tau\right)^{2}}[\ _{1}^{tot}\Lambda(\tau)+\
_{1}^{1}\Lambda(\tau)+\ _{2}^{tot}\Lambda(\tau)+\
_{2}^{1}\Lambda(\tau)-\frac{1}{\tau}]\ \delta\ ^{q}\mathcal{V}(\tau),$ $\ \
{}^{q}\widehat{S}(\tau)$ $\displaystyle=$ $\displaystyle-\
^{q}\widehat{W}(\tau)=-\int_{\widehat{\Xi}}\frac{1}{8\left(\pi\tau\right)^{2}}[\tau(\
_{1}^{tot}\Lambda(\tau)+\ _{1}^{1}\Lambda(\tau)+\ _{2}^{tot}\Lambda(\tau)+\
_{2}^{1}\Lambda(\tau)-2]\delta\ ^{q}\mathcal{V}(\tau).$

To compute the volume form $\delta\ ^{q}\mathcal{V}(\tau)$ (66) is better to
consider the equivalent d-metric (3.3.2) with $\eta$–polarization functions,
or (62) for $\eta$–polarization functions, and including data for nonmetric
generating sources. Respectively, we can write

$\ {}_{2}\Phi(\tau)$ $\displaystyle=$ $\displaystyle 2\sqrt{|[\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]\ g_{4}(\tau)|}=\ 2\sqrt{|\ [\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]\ \eta_{4}(\tau)\
\breve{g}_{4}(\tau)|}$ (69) $\displaystyle\simeq$ $\displaystyle 2\sqrt{|\ [\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]\ \zeta_{4}(\tau)\
\breve{g}_{4}|}[1-\frac{\varepsilon}{2}\chi_{4}(\tau)].$

For simplicity, we shall elaborate on nonholonomic evolution models with
trivial integration functions $\ {}_{1}n_{k}=0$ and $\ {}_{2}n_{k}=0.$
Introducing formulas (69) in (66), then separating terms with shell
$\tau$-running cosmological constants, we express:

$\displaystyle\ \delta\ ^{q}\mathcal{V}$ $\displaystyle=$
$\displaystyle\delta\mathcal{V}[\tau,\ \ _{1}^{tot}\Lambda(\tau)+\
_{1}^{1}\Lambda(\tau),\ _{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau);\
_{h}^{q}\Upsilon(\tau),\ ^{q}\Upsilon(\tau);\psi(\tau),\ g_{4}(\tau)]$
$\displaystyle=$ $\displaystyle\delta\mathcal{V}(\ \ _{h}^{q}\Upsilon(\tau),\
^{q}\Upsilon(\tau),\ \ _{1}^{tot}\Lambda(\tau)+\ _{1}^{1}\Lambda(\tau),\
_{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau),\eta_{4}(\tau)\breve{g}_{4})$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{|[\ \ _{1}^{tot}\Lambda(\tau)+\
_{1}^{1}\Lambda(\tau)][\ _{2}^{tot}\Lambda(\tau)+\ _{2}^{1}\Lambda(\tau)]|}}\
\delta\ _{\eta}\mathcal{V},\mbox{ where }\ \delta\ _{\eta}\mathcal{V}=\
\delta\ _{\eta}^{1}\mathcal{V}\times\delta\ _{\eta}^{2}\mathcal{V}.$

In these formulas, we use the functionals:

$\displaystyle\delta\ _{\eta}^{1}\mathcal{V}$ $\displaystyle=$
$\displaystyle\delta\ _{\eta}^{1}\mathcal{V}[\ _{1}^{tot}\Lambda(\tau)+\
_{1}^{1}\Lambda(\tau),\eta_{1}(\tau)\ \breve{g}_{1}]$ $\displaystyle=$
$\displaystyle e^{\widetilde{\psi}(\tau)}dx^{1}dx^{2}=\sqrt{|\ \ \
_{1}^{tot}\Lambda(\tau)+\
_{1}^{1}\Lambda(\tau)|}e^{\psi(\tau)}dx^{1}dx^{2},\mbox{ for }\psi(\tau)\mbox{
being a solution of }(\ref{eq1}),$ $\displaystyle\delta\
_{\eta}^{2}\mathcal{V}$ $\displaystyle=$ $\displaystyle\delta\
_{\eta}^{2}\mathcal{V}[\ ^{q}\Upsilon(\tau),\eta_{4}(\tau)\ \breve{g}_{4}]$
$\displaystyle=$ $\displaystyle\frac{\partial_{3}|\ \eta_{4}(\tau)\
\breve{g}_{4}|^{3/2}}{\ \sqrt{|\int dy^{3}\ \
{}^{q}\Upsilon(\tau)\\{\partial_{3}|\ \eta_{4}(\tau)\
\breve{g}_{4}|\\}^{2}|}}[dy^{3}+\frac{\partial_{i}\left(\int dy^{3}\ \
{}^{q}\Upsilon(\tau)\partial_{3}|\ \eta_{4}(\tau)\
\breve{g}_{4}|\right)dx^{i}}{\ \ {}^{q}\Upsilon(\tau)\partial_{3}|\
\eta_{4}(\tau)\ \breve{g}_{4}|}]dt,$

where numeric coefficients were used for re-defining the generating functions.
We note that we can define $\widetilde{\psi}(\tau)$ as a $\tau$–family of
solutions of 2-d Poisson equations with effective source $\
{}_{1}^{tot}\Lambda(\tau)+\ _{1}^{1}\Lambda(\tau),$ or use $\psi(\tau)$ for a
respective source $\ {}_{h}^{q}\Upsilon(\tau).$ Integrating on a closed
hypersurface $\widehat{\Xi}$ such products of $h$\- and $v$-forms, we obtain a
running phase space volume functional

$\ {}_{\eta}^{\shortmid}\mathcal{\mathring{V}}(\tau)=\int_{\
\widehat{\Xi}}\delta\ _{\eta}\mathcal{V}(\ _{h}^{q}\Upsilon(\tau),\
^{q}\Upsilon(\tau),\ \breve{g}_{\alpha})$

determined by prescribed classes of generating $\eta$-functions, effective
generating sources $\left[{}_{h}^{q}\Upsilon(\tau),\
^{q}\Upsilon(\tau)\right],$ coefficients of a prime s-metric $\
\mathring{g}_{\alpha}$ and nonholonomic distributions defining the hyper-
surface $\widehat{\Xi}.$ The explicit value of $\
{}_{\eta}^{\shortmid}\mathcal{\mathring{V}}(\tau)$ depends on the data we
prescribe for $\widehat{\Xi}$ the type of $Q$-deformations (via $\eta$\- or
$\zeta$-polarizations) we use for deforming a prime wormhole d-metric into
quasi-stationary ones as we considered in section 3. We emphasize that it is
always possible to compute $\
{}_{\eta}^{\shortmid}\mathcal{\mathring{V}}(\tau)$ for certain nonlinear
solitonic waves/ distributions and some general $Q$-deformations. The
thermodynamic variables depend on the $\tau$-running effective cosmological
constants.

##  5 Conclusions and open questions

In this work we elaborated on the nonmetric geometric flow theory of metric-
affine spaces and applied it to modified gravity theories, MGT, as in [5, 10,
6, 7, 8]. The approach was generalized in nonholonomic form [11, 13, 16] with
the aim to apply the anholonomic frame and connection deformation method,
AFCDM, for constructing physically important exact and parametric solutions in
geometric flow and gravity theories with nonmetricity. Such solutions are
defined by generic off-diagonal metrics and generalized (non) linear
connections and, in general, do not possess hypersurface/ duality /
holographic configurations which would allow to treat them in the framework of
Bekenstein-Hawking paradigm [42, 43, 44, 45]. In another turn, as we have
shown in this paper, the G. Perelman statistical and geometric thermodynamic
paradigm [15] can be applied for all types of solutions in MGTs including
nonmetric geometric flow evolution models as we considered in the previous
section. In addition to gaining a more complete understanding of gravity
theories with nonmetricity, we also studied in this article certain new
classes of wormhole and solitonic solutions encoding nonmetric data. This
included such new and original results:

  1. 1.

In section 2.1, the metric-affine geometry was formulated in nonholonomic
dyadic variables for nonmetric $Q$-deformed 4-d Lorentz manifolds. Such a
formulation allows us to prove general decoupling and integration properties
of nonmetric geometric flow equations and modified Einstein equations in MGTs
as we outlined in Appendix A.

  2. 2.

The Obj1 of this work was completed in section 2.2 where Lyapunov type F- and
W-functionals are defined for nonholonomic variables encoding
$Q$-deformations. This allowed us to formulate nonmetric geometric flow
models, which for self-similar configurations define nonmetric Ricci solitons
containing as particular cases, for instance, nonmetric gravitational
equations studied in [10, 6, 7].

  3. 3.

In section 3, we solved the goals of Obj2 by constructing in explicit form two
classes of physically important quasi-stationary solutions of nonmetric
geometric flow equations which for fixed flow parameters define $Q$-deformed
Einstein spaces. We proved that such generic off-diagonal solutions can be
described in general form by respective solitonic hierarchies and solitonic
distributions (see subsection 3.2, when the necessary concepts and formulas
are outlined in Appendix B).

  4. 4.

Wormhole solutions present an important tool for testing MGTs and applications
in modern quantum computer science as we show by constructing and analyzing
possible nonmetric effects in subsection 3.3. Such configurations can be with
nonholonomic solitonic $Q$-deformations and gravitational polarizations,
locally anisotropic, in particular, ellipsoid deformations of throats, when
nonmetricity makes such configurations to be transversable.

  5. 5.

In general, the quasi-stationary solutions encoding nonmetricity do not
involve hypersurface / holographic configurations or certain duality
conditions when the concept of Bekenstein-Hawking entropy could be applicable.
As in GR and other MGTs, general classes of exact/ parametric solutions can be
characterized thermodynamically in the framework of corresponding
generalization of G. Perelman paradigm with W-entropy. In section 4, we show
how such constructions can be performed for $Q$-deformations, which presents a
solution of Obj3.

The quasi-stationary solutions constructed in this work are characterized by a
$\tau$-family of Weyl d-vectors $\mathbf{q}_{\alpha}(\tau)$ subjected to
conditions of type (23). For such nonholonomic configurations, we can computed
respectively induced $\tau$-families of nonmetricity d-tensors,
$\mathbf{Q}_{\alpha\beta\gamma}(\tau)=\mathbf{q}_{\alpha}(\tau)\mathbf{g}_{\beta\gamma}(\tau)$,
and d-torsion,
$\mathbf{T}_{\mu\nu\alpha}(\tau)=\mathbf{A}_{\nu}(\tau)\mathbf{g}_{\mu\alpha}(\tau)-\mathbf{A}_{\alpha}(\tau)\mathbf{g}_{\mu\nu}(\tau),$
for $\mathbf{A}_{\mu}(\tau)=q\mathbf{q}_{\mu}(\tau),$ when $q=const$ or depend
on $\tau.$ Such a $q$ can be chosen as a small parameter (like $\varepsilon$
in (A.17)) for $Q$-deformations which allow to construct parametric
deformations of physically important solutions in GR.

Let us discuss the legacy of using $Q$-deformed Perelman’s F- and
W-functionals to formulate and prove analogues of Poincaré–Thorston conjecture
[15, 20, 21, 22] for nonmetric geometric flows. For general $Q$-deformations
this is an un-defined mathematical problem similar to those for an infinite
number of noncommutative/ nonassociative differential and integral calculuses
and geometric theories, see discussions and respective variants of solutions
in [16, 13, 23]. For metric-affine spaces, an infinite number of topological
and nonmetric geometric models can be formulated because of an infinite number
of nonlinear and linear connection structures that can be used. So, it is not
possible to formulate a general mathematical framework involving only some
fundamental topological theories and nonmetric geometric analysis.
Nevertheless, self-consistent generalizations of the statistic and nonmetric
geometric thermodynamics are possible if there are used $Q$-deformations as in
sections 2.2.1 and 4. They encode nonmetric geometric data and result in
nonholonomic Ricci soliton configurations and $Q$-modified Einstein equations.
Such systems of nonlinear PDEs can be solved in some general forms as we show
in Appendix A and provide explicit examples in section 3. For instance, we can
associate and compute for such generic off-diagonal solutions respective
Perelman-like nonmetric geometric thermodynamic variables, see respective
formulas (67), (68) and (LABEL:thermvar2). Thus, such nonmetric geometric flow
and MGTs and their associated thermodynamic theories can be formulated in a
self-consistent form as $\tau$-parametric $Q$-deformations of Lorentz
manifolds geometries, and this is possible even if we are not able to
formulate in general form a rigorous version of metric-affine Poincarè
hypothesis. Here we also note that the concept of Bekenstein-Hawking entropy
is not applicable for the classes of nonmetric solitonic and wormholes
solutions considered in section 3. However, the concepts of Perelman’s
W-entropy and related statistical thermodynamics can be generalized for
various classes of nonmetric theories and their solutions.

The results of this work support the Hypothesis from the Introduction section
in such senses:

  1. 1.

We constructed in explicit form certain models of metric-affine geometric flow
and MGTs which are exactly/ parametric solvable in certain general off-
diagonal forms in nonholonomic dyadic variables.

  2. 2.

The solutions with $\tau$-running effective cosmological constants can be used
for modelling DE physical effects and other type configurations with
generating sources for effective matter (all such solutions encoding
nonmetricity data) for modelling DM physics.

  3. 3.

In this paper, we elaborated only on nonmetric quasi-stationary configurations
which can be described as solitonic hierarchies or nonmetric wormhole
solutions and certain nonlinear $Q$-deformations of such generic off-diagonal
solutions subjected to respective nonlinear symmetries.

  4. 4.

Perelman type nonmetric geometric thermodynamic variables were defined and
computed in explicit form for the mentioned classes of quasi-stationary
solutions.

Nevertheless, there is a series of important fundamental problems that should
be investigated and solved in future works. Here we outline four of the most
important open questions on nonmetric geometric and information flow theories
and gravity (QNGIFG) and cite some relevant previous works:

  * •

QNGIFG1: To elaborate full and viable classical and quantum theories on
metric-affine spacetimes we have to formulate a theory of spinors and
$Q$-deformed Dirac operators, which is not possible in general form for
arbitrary nonmetric structures. This problem is discussed in more general
forms in [9, 12] for phase or Finsler-Lagrange-Hamilton theories on (co)
tangent Lorentz bundles. Corresponding conceptual and technical difficulties
exist for metric-affine generalizations of Lorentz manifolds. Certain
solutions can be found as $Q$-deformed off-diagonal Einstein-Dirac systems,
see previous results [14].

  * •

QNGIFG2: One of the next steps is to study models of $Q$-deformed Einstein-
Yang-Mills-Higgs systems. If such systems are derived as star-product R-flux
deformations in string theory, the obtain nonholonomic geometric structures
with nonsymmetric metrics and $Q$-deformed Einstein-Eisenhart-Moffat theories,
see details and references to [13, 10].

  * •

QNGIFG3: $Q$-deformed off-diagonal cosmological systems can be considered as
certain dual ones to quasi-stationary configurations as stated in [11]. Such
solutions involve, for instance, various quasi-periodic (cosmological time
quasi-crystals etc.) and time-solitonic hierarchies which can be exploited for
modelling DE and DM effects.

  * •

QNGIFG4: Finally, we point to the possibility to extend the geometric and
quantum information flow theory [16, 23] to certain $Q$-deformed versions with
nonmetric qubits, nonmetric entanglement and respective generalizations of
conditional entropies with $Q$-modified Perelman’s functionals.

We shall report on progress to answers for above questions in future works.

Acknowledgments and Historical Remarks:

SV activity on nonmetric geometry and constructing off-diagonal solutions in
MGTs was inspired by many discussions during 1986-1988 with Ascar Aringazin,
Sergei Ponomarenko and Anatolii Mikhailov who were postgraduate students of
Prof. G. A. Asanov at the department of theoretical physics of M. Lomonosov
Moscow State University. Those ideas and nonholonomic geometric formalism
resulted in elaborating the AFCDM in a series of works by SV and his students
from R. Moldova, for metric affine Finsler-Lagrange-Hamilton geometry and
nonmetric locally anisotropic gravity theories [10], see Introduction,
Conclusion and Appendix B to [12] for historical remarks, main references and
reviews of results. This research on nonmetric geometric flows and
applications extends SV and AZ visit programs, supported respectively by
Fulbright USA-Romania and the Ministry of Education and Science of the
Republic of Kazakhstan, at the physics department at California State
University at Fresno, USA. The program on the AFCDM for theories with
effective sources encoding nonmetricity consists also a generalization of some
objectives for a SV’s visiting program at CAS LMU in Munich, Germany. Co-
affiliations for the universities in Germany, Ukraine and USA are motivated by
the mentioned fellowships. Authors are grateful to Prof. D. Lüst, Prof. D.
Singleton and Dr. Lena Bouman for respective support of their research
activity.

##  Appendix A Decoupling and integrability of nonmetric quasi-stationary
geometric flow equations

In this appendix, we summarize necessary formulas which allow us to prove some
general decoupling and integration properties of nonmetric geometric flow
equations (19) with generating sources (28). There are studied generic off-
diagonal solutions parameterized by a quasi-stationary ansatz (26). Details on
nonholonomic geometric methods can be found in generalized Finsler-Lagrange-
Hamilton and metric–affine forms in [10, 11, 12].

###  A.1 Decoupling of nonlinear PDEs with nonmetricity fields

We provide formulas for $\tau$-depending coefficients of fundamental geometric
objects describing nonmetric geometric flows in such a form that for any fixed
$\tau_{0}$ we shall generate solutions for nonmetric Ricci solitons.

####  A.1.1 The coefficients of quasi-stationary ansatz and canonical Ricci
d-tensor

To generate quasi-stationary $\tau$-evolving solutions of the system of
nonlinear PDEs (29) we consider a generic off-diagonal ansatz with N-adapted
coefficients of type (26), when the nonlinear quadratic element is

$\displaystyle\mathbf{\hat{g}}(\tau)$ $\displaystyle=$ $\displaystyle
g_{i}(\tau,x^{k})dx^{i}\otimes
dx^{i}+h_{3}(\tau,x^{k},y^{3})\mathbf{e}^{3}(\tau)\otimes\mathbf{e}^{3}(\tau)+h_{4}(\tau,x^{k},y^{3})\mathbf{e}^{4}(\tau)\otimes\mathbf{e}^{4}(\tau),$
(A.1)
$\displaystyle\mathbf{e}^{3}(\tau)=dy^{3}+w_{i}(\tau,x^{k},y^{3})dx^{i},\
\mathbf{e}^{4}(\tau)=dy^{4}+n_{i}(\tau,x^{k},y^{3})dx^{i}.$

This d-metric possess a Killing symmetry on the time like coordinate
$\partial_{4}=\partial_{t}$. Such d-metric and N-connection coefficients are
functions of necessary smooth class on respective coordinates. We put a "hat"
label for a family of d-metrics $\mathbf{\hat{g}}(\tau)$ in order to emphasize
that such d-metrics are with Killing symmetry on $\partial_{t}.$ It is
supposed that such a parametrization can be obtained for corresponding classes
of frame/ coordinate transforms for a general family
$\mathbf{\hat{g}}(\tau,u)$ depending on all spacetime coordinates for other
systems of references.

Tedious computations of N-adapted coefficients of the canonical d-connection
and respective Ricci d-tensors for (A.1) result in such formulas for the
system of nonlinear PDEs (29):

$\displaystyle\widehat{R}_{1}^{1}(\tau)$ $\displaystyle=$
$\displaystyle\widehat{R}_{2}^{2}(\tau)=\frac{1}{2g_{1}g_{2}}[\frac{g_{1}^{\bullet}g_{2}^{\bullet}}{2g_{1}}+\frac{(g_{2}^{\bullet})^{2}}{2g_{2}}-g_{2}^{\bullet\bullet}+\frac{g_{1}^{\prime}g_{2}^{\prime}}{2g_{2}}+\frac{\left(g_{1}^{\prime}\right)^{2}}{2g_{1}}-g_{1}^{\prime\prime}]=-\
\ _{1}^{q}\mathbf{\Upsilon}(\tau),$ $\displaystyle\widehat{R}_{3}^{3}(\tau)$
$\displaystyle=$
$\displaystyle\widehat{R}_{4}^{4}(\tau)=\frac{1}{2h_{3}h_{4}}[\frac{\left(h_{4}^{\ast}\right)^{2}}{2h_{4}}+\frac{h_{3}^{\ast}h_{4}^{\ast}}{2h_{3}}-h_{4}^{\ast\ast}]=-\
\ _{2}^{q}\mathbf{\Upsilon}(\tau),$ (A.2)
$\displaystyle\widehat{R}_{3k}(\tau)$ $\displaystyle=$ $\displaystyle\frac{\
w_{k}}{2h_{4}}[h_{4}^{\ast\ast}-\frac{\left(h_{4}^{\ast}\right)^{2}}{2h_{4}}-\frac{(h_{3}^{\ast})(h_{4}^{\ast})}{2h_{3}}]+\frac{h_{4}^{\ast}}{4h_{4}}(\frac{\partial_{k}h_{3}}{h_{3}}+\frac{\partial_{k}h_{4}}{h_{4}})-\frac{\partial_{k}(h_{3}^{\ast})}{2h_{3}}=0;$
$\displaystyle\widehat{R}_{4k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{h_{4}}{2h_{3}}n_{k}^{\ast\ast}+\left(\frac{3}{2}h_{4}^{\ast}-\frac{h_{4}}{h_{3}}h_{3}^{\ast}\right)\frac{\
n_{k}^{\ast}}{2h_{3}}=0.$

For simplicity, we use brief notations of partial derivatives when, for
instance,
$\partial_{1}q(u^{\alpha}):=q^{\bullet},\partial_{2}q(u^{\alpha}):=q^{\prime},\partial_{3}q(u^{\alpha}):=q^{\ast}$
for an arbitrary function $q(u^{\alpha}).$ In abstract geometric form, such
formulas are written in similar forms as in various MGTs but in this work the
generating sources encode nonmetricity terms as we explained for (28).

####  A.1.2 Decoupling of nonmetric geometric flow equations

Let us express $g_{i}(\tau)=e^{\psi(\tau,x^{k})}$ and introduce the
coefficients
$\alpha_{i}(\tau)=h_{4}^{\ast}\partial_{i}[\varpi(\tau)],\beta(\tau)=h_{4}^{\ast}(\tau)[\varpi(\tau)]^{\ast},$
$\gamma(\tau)=(\ln|h_{4}(\tau)|^{3/2}/|h_{3}(\tau)|)^{\ast},$ for
$\varpi(\tau)=\ln|h_{4}^{\ast}(\tau)/\sqrt{|h_{3}(\tau)h_{4}(\tau)}|;$ and
considering $\ \Psi(\tau)=\exp[\varpi(\tau)]$ as a family of generating
functions. The equations (A.2) transform into:

$\displaystyle\psi^{\bullet\bullet}+\psi^{\prime\prime}$ $\displaystyle=$
$\displaystyle 2\ \ _{1}^{q}\mathbf{\Upsilon}(\tau),$ (A.3)
$\displaystyle(\varpi)^{\ast}h_{4}^{\ast}$ $\displaystyle=$ $\displaystyle
2h_{3}h_{4}\ {}_{2}^{q}\mathbf{\Upsilon}(\tau),$ (A.4) $\displaystyle\beta
w_{j}-\alpha_{j}$ $\displaystyle=$ $\displaystyle 0,$ (A.5) $\displaystyle\
n_{k}^{\ast\ast}+\gamma n_{k}^{\ast}$ $\displaystyle=$ $\displaystyle 0,$
(A.6)

where the explicit dependence of coefficients on respective $(\tau,x^{k})$ or
$(\tau,x^{k},y^{3})$ is omitted. This system of equations together with the
previous one possess an explicit decoupling property. In brief, this means
that $g_{i}(\tau)$ are related to a $\tau$-family of 2-d Poisson equations
(A.3); then $h_{3}(\tau)$ and $h_{4}(\tau)$ are related via nontrivial
$\varpi(\tau)$ and $\ {}_{2}^{q}\mathbf{\Upsilon}(\tau)$ as in (A.4). Finding
any solution for $h_{a}(\tau),$ we can compute the families of coefficients
$\beta(\tau)$ and $\alpha_{i}(\tau)$ and solve respective linear equations for
$w_{j}(\tau)$ from (A.5). To find solutions for $n_{k}(\tau)$ we have to
integrate two times on $y^{3}$ in (A.6) when $\gamma(\tau)$ is determined by
$h_{3}(\tau)$ and $h_{4}(\tau).$

###  A.2 Off-diagonal solutions for nonmetric quasi-stationary configurations

We can generate $\tau$-families of solutions of nonmetric geometric flow
equations by integrating recurrently the decoupled system of nonlinear PDEs
(A.3)-(A.6). Any generic off-diagonal metric (A.1) (if the N-coefficients
vanish for certain coordinate transforms, we generate diagonal metrics) is