Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids

7/30/2019 Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids

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arXiv:submit/072418

2[math-ph]25M

ay2013

Almost Khler Ricci Flows and Einstein and

LagrangeFinsler Structures on Lie Algebroids

Sergiu I. Vacaru

Rectors Office, Alexandru Ioan Cuza University, Alexandru Lapuneanustreet, nr. 14, UAIC Corpus R, office 323; Iai, Romania, 700057

May 22, 2013

Abstract

In this work we investigate Ricci flows of almost Khler structureson Lie algebroids when the fundamental geometric objects are com-pletely determined by (semi) Riemannian metrics, or (effective) reg-ular generating Lagrange/ Finsler, functions. There are constructedcanonical almost symplectic connections for which the geometric flowscan be represented as gradient ones and characterized by nonholonomicdeformations of Grigory Perelmans functionals. The first goal of this

paper is to define such thermodynamical type values and derive almostKhler Ricci geometric evolution equations. The second goal is tostudy how fixed Lie algebroid, i.e. Ricci soliton, configurations canbe constructed for Riemannian manifolds and/or (co) tangent bun-dles endowed with nonholonomic distributions modelling (generalized)Einstein or Finsler Cartan spaces. Finally, there are provided someexamples of generic offdiagonal solutions for Lie algebroid type Riccisolitons and (effective) Einstein and LagrangeFinsler algebroids.

Keywords: Ricci flows, almost Khler structures, Lie algebroids,Lagrange mechanics, Finsler geometry, effective Einstein spaces.

MSC: 53C44, 53D15, 37J60, 53D17, 70G45, 70S05, 83D99, 53B40, 53B35

Contents

1 Introduction: Almost Khler Models for Einstein & Lagran-

geFinsler Spaces 2

1.1 Preliminaries: nonholonomic manifolds and bundles . . . . . . 31.2 Canonical almost Khler variables for semiRiemannian and

LagrangeFinsler spaces . . . . . . . . . . . . . . . . . . . . . 6

All Rights Reserved c 2013 Sergiu I. Vacaru; [email protected];http://www.uaic.ro/uaic/bin/view/Research/AdvancedTheoretical

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2 Almost Khler Lie Algebroids & Nconnections 9

2.1 Distinguished Lie algebroids and prolongations . . . . . . . . 92.2 Canonical structures on Lie dalgebroids . . . . . . . . . . . . 12

2.2.1 dconnections and dmetrics on TEP . . . . . . . . . 122.2.2 The canonical dconnection . . . . . . . . . . . . . . . 13

2.3 Almost symplectic geometric data . . . . . . . . . . . . . . . . 142.3.1 Semispray configurations and Nconnections . . . . . 142.3.2 RiemannLagrange almost symplectic structures . . . 162.3.3 Nadapted symplectic connections . . . . . . . . . . . 17

2.4 Almost Khler Einstein and Lagrange Lie dalgebroids . . . . 19

3 Almost Khler Ricci Lie Algebroid Evolution 20

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Perelmans functionals in almost Khler variables . . . . . . . 213.3 Nadapted almost symplectic evolution equations . . . . . . . 233.4 Functionals for entropy and thermodynamics KEE . . . . . . 25

4 Ricci Solitons with Lie Algebroid Symmetries 27

4.1 Preliminaries on Lie dalgebroid solitons . . . . . . . . . . . . 274.2 Generalized Einstein equations encoding Lie dalgebroids . . 284.3 Generating offdiagonal solutions . . . . . . . . . . . . . . . . 304.4 On Lie algebroid & almost Khler Finsler Ricci solitons . . 32

A Formulas in Coefficient Forms 33

A.1 Torsions and curvatures on TEP . . . . . . . . . . . . . . . . 33A.2 Nadapted coefficients for the canonical dconnection . . . . . 34A.3 Lie algebroid mechanics and KernMatsumoto models . . . . 36A.4 The torsion and curvature of the normal dconnection . . . . 37

1 Introduction: Almost Khler Models for Einstein

& LagrangeFinsler Spaces

Various theories of geometric flows have been studied intensively in thepast decade. The most popular is the Ricci flow theory [ 1, 2, 3] finally elab-

orated by G. Perelman [4, 5, 6] in the form which allowed proofs of theThurston and Poincar conjectures (details and proofs are given in [7, 8, 9]).The main results are related to the evolution of Riemannian and Khler met-rics and symplectic curvature flows, see additional references in [10, 11]. Itshould be also emphasized that the Ricci flow theory has an increasing im-pact in physical mathematics. One considers that such a geometric evolution1) determines fundamental spacetime topological and geometric properties;2) the geometric flows are necessary for understanding important issues on

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renormalization of quantum field theories; 3) Perelman type functionals pro-

vide alternative thermodynamical values characterizing the evolution andsymmetries of nonlinear systems; 4) for fixed/ stationary configurations (forinstance, for Ricci solitons), the metrics define generalized Einstein spacesand modified gravity theories. In a more general context, we were interestedto study Ricci flow evolution models for nonRiemannian geometries, forinstance, of nonholonomic manifolds endowed with compatible metric andnonlinear connection structures [12], metric compatible and noncompatibleRiemannFinslerLagrange spaces [13, 14], noncommutative geometries andgeneralizations [15], fractional and/or diffusion Ricci flows [16] etc.

This work is the second one in a series of papers devoted to the Ricciflow evolution and geometric and physical models on Lie algebroids. The

first partner [17] is on algebroid Lagrange evolution when almost symplec-tic variables are not considered. The general purpose of this article is toformulate a geometric approach for the almost Khler Ricci flows on/of Liealgebroids when the fundamental geometric objects are completely definedby a regular Lagrange, L, generating function, or a (generalized) Einsteinmetric g with conventional integer (n + m)dimensional splitting. Simi-lar constructions can be performed for certain effective (analogous) modelsand/or, in particular, for Finsler, F, fundamental functions.1 Our interestin almost Khler structures is motivated from the fact that we can quantizesuch configurations following methods of deformation quantization [19, 20]or Abrane quantization [21]. In our further works, we plan to extend such

constructions to commutative and noncommutative algebroid configurations.

1.1 Preliminaries: nonholonomic manifolds and bundles

To begin, let us fix a real manifold V of dimension n 2 and necessarysmooth class and consider a conventional horizontal (h) and vertical (v)splitting determined by a Whitney sum for its tangent bundle T V,

N : T V = hT V vTV. (1)

Definition 1.1 A h v splittingN (1) defines a nonlinear connection, Nconnection, structure.

We shall use boldface symbols in order to emphasize that the geometricobjects on a space V = (V,N,g),TV, or TM = (T M ,N,L), are adapted

1In this paper, a nonholonomic manifold (V,N) , [dim V = n + m with finite n, m 2]is modelled as a (semi) Riemannian one of necessary smooth class and endowed with anonintegrable (nonholonomic, equivalently, anholonomic) distribution N. For geomet-ric models of LagrangeFinsler spaces, we can work with nonholonomic vector/ tangentbundles, when V = T M is the total space of the tangent bundle on a real manifoldM, dim M = n, see details and references in [18].

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to a Nsplitting (1).2 In global form, the concept of Nconnection was for-

mulated by C. Ehresmann [22] in 1955. E. Cartan used such a geometricobject in his works on Finsler geometry beginning 1935 (see [18] for refer-ences and an alternative definition of Nconnections via exact sequences)but in coordinate form, N = Nai (x, y)dx

i /ya. On (semi) Riemannianmanifolds, a h vsplitting of type (1) can be defined by a correspondingclass of nonholonomic frames.

Let us consider a nonholonomic distribution on V defined by a generat-ing function L(u), with nondegenerate Hessian hab = 12

2Lyayb

, det |hab| = 0.If V = T M, and n = m, we can consider L = L(x, y) as a regular La-grangian defining a Lagrange space [23]. For L = F2(x, y), where F(x,y) =F(x, y), > 0, we get a homogeneous Finsler generating function (addi-

tional conditions are imposed on F for different models of Finsler like ge-ometries), see section 4.4. We shall work with arbitrary L considering theFinsler configurations as certain particular ones distinguished by homogene-ity conditions and additional assumptions.

Theorem 1.1 The EulerLagrange equations ddLyi

Lxi

= 0, where yi =

dxi/d, for xi() depending on real parameter , are equivalent to the "non-linear" geodesic equations dx

a

d +2Ga(x, y) = 0, i.e. to the paths of a canonical

semispray S = yi Lxi

2Ga ya , when Ga = 14 ha n+i( 2L

yn+ixkyn+k L

xi),

where hab is inverse to hab, and the set of coefficients Nai =

Ga

yn+idefines a

Nconnection structure (1).

An explicit proof consists from straightforward computations. We put"tilde" on some geometric objects in order to emphasize that they are gen-erated by L.Corollary 1.1 Any prescribed generating function L onV defines a canon-ical distinguished metric, dmetric g, which is adapted to the Nconnection

2Notational Remarks: All constructions in this work can be performed in coordi-

nate free form. Nevertheless, to find/generate explicit examples of solutions of systemsof partial differential equations, PDEs, is necessary to consider some adapted frame andcoordinate systems. Working with different classes of geometries and spaces, it is moreconvenient to treat indices also as abstract labels. This simplifies formulas and sugests

ideas how certain proofs are possible when local constructions have an "obvious" globalextension. For instance, index type formulas are largely used in G. Perelmans works andfor applications in mathematical relativity and/or geometric mechanics.

For a conventional n + m splitting, the local coordinates u = (x, y) can be labelled inthe form u = (xi, ya), where i,j,k,... = 1, 2, ...n and a,b,c... = n + 1, n + 2,...,n + m,where xi and ya are respectively the h and vcoordinates. A general local base is writ-ten e = (ei , ea) for some frame transforms e = e

(u), where = /u

=(i = /x

i, a = /ya), and define corresponding dual transforms with inverse matri-

ces e

(u), when e = e

(u)du. Calligraphic symbols will be considered for algebroid

configurations. The left "uplow" indices will be considered as abstract labels withoutsummation rules. Finally, we note that the Einstein summation rule on "low - up" re-peating indices will be applied if the contrary is not stated.

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splitting determined by data hab and Nai , i.e. g = hg + vg, when

g = gijdxi dxj + habea eb, gij = hn+i n+j, (2)e = (ei = i Nai a, ea = a), e = (ei = dxi, ea = dya + Nai dxi)(3)The proof is a similar to that for the Sasaki lift from M to T M, see [24].

It should be emphasized here that any metric g = {g} on V, or T M,can be expressed in a form g = [ gij, hab] (2), i.e. as a dmetric, via

corresponding frame transforms, e = e

e and ge

e

= g.3 For

simplicity, we shall write (g,N) instead ofg, N

if that will not result in

ambiguities. It is possible always to chose a necessary type L and encodeequivalently the geometric data with respect to a basis generated by N[

L],

or other conventional hvsplitting.

Definition 1.2 A distinguished connection (dconnection) D = (hD; vD)is a linear connection preserving under parallelism the Whitney (1).

A dconnection is metric compatible ifDg = 0. We can also considerhvsplitting (Nadapted decompositions) of tensors, vectors, differentialforms, called respectively distinguished tensors, dtensors, distinguished vec-tors, dvectors, etc. For instance, a dvector Y = hY + vY.

Definition 1.3 The torsion and curvature dtensors of a metric compatibledconnectionD are defined correspondingly

T(X,Y) := DXY DYX+ [X,Y] and (4)R(X,Y) := DXDY DYDX D[X,Y]. (5)

The Ricci, Ric, and Einstein, E, dtensors ofD are constructed in stan-dard form as for any linear connection, see relevant details in Appendix.4

3Prescribing a nonholonomic distribution L, we compute g and then determine e

as solutions of algebraic equations for any given set g . The term nonholonomic (equiv-alently, anholonomic, or nonintegrable) manifold is related not only to the nonholonomicmechanics but also to the property that the components of a general local basis e satisfythe anholonomy relations [e, e] = ee ee = w

e. For any data N

bi , the non-

trivial anholonomy coefficients wbia = aNbi and w

aji =

aij , where

aij = ej (N

ai ) ei(N

aj )

are used for the coefficients of Nconnection curvature defined as the Neijenhuis tensor.If w = 0, we get holonomic / integrable configurations.4Coefficient formulas for a dconnection = { = (L

ijk ,

vLabk; Cijc ,

vCabc)} can be

derived for a 1form := e

. The values are computed with respect to

e and e of type (3). The coefficients of dtorsion, T = {T}, and dcurvature,

R = {R}, are found respectively from

T:=De = de+e = Te

e, R :=D = d

= R

e

e,

see details in [17]. A Ricci dtensor Ric = {R := R;R = R} splits into h

vcomponents R = {Rij := Rkijk , Ria := R

kika, Rai := R

baib, Rab := R

cabc}. By

definition, the scalar curvature of a dconnection D is sR := gR = R + S = gijRij +

habRab, with R = gijRij and S = h

abRab. The Einstein dtensor is E := Ric 12g

sR.

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Theorem 1.2 For any given g andN, or g g andN N (such valuesare equivalent up to frame transforms if a generated function L is fixed onN), on a nonholonomic manifoldV, we can construct three metric com-patible linear connections determined completely by the metric field. Suchconnections are defined respectively by the conditions:

g = g : g = 0; T = 0, the LeviCivita connection ;D : Dg = 0; hT = 0, vT = 0, the canonical dconnection ;D : Dg = 0; hT = 0, vT = 0, the Cartan dconnection .

Proofs are similar to those for constructing the LeviCivita connection.

Remark 1.1 The dtorsions

T and

T of respective dconnections

D andD

are not zero. Nevertheless, the nontrivial coefficients are completely defined

by g, or g, for a chosen NsplittingN, or L and corresponding N. Suchtorsion fields are induced by nonholonomic distributions. They are different

from the torsion in RiemannCartan, or from the completely anti-symmetrictorsion in string theory, for which we need additional fields and equations.

The LeviCivita connection is not a dconnection because it does notpreserve under parallelism the Nconnection splitting. One holds

Corollary 1.2 There are unique distortion relations

D = +

Z and D = + Z, (6)

where all connections and distortion dtensors Z and Z are completely de-fined byg for a prescribedN.

A proof follows form the computation of coefficients of such values withrespect to Nadapted bases (3). The coefficients of distortion dtensors Zand Z are certain algebraic combinations of the coefficients of, respectively,torsion T and T (see details in [17]).1.2 Canonical almost Khler variables for semiRiemannian

and LagrangeFinsler spaces

For any

Land induced N, we can define a canonical almost complex

structure J when J(ei) = e2+i and J(e2+i) = ei, for J J = I, for unitymatrix I.5 The Neijenhuis tensor field can be computed for any almost com-plex structure J, J(X,Y) := [X,Y] + [JX,JY] J[JX,Y] J[X,JY].On TV, any X = Xe = X

iei + Xaea is a Nadapted dvector.

5In component form, we compute

J = J e e = J

u du = J

e e = e2+i e

i + ei e2+i

=

y i dxi +

xi N2+ji

yj

dyi + N2+ik dxk

.

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Definition 1.4 An almost symplectic structure is defined by a nondegener-

ate 2form = 12(u)e e.In Nadapted form, = 12ij(u)e

i ej + 12ab(u)ea eb.Proposition 1.1 -Definitions: Any data (g,J) define an almost symplec-tic structure, (X,Y) := g (JX,Y) , for any dvectorsX andY.

An almost Hermitian model of a nonholonomic Riemannian manifold(V,g,N) is defined by a tripleHn+n = (V, (, ) := g (J, ) ,J).

A spaceHn+n is almost Khler, denotedKn+n, if and only if d = 0.

Proofs consist from straightforward verifications.

Lemma 1.1 Prescribing a generating function L(x, y) onV, we can modelequivalently this nonholonomic manifold as a canonical almost Khler space.

Proof. For any g = g, N = N and J = J canonically determined by L,we can define (, ) := g

J,

. In Nadapted form,

=1

2(u)du

du = 12

(u)e e (7)

= gij(x, y)en+i dxj = gij (x, y)(dyn+i + Nn+ik dxk) dxj,

where ab = n+i n+j are respectively the coefficients ij. We obtain d =dd = 0 if = d, for := 12

Lyi

dxi. (end proof).

Due to M. Matsumoto [25] (the original results were proven for FinslerCartan spaces and then extended to Lagrange geometry and nonholonomicmanifolds, see references in [20]) we can prove this theorem.

Theorem 1.3 The Cartan dconnection D is a unique almost symplecticdconnection which satisfies the conditions D = 0 and DJ = 0.

Let us denote by Ric, Ric and Ric the Ricci tensors for, respectively,linear connections

,

D and D all defined by the same metric g and corre-

sponding N, or L.Theorem 1.4 An Einstein manifold is defined by a metricg as a solutionof the vacuum gravitational field equations with cosmological constant ,

Ric = g; (8)

or of equivalent equations in nonholonomic canonical variables,

Ric = g and (9)

Z = 0, nonholonomic constraints; (10)

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or, in Cartan type almost Khler variables,

Ric = g and (11)

Z = 0, nonholonomic constraints. (12)

Proof. Details of proofs can be found in Refs. [20, 18].

There is a series of arguments to consider reformulations of gravitationalfield equations in various nonholonomic variables. For instance, Ric = Ric+Zic, where the distorting tensor Zic is computed using the distortion of Dfrom (6). With respect to Nadapted frames, we can decouple the equa-tions (9) and find generic offdiagonal solutions depending on all coordi-nates. Such solutions are with nontrivial torsion determined by the metric

structure g and Nstructure. We can extract certain subvarieties of solu-tions by imposing at the end the nonholonomic constraints (10) resulting inzero torsion. It is not possible to decouple the vacuum Einstein equations(8) with cosmological constant if we work from the very beginning with theLeviCivita connection. This is related to a generic nonlinear character ofthe Einstein equations. Different ansatzes and/or nonintegrable constraints(for instance, before finding solutions, during the process of constructing suchsolutions, or at the end) result in different classes of solutions.

In a similar form, we can prove the equivalence of the system (11) and(12) to (8). The priority of such equations and D is that we can applydirectly certain methods of deformation and Abrane quantization which

were formulated for almost Khler geometries [20, 21].If the equations (11) are formulated on V = T M, we model a theory

of LagrangeFinsler gravity with fundamental generating function L(x, y),or Finsler metric F(x, y), when the dconnection structure is fixed to bethe Cartan one. Such variables can be introduced also in general relativ-ity for a socalled nonholonomic 2 + 2 splitting mimicking a local fiberedstructure. This way we elaborate an unified geometric formalism using non-holonomic distributions both for (modified) gravity theories and LagrangeFinsler geometries. The approach can be developed for noncommutativegeometric models [15], CliffordLie algebroid and LagrangeHamilton struc-tures [26, 27], in deformation quantization [20], nonRiemannian Ricci flows

[12, 13, 14, 16] etc if such theories are formulated in almost Khler variables(with certain modifications for Lorentz signatures, noncommutative general-izations etc).

The theory of nonholonomic Ricci flows was extended to include certainevolution models of geometric mechanics and gravity formulated on prolon-gation Lie algebroids in Ref. [17]. In this paper, we study geometric flowsof almost Khler structures which are canonically determined by generated(semi) Riemannian / Lagrange / Finsler metrics and generating functions.We also provide examples of exact solutions for Ricci soliton equations withLie algebroid symmetries defining effective LagrangeFinsler structures or

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exact solutions in (modified) gravity. A more advanced geometric techniques

has to be elaborated and applied. It should be emphasized that it is not onlyan academic exercise and formal technical generalization to perform certainformal extensions to FinslerRicci flow and gravity theories or Lagrange dy-namics on Lie algebroids. The spacetime topology and most fundamentalgeometric and physical properties of classical and quantum theories seem tobe encoded into nonholonomic geometric evolution scenarios with generalizedLie symmetries.

Here is an outline of the rest of the work. In section 2, we survey thegeometry of Lie algebroids and prolongations endowed with nontrivial Nconnection structure and formulate an approach to the almost Khler ge-ometry on nonolonomic Lie albebroids. The Ricci flow theory for almostsymplectic geometries determined by canonical metric compatible algebroidconnections is studied in section 3. Finally, we provide a series of examplesof almost Khler Ricci soliton solutions, related to (modified) gravity modelsand LagrangeFinsler Lie algebroid mechanics in section 4. The Appendixcontains some necessary and important coefficient type formulas and proofs.

2 Almost Khler Lie Algebroids & Nconnections

In this section, we recall basic definitions on noholonomic Lie algebroidsendowed with Nconnection structure, see [17, 26, 27], and develop the ap-

proach for almost Khler Lie algebroids.

2.1 Distinguished Lie algebroids and prolongations

In Refs. [26, 27], we introduced

Definition 2.1 A Lie distinguished algebroid (dalgebroid) E= (E, , , )over a manifold M is defined by 1) a Nconnection structure, N : T E =hE vE, and 2) a Lie aglebroid structure determined by 2a) a real vectorbundle : E M together with 2b) a Lie bracket , on the spaces ofglobal sections Sec() of map and 2c) an anchor map : E T M, i.e.,a bundle map over identity and constructed such that for the homomorphism

: Sec() X(M) of C

(M)modules X this map satisfies the conditionX ,f Y = fX, Y + (X)(f)Y, X, Y Sec() and f C(M).IfN is integrable (i.e. with trivial Nconnection structure), a Lie d

algebroid is just a Lie algebroid E= (E, , , ). For "non-boldface" con-structions, see details in Refs. [28, 29, 30, 31, 32, 33]. The anchor map isequivalent to a homomorphism between the Lie algebras (Sec(), , ) and(X(M), , ) .6

6Locally, the properties of a Lie dalgebroid E are determined by the functions ia(xk)

and Ceab(xk), where x = {xk} are local coordinates on a chart U on M (or on hV if a non-

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Example 2.1 Nonholonomic Lie algebroids: IfE =T V for a nonolo-

nomic manifoldV = (V,N) with Nsplitting (1), the values like X(M) areconsidered for M hV and sections are modelled on vV. We shall con-struct Ricci soliton configurations, or Einstein manifolds, with Lie algebroidsymmetry in section 4.

Let us extend the concept of prolongation Lie algebroid [30, 31, 32, 33] (inour case) in order to include Nconnections. We consider a Lie dalgebroidE= (E, , , ) and a fibration : P M both defined over the samemanifold M. In general, E,P andTM, or TV, may be enabled with differentNconnection structures. The local coordinates are u = (xi, yA) P when{ea} will be used for a local basis of sections of E. If E = TV, we writeu

= (xi

, yI

). In our constructions, we can consider that P = E. The anchormap : E TM and the tangent map T : TP TM are all defined tobe hvadapted for nonholonomic bundles and/or fibered structures. Thesestructures can be used to construct the subset

TEs P := {(b, v) Ex TxP; (b) = Tp(v);p Px, (p) = x M} (13)

and prove this result:

Theorem 2.1 Definition. The the prolongation TEP := sSTEs P ofa nonholonomicE over is another Lie dalgebroid (the construction (13)can be considered for any set of charts covering such spaces).

Similarly to holonomic configurations, with trivial Nconnection struc-ture, the prolongation Lie dalgebroid TEP is called the (nonholonomic)Etangent bundle to , which is also a nonholonomic vector bundle over P.The corresponding projection E

Pis just onto the first factor, E

P(b, v) = b

being adapted to the Nconnection structures [17]. The elements ofTEP areparameterized by Nadapted triples (p, b, v). We shall use also brief denota-tions (p, b, v) TEP (b, v) TEP if that will not result in ambiguities.The Nadaped anchor : TEP TP is given by maps (p, b, v) = v,holnomic manifold is considered), with (ea) =

ia(x)ei and ea, eb = C

fab(x)ef, satisfying

the following equations iaei

jb

ibei

ja =

jfC

fab and

cycl (a,b,f)

iaiC

fbe + C

dbeC

fad

= 0.

Boldface operators are defined by Ncoefficients similarly to (3). For trivial Nelongatedpartial derivatives and differentials, we can use local coordinate frames when ei i,ea = a, e

i = dxi, ea dya.The exterior differential on E with nonholonomic E can be defined in standard form

with E. We introduce on E the operator d : Sec(k ) Sec(k+1 ), d2 = 0, where

is the antisymmetric product operator. The contributions of a Nconnection can be seenfrom such formulas for a smooth formula f : M R, df(X) = (X)f, for X Sec(),when dxi = iae

a and def = 12Cfabe

a eb. With respect to any section X on M, we

can define the Lie derivative LX = iX d + d iX : Sec(k ) Sec(k ), using the

cohomology operator d and its inverse iX .

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i.e. projection onto the third factor when the hcomponents transforms

in other hcomponents etc. For more special cases, we can define also theprojection onto the second factor (i.e. a morphism of Lie dalgebroids over), T : TEP E, when T(p, b, v) = b. An element (p1, b1, v1) TEPis vertical if T (p1, b1, v1) = b1 = 0, i.e. such elements are of type (p, 0, v)when v is a vertical vector (tangent to P at point p).

To understand consequences of Theorem 2.1 let us consider some localconstructions. In coefficient form, any element of a prolongation Lie dalgebroid TEP can be parameterized z = (p, b, v) TEP, for b = zaea andv = iaz

aei + vAA, for /y

A, can be decomposed z = zaXa + vAVA. Thecouple (Xa, VA) , with vertical VA, defines a local basis of sections ofTEP. Forsuch bases, we can write

Xa =

Xa(p) =

ea((p)), iaei|p

and

VA =

0, A|p,

where partial derivatives and their Nelongations are taken in a point p Sx.It is also possible to elaborate on prolongation Lie dalgebroids a Nadaptedexterior differential calculus. The anchor map (Z) = iaZ

aei + VAA is an

Nelongated operator acting on sections Z with associated decompositionsof type z. The corresponding Lie brackets are

Xa, Xb = CfabXf, Xa, VB = 0, VA, VB = 0.Denoting by

Xa, VB the dual bases to (Xa, VA) , we can elaborate a differ-ential calculus for Nadapted differential forms using

dxi = iaXa, for dXf = 1

2CfabXa Xb, and dyA = VA, for dVA = 0.

(14)Nconnections can b e introduced on TEP similarly to (1).

Definition 2.2 On a prolongation Lie dalgebroid, a Nconnection is de-fined by a hvsplitting,

N: TEP = hTEP vTEP. (15)We can consider N : TEPTEP, with N2 = id, as a nonholonomic

vector bundle, and Lie dalgebroid, morphism defining an almost productstructure on P : TP P, for a smooth map on TP\{0}, were {0} denotesthe set of null sections. Any Nconnection

Ninduces h- and vprojectors

for every element z = (p, b, v) TEP, when h(z) = hz and v(z) = vz, forh = 12(id +N) and v = 12(id N). The respective h- and vsubspaces arehTEP = ker(id N) and vTEP = ker(id +N).

Let us analyze some local constructions related to Nconnection struc-tures for prolongation Lie dalgebroids. Locally, Nconnections are deter-mined respectively by their coefficients N = {NAi } and N= {NAa }, when

N =NAi (xk, yB)dxi A and N=NAa Xa VA. (16)

Such structures on TP and TEP are compatible ifNAa = NAi ia. Using NAa ,we can generate sections a := Xa NAa VA as a local basis of hTEP.

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Corollary 2.1 Any Nconnection NAa on TEP determines a Nadaptedframe structure

e := {a = Xa NCa VC, VA}, (17)and its dual

e := {Xa, B = VB +NBc Xc}. (18)Proof. In above formulas, the overlined small Greek indices split =

(a, A) if an arbitrary vector bundles P is considered, or = (a, b) for P = E.A proof follows from an explicit construction such Nadapted frames. Thenthe formulas can be considered for arbitrary frames of references.

The Nadapted bases (17) satisfy the relations e e e e = We,

with nontrivial anholonomy coefficients W

= {Cfab, Cab, BNCc }. The cor-

responding generalized Lie brackets are

a, b = Cfabf + CabVC, a, VB = (BNCa )VC, VA, VB = 0. (19)

Definition 2.3 The curvature of NconnectionNAa is by definition the Neig-enhuis tensor hN of the operator h,

hN(, ) = h, h h h, h , h + h2 h, h = 12

CabXa Xb VC,

whereCab = b

NCa

a

NCb + C

fab

NCf . (20)

It should be noted that above formulas for TEP mimic (on sections of Efor P = E) the geometry of tangent bundles and/or nonholonomic manifoldsof even dimension, endowed with Nconnection structure. On applicationsin modern classical and quantum gravity, with various modifications, andnonholonomic Ricci flow theory, see Refs. [18, 12, 15]. IfP = E, we modelnonholonomic vector bundle and generalized Riemann geometries on sectionsofTEE.

2.2 Canonical structures on Lie dalgebroids

Almost Khler Lie algebroid geometries can be modelled on prolonga-tion of Lie dalgebroids following the method outlined in section 1.2 forvector/tangent bundles and nonholonomic manifolds.

2.2.1 dconnections and dmetrics on TEPDefinition 2.4 A distinguished connection, dconnection, D = (hD, vD),onTEP is a linear connection preserving under parallelism the splitting (15).

The formulas for distinguished torsion (4) and curature (5) can be gen-eralized for prolongation Lie dalgebroids.

12


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14/40

Proof. See proof in section A.2.

For trivial algebroid structures, D D following the conditions ofTheorem 1.2.

Remark 2.1 There is a canonical distortion relation D = + Z,when both linear connections D and (the last one is the LeviCivitaconnection) and the distorting dtensor Zare defined by the same data(N,g). This is a generalization on TEP of the the formula (6). Inexplicit form, the Nadapted coefficients of such values are computed

following formulas (A.5) and (A.6).

We note that h

T = 0 for

D onTM but h

T = 0 for

D onTEP. The

formulas for Labf in (A.5) contain additional terms with Cabf whichresults in nontrivial Tabf = Cabf and additional terms in NadaptedcoefficientsRaebf andR

ABbf of curvature (A.2).

In order to generate exact solutions, it may be more convenient to workwith an auxiliary dconnection c D := + c Z for which h cT = 0 andv cT = 0. In Nadapted form this results in cTabf = 0 and cTABC = 0 but,in general, R = c R and T = cT. The nontrivial Lie dalgebroid structure isencoded in c Zvia structure functions ia and Cabf and Nelongated frames(17) and (18). The Nadapted coefficients of c D are computed cLabf =12g

ae (fgbe + bgf e

egbf) and

c

LABf = L

ABf,

c

BC = B

C,

c

BABC = B

ABC

are those from (A.5).

2.3 Almost symplectic geometric data

2.3.1 Semispray configurations and Nconnections

We show how a canonical almost symplectic structure can be generatedon TEP by any regular effective Lagrange function L.

Lemma 2.1 Prescribing any (effective) Lagrangian L, we can construct acanonical Nconnection qN := LiqS defined by a semispray q = yaXa +qA

VA and Lie derivative

Liq acting on any X

Sec(TE) following formula

qN(X) = q,SX + Sq, X .Proof. We use the semispray formula Sq = with the operators S

and from (A.9) and computeqN(Xa) = q, S(Xa) + Sq, Xa = Xa + (aqb + yfCbfa)Vb.

For qN(Va) = Va and qN(Xa) = Xa 2 qNfa (x, y)Vf, we define theNconnection coefficients Nf = 12(aqf + ybCfba), see formulas (16).

We can formulate on Lie daglebroids the analog of Theorem 1.1.

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Theorem 2.3 Any effective regular LagrangianL C(E) defines a canon-ical Nconnection on prolongation Lie algebroid T

E

E,

N= { Nfa = 12 (af + ybCfba)}, (21)determined by semispray configurations encoding the solutions of the EulerLagrange equations (A.12).

Proof. It is a straightforward consequence of above Lemma and (A.11).To generate Nconnections, we can use sections L = y

aXa + aVa, withqe = e(xi, yb) (A.11). Let us consider the value S = yaia

Lxi

2GA yA

,

where GA is associated to the "nonlinear" geodesic equations for sections,dyA

d + 2GA

= 0, depending on real parameter , and define a Nconnectionstructure NFa =

GF

ya of type (16). For P = E, such sections can be relatedto the integral curves of the EulerLagrange equations (A.12) if we chose

the sections GF Gf and f(xi, yb) in such forms that Nfa = 12(af +ybCfba) =

Gf

ya . The constructions can be performed on any chart coveringsuch spaces, i.e. we can prove that the coefficients (21) define a Nconnectionstructure (15).

Proposition 2.3 Any metric structure on TEP can be represented in Nadapted form as a dmetric g=hgvg constructed as a formal Sasaki liftdetermined by an effective regular generating function

L.

Proof. Sasaki lifts are used for extending certain metric structures froma base manifold, for instance, to the total space of a tangent bundle, seedetails in [24]. For vector bundles, the formulas (2) provide a typical ex-ample of such a construction using canonical Nconnection structure N.The method can be generalized for prolongation Lie dalgebroids. At thefirst step, we use the canonical Nconnection N= { Nfa } (21) and constructNadapted frames of type (17) and (18), respectively,

e := {a = Xa Nfa Vf, Vb} and e := {Xa, b = Vb + NbfXf}. (22)Then (the second step) we define a canonical dmetric

g := ge e = gab Xa Xb + gab a b (23)using the Hessian (A.10). Considering an arbitrary dmetric g = {g

}

(A.3), we can find a regular L and certain frame transforms e = e ewhen g

= ee

g. We can work equivalently both with g and/or g

if a nonholonomic distribution L is prescribed and the vielbein coefficientse

are defined as solutions of the corresponding algebraic quadratic system

of equations for some chosen data g

and g.

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2.3.2 RiemannLagrange almost symplectic structures

Let us consider canonical data defined respectively by Nelongated framese = (ea = a, VA) (22), Nconnection N (21) and dmetric g = g (23).Proposition 2.4 Definition. For any regular effective Lagrange structureL, we can define a canonical almost complex structure on TEE following

formulas J(ea) = Vm+a and J(Vm+a) = ea, when J J= IProof. It follows from an explicit construction of a dtensor field

J= Je e = Vm+a Xa + ea a.

Using vielbeins e

and their duals e

, we can compute the coefficients ofJ with respect to any e and e on TEE, when J = ee J .In general, we can define an almost complex structure J on TEE for

an arbitrary Nconnection N (15) by using Nadapted bases (17) and (18)which are not necessarily induced by an effective Lagrange function L. Thisallows us to generate almost Hermitian models and not almost Khler ones.

Definition 2.6 The Nijenhuis tensor field for any almost complex structureJ on TEE determined by a Nconnection N (equivalently, the curvature ofNconnection N) is by definition

J(x, y) := [x, y]+[Jx, Jy]J[Jx, y]J[x, Jy], (24)for any sections x, y of TEE.

We can introduce an arbitrary almost symplectic structure as a 2formon a prolongation Lie dalgebroid.

Definition 2.7 An almost symplectic structure onTEP is defined by a non-degenerate 2form = 12(x

i, yB)e e.Using frame transforms, we can prove

Proposition 2.5 For any on TE

P when h(x,y) := (hx,hy), v(x,y) :=(vx,vy), there is a unique Nconnection N= {NAa } (16) when

= (hx,vy) = 0 and = h + v. (25)

Proof. In Nadapted form,

=1

2ab(x

i, yC)Xa Xb + 12

AB(xi, yC)A B, (26)

where the first term is for h and the second term is v, i.e. we get thesecond formula in (25).

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Definition 2.8 a) An almost Hermitian model of a prolongation Liedalgebroid T

E

E equipped with a Nconnection structure N is definedby a triple HEE = (TEE, , J), where (x,y):= g(Jx,y).

b) A Hermitian prolongation Lie dalgebroid HEE is almost Khler,denoted KEE, if and only if d = 0.

For effective regular Lagrange configurations, we can formulate:

Theorem 2.4 Having chosen a generating function L, we can model equiv-alently a prolongation Lie dalgebroid TEE as an almost Khler geometry,i.e. HEE = KEE.

Proof. For the canonical geometric data (g = g, N, J), we define thesymplectic form (x,y):= g(Jx,y) for any sections x, y of TEE. In localNadapted form, = gab

a Xb. Let us consider the form := 12 Lym+aXa.Using Proposition 2.5 and Nconnection N (21), we prove that = d,which means that d = dd = 0. The constructions can be redefined inarbitrary frames,

= ee

, for a 2form of type (26),

=1

2ab(x

i, yC)Xa Xb + 12

AB(xi, yC)A B. (27)

2.3.3 Nadapted symplectic connections

Let us consider how dconnection structures can be defined on HEEand/or KEE.Definition 2.9 An almost symplectic dconnection D for a modelHEE ofTEE, or (equivalently) a dconnection compatible with an almost symplecticstructure , is defined such that this linear connection is Nadapted, i.e. adconnection, and Dx = 0, for any section x of TEE.Lemma 2.2 We can always fix a dconnection D on TEE and then con-struct an almost symplectic

D.

Proof. Let us consider a in Nadapted form (26). Introducing

D = {h D = ( hDa, vDa); v D = ( hDA, vDA)}= {

= ( Labf,

LABf ;BabC,

BABC)},we can verify that

D =

h D = ( hDa, vDa); v D = ( hDA, vDA)

= {

= ( Labf,LABf ;

BabC,BABC)}, with

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Labf =Labf +

1

2

ae hD

fbe,LABf =

LABf +1

2

AE vD

fEB ,(28)

BabC =BabC +

1

2ae hDCbe, BABC = BABC +

1

2AE vDCEB ,

satisfies the conditions hDabe = 0, vDaAB = 0, hDAbe = 0, vDAAB = 0,which are h- and vprojections of Dx = 0 from Definition 4.1.

Let us introduce the operators

abcd =1

2(ac

bd cdab) and ABCD =

1

2(AC

BD CD AB). (29)

Theorem 2.5 The set of dconnections s

= ( sLabf,sLABf;

sBabC,sBABC)

labeled by an abstract left index "s", compatible with an almost symplecticstructure (26), are parameterized by

sLabc =Labc +

dabe Y

edc,

sLABc =LABc +

CABE Y

ECc , (30)

sBabC =BabC +

eabf Y

feC,

sBABC =BABC +

EABF Y

FEC,

where the Nadapted coefficients are given by (28), theoperators are those

from (29) andY

= (Yedc,YECc ,Y

feC,Y

FEC) are arbitrary dtensor fields.

Proof. It follows from straightforward Nadapted computations.

Remark 2.2 The Lie algebroid structure functionsCdbf in (A.5) can be con-

sidered as an example of dtensor fieldsY in (30). On TEE, we canwork as onTM but for differnet classes of nonholonomic distributions forsections. The dconnections D, D can be constructed for correspondinglydefined Nconnection structures N, N when the main geometric propertiesare similar to some geometric models with D, D and certain N, N. Thenonholonomic frame structures on Lie dalgebroids are different from thoseon nonholonomic tangent bundles because in the first case the vierbein fieldsencode the ancor and Lie type structure functions.

We can select a subclass of metric and/or almost symplectic compatibledconnections on

TEE which are completely defined by g and prescribed by

an effective Lagrange structure L(x, y).Theorem 2.6 On TEE, there is a unique normal dconnection nD =

{h nD = ( nhDa = Da, nv Da = Da); v nD = ( nhDA = DA, nv DA = DA)}= { n

= ( nLabf =

Labf, nLA=m+aB=m+b f = Labf;nBa=Amb=Bm C =

BABC, nBABC = BABC)}, (31)which is metric compatible, Dagbc = 0 and DAgBC = 0, and completelydefined by g = g and a fixed L(x, y).

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Proof. We provide a proof constructing such a normal dconnection in

explicit form an example when nD = D generalizes the concept of Cartandconnection from Theorem 1.3. Such a dconnection is completely defined

by couples of h and vcomponents D = ( Da, DA), i.e. = (Labf, BABC).Let us chose

Labf = 12gae (fgbe + bgfe egbf), BABC = 12gAD (VCgBD + VBgCD VDgBC),(32)

where the Nelongated derivatives are taken in the form (22) and gab =gA=m+a B=m+b are generated by canonical values using the Hessian (A.10)and (23) induced by a regular L(x, y), we can prove that this dconnectionis unique and satisfies the conditions of the theorem. Via frame transforms,

we can consider any metric structure g g. Finally, we formulate this very important for our purposes result:

Theorem 2.7 The normal dconnection nD = D defines a unique almostsymplectic dconnection, D D, see Definition 2.9, which is Nadaptedand compatible to the canonical almost symplecti form (27), i.e. D=0and Tacb = TACB = 0, see torsion coefficients (A.16).

Proof. Using the coefficients (32), we can check that such a normaldconnection satisfies the conditions of this theorem.

Conclusion 2.1 Prescribing an effective generating function L on TE

E,we can transform this prolongation Lie dalgebroid into a canonical almostKhler one, KEE. It is possible to work equivalently with any geometric data

g,N, D = + Z g,L, N, D (, ):= g( J ,), D .The Lie algebroid structure functions (ia, C

fab) are encoded into nonholo-

nomic distributions on TEE determining such equivalent prolongation Liedalgebroid configurations.

2.4 Almost Khler Einstein and Lagrange Lie dalgebroids

We can formulate analogs of Einstein equations for different classes ofdconnections on prolongation Lie dalgebroids and almost Khler models.

Corollary 2.2 Definition The Ricci tensor of a dconnection D on aTEP endowed with dmetric structure g is defined following formula Ric ={R := R}, see the coefficients (A.2) for Riemannian dtensor R ={R

}, and characterized by Nadapted coefficients

R = {Rab := Rcabc, RaA := Rc acA, RAa := RBAaB, RAB := RCABC}.(33)

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Proof. The formulas for hvcomponents (33) are respective contrac-

tions of the coefficients (A.2). The scalar curvature sR of D is by definition

sR := gR = gabRab + g

ABRAB . (34)

Using (33) and (34), we compute the Einstein dtensor E := R 12g

sR of D. Such a tensor can be used for modeling effective gravitytheories on sections ofT EP with nonholonomic frame structure [26, 17, 27].

The conditions of Theorem 1.3 can be reformulated and proven on TEE,or KEE, respectively, for

D and

D =

D. Prescribing the geometric data

(

L; ia, C

f

ab

), any solution of the gravitational field equations (9) and (10) canbe transformed into solutions of

Ric = g, with possible nonholonomic constraints Z= 0. (35)Such PDEs can be rewritten via nonholonomic frame transforms and de-formations into almost Khler variables,

Ric = g, with possible nonholonomic constraints Z= 0. (36)In Nadapted coefficients, the formulas (A.19) present an example of distor-tion relations which became trivial if

Z= 0.

The solutions of above equations define are stationary configurations inthe theory of Ricci flows and define Ricci Lie dalgebroid solitons. In general,we can consider solutions with nontrivial dtorsion when Z and/or Zarenot zero. We shall provide such examples in section 4.

3 Almost Khler Ricci Lie Algebroid Evolution

3.1 Motivation

A model of theory of LagrangeRicci flow evolution on TEE was elab-orated in Section 4 of Ref. [17] following an approach to geometrization

of regular Lagrange mechanics (due to the KernMatsumoto [23, 25]) andanalogous gravity on (prolongation) Lie dalgebroids [26, 27]. Similar con-structions were considered for noncommutative and/or fractional derivativemodifications, see explicit examples and details in [12, 13, 14, 15, 16].

The goal of this section is to prove that Nadapted Ricci flow theories foralmost Khler models of prolongation Lie algebroids, KEE, can be formu-lated as models of generalized gradient nonholonomic flows. We extend theGrisha Perelmans geometric thermodynamic functional approach [4, 5, 6]and show how modified R. Hamilton type equations [1, 2] can be derived forthe almost Khler evolution of Lie dalgebroids.

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Symplectic and almost symplectic geometric flows have been studied in

modern Ricci flow theory, see a series of examples and reviews of results in[8, 9, 10, 11]. Our approach is very different from those with "pure" complexand/or symplectic forms and connections. Our main idea is to elaborateand study such almost symplectic models which are equivalent to certainmetric compatible theories induced in canonical form following welldefinedgeometric/ physical principles and completely defined by the metric fieldsand nonholonomic distributions. The constructions can be generalized forgeometric flows of nonlinear systems with generalized symmetries and, forinstance, effective mechanical and/or analogous (modified) gravity theorieson nonholonomic Lie algebroids and vector/tangent bundle spaces.

It should be noted that for realistic physical theories we have to consider

semiRiemannian metrics7

when a theory of Ricci flow evolution for Lorentzmanifolds in four (and higher) dimensions has to be formulated. We do notconcern such sophisticate problems in this work and restrict our methodsand classes of solutions only to fixed configurations for Ricci evolution ofalmost Khler Lie dalgebroids which are defined by modified/generalizedEinstein equations. Generic offdiagonal solutions of such equations can beconstructed both with Riemannian and/or Lorentz signatures and charac-terized by generalized Lie symmetries.

3.2 Perelmans functionals in almost Khler variables

There are both conceptual and technical difficulties which do not allow usto formulate a generalized Ricci flow theory for nonRiemannian geometrieswith independent metric and linear connection structures, or their almostsymplectic analogs. Nevertheless, unified geometric evolution theories canbe constructed for certain classes of nonholonomic manifolds and LagrangeFinsler spaces [12, 13, 14] both with metric compatible and noncompatibleNadapted connections if the fundamental geometric objects are determinedin unique forms by distortion relations of type (6). In such geometric evolu-tion models, all involved linear connections and distorting tensors are deter-mined by the same metric (almost symplectic) structure. We can begin with"standard" Ricci flows of Riemannian metrics and LeviCivita connections

modelled and consider further nonholonomic deformations into, for instance,(modified) Einstein, LagrangeFinsler, almost Khler and other type geome-tries on tangent bundle and/or Lie algebroid geometries.

Remark 3.1 Various proofs of theorems for almost Khler models on pro-longation Lie dalgebroids onTEE and/orKEE can be obtained by nonholo-nomic deformations/ transforms and Nadapting of geometric constructions

7physicists use the term pseudoRiemannian; to work with a pseudo/ semiEuclideansignature of type (,+,+,+) is necessary if we wont to elaborate theories with finite speedof light, causality etc

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for the standard LeviCivita connection , wheng g,L, N, D (, ):= g( J ,), D = + Z .

The functions determining nonholonomic distributions, Nconnection coeffi-cients and Lie algebroid structure functions are considered to be of the samesmooth class as the coefficients ofg and .

The theory of LagrangeRicci flows on TEE is formulated for evolvingnonholonomic dynamical systems on the space of equivalent geometric dataL : g, , when the functionals Fand Ware postulated to be of Lyapunovtype, see below formulas (40) and (41). Ricci solitonic configurations aredefined as fixed on points of the corresponding dynamical systems. The

"stationary" variational conditions depend on what type of the Ricci tensorwe use, for instance, that one for the connections or D. We can elaborateNadapted almost Khler scenarios if the Perelmans functionals are redefined in terms of geometric data (g, D) and the derived flow equations areconsidered in Nadapted variables. Both approaches are equivalent if thedistortion relations are considered for the same family of metrics, g() = g()correspondingly computed for a set L().Lemma 3.1 For the scalar curvature and Ricci tensor determined by thedistortion relation

D =

D = +

Z, (37)

the Perelmans functionals are defined equivalently in almost Khler canoni-cal Nadapted variables,

F(g, D, f) = V

( s R+ |h Df|2 + | v Df|)2)ef dv, (38)W(g, D, f , ) =

V

[( s R+ |h Df| + |v Df|)2 + f 2m]dv, (39)where the new scaling function f is intorduced for

V dv = 1 with volume

element dv, = (4)m ef and > 0, where = h = v for a coupleof possible h and vflows parameters, = ( h, v).

Proof. On TE

P, evolution models can be formulated as in standardtheory for Riemann metrics [1, 2, 4, 5, 6] but for a family of geometricdata

g(), () induced by a family of regular L() C(P) with a flow

parameter [, ] R, when > 0 is taken sufficiently small. For P = E,we can postulate on the space of Sec(E), for : E M, dim E = n+ m 3and dim M = n 2, the (Perelmans) functionals

F(g, , f , ) =V

R +

f2 ef dv, (40)W(g, , f , ) =

V

R +

f2 + f 2m) dv, (41)22


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where the volume form dv and scalar curvature R of are computed for setsoffdiagonal metrics g (A.4) with Euclidean singature. The integrationis taken over compact regions V TEE, dim V = 2m, corresponding tosections over a U M. We can fix Vdv = 1, with = (4)m ef anda real parameter > 0. We introduce a new function f instead of f. Thescalar functions are redefined in such a form that the subintegral formula(40) under the distortion of the Ricci tensor (A.19) is rewritten in terms of

geometric objects derived for the canonical dconnection, ( R+f2)ef =

( s R + | Df|2)ef + . We obtain the Nadapted functional (38). For thesecond functional (41), we rescale and write

( R +

f

)2 + f

2m

= [( s

R+

|h

Df

|+

|v

Df

|)2 + f

2m] +

1,

for some and 1 for which V dv = 0 and V 1dv = 0. Finally, we getthe formula (39).

In this section, we shall only sketch the key points for proofs of theoremswhen the geometric constructions are straightforward consequences of thosepresented for the LeviCivita connection in [4, 5, 6, 7, 8, 9] and extended tononholonomic configurations in [12, 13, 14, 15, 16, 17]. For our models, weconsider operators which up to frame transforms are defined by L via onTEE. Following Remark 3.1 for distortions completely defined by g, we canstudy canonical Cartan and almost Khler Ricci flows on prolongation Liealgebroids as nonholonomic deformations of the Riemannian evolution.

Using (,):= g(J ,), we can define the canonical (almost symplectic)Laplacian operator, := D D, and (the LeviCivita) Laplace operator, =, and consider a parameter (), / = 1. For simplicity, we shallnot include normalized terms or values of type

V1dv = 0 if those values

can be generated, or transformed into gradient type ones, via nonholonomicdeformations. Inverting the distortion relations (37), we can compute

= + Z, Z = Z Z [ D( Z) + Z D]; (42)R =

R + Zic, R = s R+ gZic = s R+ sZ,s

Z =

g

Zic = h

Z+ v

Z, h

Z =

gab

Zicab, v

Z =

gAB

ZicAB ;

sR = hR + vR, hR := gab Rab, vR = gABRAB,where the terms with left up label "Z" are determined by Z(for instance,Zic are components of respective deformations of the Ricci dtensor).For convenience, the capital indices A,B,C... are used for distinguishingvcomponents even the prolongation Lie algebroid is constructed for P = E.

3.3 Nadapted almost symplectic evolution equations

Let us consider the symmetrization and antisymmetrization operators,for instance, R() :=

12(R +R ) and R[] :=

12(R R ). Using

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deformations of type (42) of corresponding geometric values in the proof, for

instance, given in for Proposition 1.5.3 of [7], we obtain

Theorem 3.1 a) The canonical Nadapted Ricci flows for the Cartan dconnection D preserving a symmetric metric structure g on TEE can becharacterized by this system of geometric flow equations:

gab

= (Rab + Zicab), gAB

= (RAB + ZicAB), (43)R aA = ZicaA, R Aa = ZicAa, (44)

f

= (

+ Z

)

f +

D

Zf2 s

R s

Z, (45)

and the property that

F(g, D, f) =

V

[|Rab + Zicab + ( Da Za)( Db Zb) f|2 +|RAB + ZicAB + ( DA ZA)( DB ZB) f|2]efdv,

V

efdv = const.

b) In almost symplectic variables with = gaba Xb and almost Khler

dalgebroids KEE, and for redefined scaling function f , up to normalizingterms, the h- and vevolution equations are written in equivalent form

ab

= R[ab], AB = R[AB]. (46)For different classes of distortions of type (42), we can redefine the scaling

functions from above Lemma and write the evolution equations (43) in theform (46) for symplectic variables with (27). On TEE, the correspondingsystem of Ricci flow evolution equations can be written for D,

gab

= 2Rab , gAB

= 2RAB , (47)

R aA = 0, R Aa = 0,

f

=

f + Df2

s

R,which can be derived from the functional F(g, D, f) = V( sR + |Df|2)e

f dv. We note that the conditions of type RA = 0 and RA = 0 must beimposed in order to model Nadapted evolution scenarios only with symmet-ric metrics. In general, a nonholonomically constrained evolution can resultin nonsymmetric metrics (see examples in [36]).

Corollary 3.1 The geometric almost Khler dalgebroid evolution definedin Theorem 3.1 is characterized by corresponding flows (for all time

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[0, 0)) of Nadapted frames, e() = e

(, xi, yC) , which up to frame/

coordinate transforms are defined by the coefficients

e

(, xi, yC) =

e a

a (, xi, yC) NBb (, x

i, yC) e a

B (, xi, yC)

0 e A

A (, xi, yC)

,

e(, xi, yC) =

eaa =

aa e

bi = NBb (, xi, yC) ba

eaA = 0 eA

A = AA

,

with gab() = ea

a (, xi, yC) e b

b (, xi, yC)ab and

gAB() = eA

A (, xi, yC) e B

B (, xi, yC)AB , where ab = diag[1,..., 1] and

AB = diag[1, ..., 1] in order to fix a Riemannian signature of g[0](x

i, yC),

is given by equations

e

= g R e if we prescribe that thegeometric constructions are derived by the Cartan dconnection.

The proof of this Corollary for KEE is similar to those presented in Nadapted forms for nonholonomic Ricci flows and/or FinslerRicci evolution,or on TEE, see [12, 13, 14, 15, 17]. All constructions depend on the type ofdconnection we chose for our considerations.

3.4 Functionals for entropy and thermodynamics KEEFor three dimensional Ricci flows of Riemannian metrics, the value W

(41) was introduced by G. Perelman [4] as a "minus entropy" functional.

We can consider that W(39) has a similar interpretation but in almost sym-plectic variables and on prolongation Lie dalgebroids. The main equationsstated by Theorem 3.1 for F(38) can be proven in equivalent form.Theorem 3.2 The Ricci flow evolution equations with symmetric metricsand respective almost symplectic forms on TEE and, correspondingly, KEE,see (43), (44) and (46), and functions f() and () being solutions of

f

= ( + Z)f + ( Da Za)f2 s R + 2m

,

= 1,

can be derived for a functional Wsatisfying the condition

W(g(),f(), ()) = 2

V

[|R Zic + ( D Z)( D Z)f 1

2g|2](4)mefdv,

forVe

fdv = const. Such a functional is Nadapted and nondecreasing ifit is both h and vnondecreasing.

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Proof. For the LeviCivita connection on TEE, the proof is similar tothat in Proposition 1.5.8 in [7] containing the details of the original resultfrom [4]. Using Nadapted deformations, the geometric constructions areperformed in almost symplectic variables on KEE.

Let us remember some main concepts from statistical thermodynamics.It is considered a partition function Z =

exp(E)d(E) for a canonical

ansamble at temperature 1. Such a temperature is defined by the mea-sure determined by the density of states (E). We can provide a statisticalanalogy computing respective thermodynamical values. In standard form,there are introduced E := log Z/, the entropy S := E + log Zand the fluctuation :=

(E E)2

= 2 log Z/2. The original idea of

G. Perelman was to use such values for characterizing Ricci flows of Rieman-nian metrics [4]. The constructions can be elaborated in Nadapted formfor geometric flows subjected to nonintegrable constraints on various spacesendowed with nonholonomic distributions of commutative and noncommu-tative type, Lie algebroids etc [12, 13, 14, 15, 16, 17].

Theorem 3.3 The Nadapted metric compatible (with symmetric metrics)Ricci on TEE are characterized by a) canonical thermodynamic valuesE = 2

V( sR+ |Df|2 m

) dv,S = V[( sR+ |Df|2) + f 2m] dv, = 2 4

V[|R Zic + ( D Z)(D Z)f 12g|2] dv

b) and/or by effective Lagrange and/or almost Khler Ricci flows

E

= 2V

( s R+ | Df|2 m

) dv,

S = V

[( s R+ | Df|2) + f 2m] dv, = 2

4 V[|R + D D f1

2g|2

] dv,

where all values are constructed equivalently in Cartan and/or almost sym-plectic variables on KEE.

Proof. Similar proofs in coordinate and/or Nadapted forms are given in[7, 12, 13, 14, 15, 16, 17]. We have to use the corresponding partition function

Z = exp

V[f + m] dv

for computations on KEE. The formulas inthe conditions of Theorem depend on the type of dconnection, D,

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or D, which is chosen for nonholonomic deformations. Corresponding

re-scaling f f , or f , and , or , have to be considered. Finally, we note that Ricci flows with different dconnections are charac-

terized by different thermodynamical values and stationary configurations.

4 Ricci Solitons with Lie Algebroid Symmetries

The problem of constructing distinguished metric and connection struc-tures encoding generalized symmetries and describing evolution processesand stationary/fixed p oints is important in geometry and physics. The socalled Ricci solitons and, in particular, (modified) Einstein spaces, or ef-

fective LagrangeFinsler geometries are candidates for constructing exactsolutions. We refer the readers to [37, 38, 7] and references therein for de-tails on HamiltonPerelman evolution and Ricci solitons with Riemannianmetrics. Recently, some models of Ricci solitons in the semiRiemannian set-ting have been elaborated with special attention to Lorentzian Lie groups,algebraic classifications and homogeneous metrics, surface projections etc,see [39, 40, 42, 41, 43, 44].

In this section, we shall construct in explicit form some examples ofexact solutions for Ricci soliton Lie dalgebroid configurations. The fistclass of models describes generalized Einstein spaces with nonholonomic (forinstance, almost symplectic) variables and the second one is determined by

LagrangeFinsler generating functions.

4.1 Preliminaries on Lie dalgebroid solitons

Lie dalgebroid Ricci solitons can be viewed as fixed points of generalizedRicci flows with a functional W (39) satisfying the conditions of Theorem3.2. Such nonholonomically constrained dynamical systems correspond toselfsimilar solutions describing Nadapted geometric evolution models.

Definition 4.1 The geometric data [g g,L, N, D] [(, ):= g( J ,),

D = +

Z] for a complete Riemannian metric g on a smooth TEE and

corresponding

KEE define a gradient almost KhlerRicci dalgebroid soliton

if there exists a smooth potential function on (xi, yC) such that

R + D D = g . (48)Using the almost symplectic form (27), these equations can be written equiv-alently in the form

R [] + D[ D] = .There are three types of such Ricci solitons determined by = const : steadyones, for = 0; shrinking, for > 0; and expanding, for < 0.

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The above classification is important because shrinking solutions for the

Riemannian LeviCivita solitons helps us to understand the asymptotic be-haviour of ancient solutions of Ricci flows (see, for instance, Proposition 11.2in [4] and/or Theorem 6.2.1 in [7]). In general, complete gradient shrinkingRicci solitions describe possible Type I singularity models in the Ricci flowtheory. If = const, the equations (48) transform into the distorted Einsteinequations (36) but for Ricci solitonic configurations.

Proposition 4.1 Let (g g,L, N, D; ) be a complete shrinking solitonon TEE and/ or KEE. Using nonholonomic frame deformations, we canconstruct a redefined potential function (xi, yC), for g g, when (48) areequivalent to

R + DD = g . (49)Proof. Using Conclusion 2.1 and contracting indices in (48), we obtain

that s R + | D|2 = const. Distortion relations of type D = D+ Zallows us to compute sR + sZ + |( D + Z)|2 = const, which canbe rewritten as sR + | D|2 = const for certain nonlinear transform . In general, the systems (49) and (48) have different solutions. Nev-ertheless, conditions of type Z= 0 and/or Z= 0 result in the LeviCivitaconfigurations and equivalent classes of solutions.

4.2 Generalized Einstein equations encoding Lie dalgebroids

We can construct very general classes of offdiagonal solutions of ( 49) ifwe impose the condition that in some Nadapted frames

D = e = = const, (50)i.e. a = Xa NCa C = 0 and VA = A.

The information from potential functions is encoded into the data for Nconnection structure with coefficients NCa .

For simplicity, we shall consider in this section nonholonomic distri-butions on a nonholonomic E = P with 2 + 2 splitting when a,b,... =1, 2; i, j,... = 1, 2 and A,B,... = 3, 4. The local coordinates are parame-terized in the form u = (xi, ya) = (x1, x2, y3, y4). We study nonholonomicdeformations of a dmetric g on E into a target metric g (A.3) on TEE,g g, which results in solutions of the Ricci solitonic equations (49) and(50). The prime metric is parameterized

g = g(u)e e = gi(x)dxi dxi + ha(x, y)ea ea,

for e = (dxi,ea = dya + Nai (u)dxi), (51)

e = (ei = /ya Nbi (u)/yb, ea = /ya).

For physical applications, we can consider that the coefficients of such met-rics are with two Killing vector symmetries and that in certain systems of

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coordinates it can be diagonalized8. In general, we can consider arbitrary

(semi) Riemannian metrics. The target Lie algebroid dmetrics are chosen

g = ge e = ga Xa Xa + gA A A (52)

= a(xk)gaXa Xa + A(xk, y3)hAA A,

where we shall construct exact solutions with Killing symmetry on /y4

(nonKilling configurations request a more advanced geometric techniques).Let us denote by h := 3 and N3a = wa(xk, y3), N4a = na(xk, y3).Proposition 4.2 The nontrivial components of the Ricci soliton dalgebroidequations (49) and (50), with respect to Nadapted bases (17), (18) and for

coordinate transforms when a Xa and VA = A for a metric (52, are R11 = R22 = 12g1g2 [X1(X1g2) X1g1 X1g22g1 (X1g2)

2

2g2(53)

+X2(X2g1) X2g1 X2g22g2

(X2g1)2

2g1] = ,

R33 = R44 = 12h3h4 [h4 (h4)

2

2h4 h

3h

4

2h3] = , (54)

R3a = wa2h4

[h4 (h4)

2

2h4 h

3h

4

2h3] +

h44h4

(Xah3

h3+

Xah4h4

) Xah4

2h4= 0, (55)

R4a =

h4

2h3

na + (h4

h3

h3

3

2

h4)na

2h3

= 0; (56)

for the equations for the potential function

Xa wa3 na4 = 0 and VA = A,when the torsionless (LeviCivita, LC) conditions Z= 0 transform into

wa = (Xa wa3) ln

|h3|, (Xa wa3) ln

|h4| = 0, (57)Xbwa = Xawb, na = 0, anb = bna.

Proof. It follows from straightforward computations of the Ricci d

tensor on TEE

using formulas (A.5), (A.2) and (35). The local frames areredefined in the form a = ea

aXa in order to include Lie algebroid anchorstructure functions and commutation relations of type (14) and (19). Fornonholonomic 2+2+2+...+ decompositions, this can be performed by localframe/coordinate transforms. Details for calculus on TV are provided in[18, 17] and references therein. We can work similarly both on TV and TEEbut with different Nadapted nonholonomic frames and Nelongated partialderivatives and differentials. The system of equations (53) - (57) possessan important decoupling property. For instance, the equation (53) is a 2d

8this include the bulk of physically important exact solutions of Einstein equations

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version of Laplace/ dAlambert equation (it depends on signature of the h

metric) with prescribed local source . Such equations can be integrated ingeneral form even the algebroid structures functions ia(x

k) are not trivial.The equation (54) is the same both on TV and TEE and contains partialderivatives only on 3 and can be also in similar form.

4.3 Generating offdiagonal solutions

We can integrate the algebroid Ricci soliton equations (49) and (50) for

a nontrivial source , ga = ae(xk), a = 1 and ha = 0.

Theorem 4.1 The system (53)(56) decouple in Nadapted form,

1X1(X1) + 2X2(X2) = 2 (58)h4 = 2h3h4 (59)

wA A = 0, (60)nA + n

A = 0, (61)

for A = h4A, = h

4

, =

ln |h4|3/2/|h3|

, (62)

where = ln |h4/

|h3h4|| (63)is considered as a generating function.

Proof. It follows from explicit computations for dmetrics (52) withKilling symmetry on 4. It is convenient to use the value := e

.

Corollary 4.1 The above systems of nonlinear partial differential equations,PDE, can be integrated in very general forms.

Proof. We should follow such a procedure:

1. The (58) is just a 2d Laplace/ dAlamber equation which can be solvedfor any given .

2. For hA

:= A

z2A

(xk, y3), A

=

1 (we do not consider summation onrepeating indices in this formula), the system of two equations (59) and(63) can be written z4 = 3z4(z3)

2 and ez3 = 24z4 . Multiplying

both equations for nonzero z4 , , zA and introducing the result instead

of the first equation, this system transforms into = 234z3z4 andz3 = 24z

4 . Introducing z3 from the second equation into the first

one, we obtain [(z4)2] = 3[

2]/4. We can integrate on y3,

h4 = 4(z4)2 = 0h4(x

k) +344

2, (64)

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for an integration function 0h4(xk). From the first equation in above

system, we compute

h3 = 3(z3)2 =

z4z4z4z4

=1

2(ln ||)(ln |h4|). (65)

Redefining the coordinates and and introducing 34 in we expressthe solutions in functional form, h3[] = (

)2/2, h4[] = 2/4.

3. To find wa we have to solve certain algebraic equations which can beobtained if we introduce the coefficients (62) in (60),

wa = Xa/ = Xa/. (66)

4. Integrating two times on y3 in (61), we find

nb = 1nb + 2nb

dy3 h3/(

|h4|)3, (67)

where 1nb(xi), 2nb(x

i) are integration functions.

5. The nonholonomic constraints for the LCconditions (57) can be solvedin explicit form for certain classes of integration functions 1nb and

2nb. We can find explicit solutions if 2nb = 0 and 1nb = Xbn witha function n = n(xk). We get (

Xa

wa3)

0 for any (xk, y3)

if wa is defined by (66). For any functional H(), we obtain (Xa wa3)H =

H (Xa wa3) = 0. It is possible to solve the equations

(Xa wa3)h4 = 0 for h4 = H(|()|). This way we solve the secondsystem of equations in (57) when (Xawa3) ln

|h4| (Xawa3)h4.We can consider a subclass of generating functions = for which(Xa) = Xa(). Then, we can compute for the left part of the secondequation in (57), (Xa wa3) ln

|h4| = 0. The first system of equa-tions in (57) can be solved in explicit for any wa determined by formulas(66), and h3[] and h4[,

]. Let us consider = (ln|h3|) for a

functional dependence h3[[]]. This allows us to obtain the formulaswa =

Xa

|

|/

|

| =

Xa

|ln|

h3

||/

|ln|

h3

||. Taking derivative 3 on

both sides of this equation, we get wa =(Xa| ln|h3||)| ln

|h3||

wa | ln|h3||| ln

|h3||

.

The condition wa = (Xa wa3) ln|h3| is necessary for the zero

torsion conditions. It is satisfied for = . We can chose wa =wa = Xa/ = Xa A, with a nontrivial function A(xk, y3) dependingfunctionally on generating function in order to solve the equations

Xawb = Xbwa from the second line in (57).

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Conclusion 4.1 The class of offdiagonal metrics of type (52) with coeffi-

cients computed following the method outlined in above Proof are determinedby quadratic elements of type ds2 =

e(xk)[1(X1)2 + 2(X2)2] + 3 (

)2

2[V3 + (Xa A[])Xa]2 + 4 2

4|| [V4 + (Xan)Xa]2.

(68)

In general, on prolongation Lie dalgebroids, the solutions defining Riccisolitons can be with nontrivial torsion.

Remark 4.1 For arbitrary and related , or , we can generate offdiagonal solutions of (53)(56) with nonholonomically induced torsion,

ds2 = e(xk)[1(X1)2 + 2(X2)2] + (69)3(z3)

2[V3 + Xa

Xa]2 + 4(z4)2[V4 + ( 1na + 2na

dy3(z3)2

(z4)3)Xa]2,

where the values z3(xk, y3) and z4(x

k, y3) are defined by formulas (65) and(64). In Nadapted frames, the ansatz for such solutions define a nontrivialdistorting tensor as in Z = {Z} in (6), see also (A.6).

Taking data ga(xk) and hA(x

k) for a prime metric (51) to define, for in-stance, a black hole solution for Einstein de Sitter spaces, and reparameterizingthe metric (68) in the form (52), we can study nonholonomic deformations

of black hole metrics into Lie algebroid solitionic configurations. In explicitform, such examples of algebroid black holes are proved in [27] and referencetherein.

4.4 On Lie algebroid & almost Khler Finsler Ricci solitons

We show how a Finsler geometry model can be nonholonomically de-formed into a Ricci soliton dalgebroid configuration. Let E = TM for atangent bundle T M on the base space M being a real C manifold of di-mension dim M = n = 2.

Definition 4.2 A Finsler fundamental, or generating, function (metric) is a

function F : T M [0, ) for which 1) F(x, y) isC on T M := T M\{0},where {0} is the set of zero sections of T M on M; 2) F(x,y) = F(x, y),

for any > 0, i.e. it is a positive 1homogeneous function on the fibers ofT M; 3) for any y TxM , the Hessian vgij(x, y) = 12

2F2

yiyjis considered as

s a vertical (v) metric on typical fiber, i.e. it is nondegenerate and positivedefinite, det | v gij | = 0.

For the conditions of Theorem 1.1, we take L = L = F2 and constructthe geometric data (g, N).

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Theorem 4.2 Any FinslerCartan geometry (equivalently modelled as an

almost KhlerFinsler space) with nonholonomic splitting 2 + 2 can be en-coded as a canonical Ricci dalgebroid soliton with metric of type (69) and arespective almost Khler dalgebroid soliton, see Definition 4.1.

Proof. It follows from explicit computations for dmetrics (52) withKilling symmetry on 4. We consider for the class of metrics (52) that upto frame/ coordinate transforms the prime configuration is determined by

a total bundle metric g

= e

eg. The target metric g (52) de-

fines generic offdiagonal solutions for prolongation Lie dalgebroids withcanonical dconnections if g

is of type (69). Such solutions of Ricci

soliton dalgebroid equaitions (49) define nonholonomic transforms (gg,L = F2, N, D; ) (g g, N, D;). We reencode a FinslerCartan

geometry into the canonical data for a Ricci soliton solution on TEE, forE = TM. Via additional dconnection distortions D = D + Z, com-pletely defined by a Ricci soliton dalgebroid solution (69), we redefinethe geometric constructions on KTMTM. For fundamental geometric val-ues, (g g, N, D;) [(, ):= g( J ,), D = D + Z].

The constructions for this theorem can be extended for nonholonomicsplitting of any finite dimension 2+2...+2.

Finally, we note that both primary and target almost Khler Finslergeometry / Lie algebroid configurations can be quantized, for instance, using

methods of deformation, or Abrane quantization [20, 21].Acknowledgments: The work is partially supported by the Program

IDEI, PN-II-ID-PCE-2011-3-0256 and contains the main results presented asa Plenary Lecture at the 11th Panhellenic Geometry Conference (May 31 -June 2, 2013; Department of Mathematics of the National and KapodistrianUniversity of Athens, Greece). The author is grateful to the OrganizingCommittee and P. Stavrinos for kind support and collaboration.

A Formulas in Coefficient Forms

In this section, we summarize some important local constructions andcoefficient formulas which are necessary for formulating Ricci evolution equa-tions and deriving exact solutions on Lie algebroid models.

A.1 Torsions and curvatures on TEPThe Nadapted components

= (Labf,L

ABf ;B

abC,B

ABC) of a dconnec-

tion 2.4 and corresponding covariant operator D = ( eD), where is theinterior product, are computed following equations

= (De)e. There

are defined the h and vcovariant derivatives, respectively, hD = {D =

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35/40


36/40

for adbf =12(

ab

df gg) and ABCD = 12(ACBD gCD gAB).

Introducing K = (A.5) into formulas (A.2), (33) and (34), wecompute respectively the coefficients of curvature, R

, Ricci tensor, R,

and scalar curvature, sR. The distortions K = + Z (A.6) allows us tocompute the distorting tensors (Z

, Z and sZ) resulting in similar val-

ues for the (pseudo) Riemannian geometry on TEP determined by (g, K) ,i.e. to define R

, R and

sR.

A.3 Lie algebroid mechanics and KernMatsumoto models

Let us briefly outline some basic constructions [17, 27] when the canoni-

cal N and dconnections and dmetric on TEE, for P = E, can be gener-ated from a regular Lagrangian L as a solution of the corresponding EulerLagrange equations. The approach was developed in geometric mechanicswith regular Lagrangians on prolongations of Lie algebroids on bundle maps,see [33, 30, 32] and references therein (the first models on mechanics on al-gebroids were elaborated in [34, 35]).

For a generating function L(xi, ya) C(E) (or Lagrangian L(xi, ya)if E = TM), we can compute dEL = ia(iL)Xa + (AL)VA. A verticalendomorphism S : TEE TEE is constructed by S(a,b,v) = V(a, b) =(a, 0, bVa ). We consider b

Va as the vector tangent to the curve a + b; the

curve parameter = 0. The vertical lift is a map V : E

TEE and the

Liouville dilaton vector field (a) = V(a, a) = (a, 0, bVa ). This allows us toconstruct a model of Lie algebroid mechanics for L which can be geometrizedon TEE in terms of three geometric objects,

the Cartan 1-section: L := S(dL) Sec((TEE));

the Cartan 2-section: L := dL Sec(2(TEE)); (A.7)the Lagrangian energy : EL := LiL L C(E),

where the Lie derivative Li is considered in the last formula. The dynamicalequations for L are geometrized

iSXL = S(iXL) and iL = S(dEL), X Sec(TE

E). (A.8)

The geometric objects (A.7) and equations (A.8) can are known for var-ious applications in coefficient forms. Using local coordinates (xi, ya) Eand choosing a basis {Xa, VA} Sec(TEE), for all a, we have

SXa = Va, SVa = 0, = yaVa, EL = yaL/ya L, (A.9)L =

2Ly ay b

Xa Vb + 12

(ib2L

xiya ia

2Lxiy b

+ CfabLy f

)Xa Xb,

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for Lie algebroid structure functions (ia, Cfab). As a vertical endomorphism

(equivalently, tangent structure) can be used the operator S := Xa Va. Aregular system is characterized by a nondegenerate Hessian

gab :=2L

y ay b, |gab| = det |gab| = 0. (A.10)

An EulerLagrange section associated with L is given by any L =yaXa + aVa Sec(TEE), when functions a(xi, yb) solve this system oflinear equations b

2Lybya

+ yb(ib2L

xiya+ Cfab

Lyf

) ia Lxi = 0. The semispray vector

e

= g

eb

(

i

b

Lxi i

a

2

Lxiy b ya

Cf

ba

Ly f ya

) (A.11)

can be found in explicit forms for regular configurations when gab is inverse togab. The section L transforms into a spray which states that the functions

b

are homogenous of degree 2 on yb if the condition [, L]E = L is satisfied.The solutions of the EulerLagrange equations for L,

dxi

d= iay

a andd

d(

Ly a

) + ybCfabLy f

iaLxi

= 0, (A.12)

are parameterized by curves c() = (xi(), ya()) E.

A.4 The torsion and curvature of the normal dconnection

Let us consider a 1form associated to the normal dconnection D =( Da, DA), see nD (31) and = (Labf, Babc) (32), ab := LabfXf + Babcc,where e = (ea = a, VA) and e := {Xa, b = Vb + NbfVf} are taken as in(22). We can prove that the Cartan structure equations are satisfied,

dXa Xb ab = hTa, dc b cb = v Tc, (A.13)and dab cb ac = Rab. (A.14)

The h and vcomponents of the torsion 2form T = hT

a, v Ta

=Tacbc b from (A.13). The Nadapted coefficients are computedhTa = BabcXb c, v Ta = 12 abcXb Xc + (Vc Nab Labc)Xb c, (A.15)

where abc are coefficients of the canonical Nconnection curvature (20) of

the canonical Nconnection, Nbf Nbf (21). In explicit form, the Nadaptedcoefficients of dtorsion (A.1) of D are

Tacb = 0,

TacB =

BacB,

TAcb =

Abc,

TAcB = VB

NAc

LAcB,

TACB = 0, (A.16)

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http://arxiv.org/abs/math/0307245http://arxiv.org/abs/math/0211159


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Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids

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