Nonholonomic Ricci Flows and Running Cosmological Constant: I. 4D Taub-NUT Metrics

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International Journal of Modern Physics AVol. 22, No. 6 (2007) 1135–1159c World Scientic Publishing Company

NONHOLONOMIC RICCI FLOWS ANDRUNNING COSMOLOGICAL CONSTANT:

I. 4D TAUB-NUT METRICS

SERGIU I. VACARUThe Fields Institute for Research in Mathematical Science,

222 College Street, Second Floor, Toronto, Ontario M5T 3J1, Canada [email protected] sergiu [email protected]

MIHAI VISINESCUDepartment of Theoretical Physics,

National Institute for Physics and Nuclear Engineering,P.O. Box M.G.-6, Magurele, Bucharest, Romania

[email protected]

Received 20 September 2006

In this work we construct and analyze exact solutions describing Ricci ows and nonholo-nomic deformations of four-dimensional (4D) Taub-NUT space–times. It is outlined anew geometric technique of constructing Ricci ow solutions. Some conceptual issues onspace–times provided with generic off-diagonal metrics and associated nonlinear connec-tion structures are analyzed. The limit from gravity/Ricci ow models with nontrivialtorsion to congurations with the Levi-Civita connection is allowed in some specicphysical circumstances by constraining the class of integral varieties for the Einsteinand Ricci ow equations.

Keywords : Ricci ows; exact solutions; Taub-NUT spaces.

PACS numbers: 04.20.Jb, 04.30.Nk, 04.50.+h, 04.90.+e, 02.30.Jk

1. Introduction

Hawking has suggested 1 that the Euclidean Taub-NUT metric might give rise tothe gravitational analogue of the Yang–Mills instanton. Also, in the long-distantlimit, neglecting radiation, the relative motion of two monopoles is described by

the geodesics of this space.2,3

The Kaluza–Klein monopole was obtained by em-bedding the Taub-NUT gravitational instanton into ve-dimensional (5D) Kaluza–Klein theory 4,5 (there are various classes of solitonic/monopole solutions, in brief,termed KK monopoles; see further developments and reviews of results in pseudo-classical models 6,7 ).

1135



1136 S. I. Vacaru & M. Visinescu

On the other hand, the recent experimental data seem to indicate that theuniverse does indeed possess a small positive cosmological constant. This induceda program of researches on generalized solutions in asymptotically (anti)-de Sitter(AdS) space–times (KK–AdS–Taub-NUT solutions 8–10 and new cosmological Taub-NUT-like solutions 11,12 ).

The problem of extending the gravitational monopole solutions to extra dimen-sions and/or on spaces with nontrivial cosmological constants is related to amore general problem of constructing and interpretation of solutions deningphysical objects self-consistently embedded in arbitrary nontrivial gravitationalbackgrounds. The usual techniques for generating new classes of monopole solu-tions with nonzero cosmological constants is to add the timelike coordinate, togeneralize the metric to time dependencies (for instance, to a cosmological model)and then to perform the Kaluza–Klein compactication on the fth dimension(in brief, 5D).

There is a more general approach of constructing exact solutions in gravityfollowing the so-called “anholonomic frame method” elaborated and developed ina series of works (see Refs. 13–15). The idea is to use certain classes of nonholo-nomic (equivalently, anholonomic) deformations of the frame, metric and connectionstructures and superpositions of generalized conformal maps in order to generate aclass of off-diagonal metric ansatz solving exactly the vacuum or nonvacuum Ein-stein equations. a The method was considered, for instance, for constructing locallyanisotropic Taub-NUT solutions 16 and investigating self-consistent propagations of three-dimensional Dirac and/or solitonic waves in such space–times. 17

A general nonholonomic transform of a “primary” metric (it can be an exactsolution or, for instance, a conformal transform of a known exact solution) into ageneric off-diagonal exact solution does not preserve the properties of former metric.Nevertheless, if there are certain satised smooth limits to already known solutions,boundary (asymptotic) and deforming symmetry conditions, one may consider that,for instance, a primary Taub-NUT conguration became locally anisotropic with

polarized constants and/or imbedded self-consistently in a nontrivial, for instance,solitonic background. The new classes of solutions can be used for testing varioustype of physical theories (string/brane gravity, noncommutative and gauge gravity,Finsler like generalizations . . .) when the physical interactions are described bygeneric off-diagonal metrics and nonholonomic constraints; in general, with non-trivial topological congurations and new type symmetries (non-Killing ones, forinstance, with generalized Lie algebra symmetries); the extra dimensions are notsubjected to the Kaluza–Klein restrictions.

a Such ansatz cannot be diagonalized by coordinate transforms but can be effectively diagonalizedwith respect to certain systems of nonholonomic local frames with associated nonlinear connectionstructures, see details in Ref. 15. This allows us to apply a well-developed geometric technique inorder to dene generalized symmetries and to integrate exactly the corresponding systems of eldequations. For instance, for 5D space–times, such solutions depend on sets of integrating functionson four and three variables.



Nonholonomic Ricci Flows and Running Cosmological Constant 1137

The anholonomic frame method seems to be effective in constructing exact solu-tions of the Ricci ow equations. 18 Such equations were introduced by Hamilton 19

in order to describe the geometric evolution of a Riemannian manifold ( V, g), whereby g we denote the metric, in the direction of its Ricci tensor Ric( g), see reviewsof results and applications in Refs. 20–22. There is a vast potential use of thisgeometric approach in theoretical physics and cosmology. 23,24,26–30

It is known that there is a unique solution for the Ricci ow with boundedcurvature on a complete noncompact manifold. 31 A very important task is to con-struct exact solutions for the Ricci ow equations and to investigate their physicalimplications. The purpose of this work is two-fold. The rst one is to elaborate ageneral method of constructing exact Ricci ow solutions for certain classes of off-diagonally deformed metric ansatz, both for connections with trivial and nontrivialtorsion. The other is to construct explicit examples of such 4D exact solutions, forTaub-NUT-like metrics, to analyze their physical properties and to show that theymay possess nontrivial limits to the Einstein spaces.

The structure of the paper is as follows: In Sec. 2, we dene a new geometricapproach in constructing exact solutions for Ricci ows of generic off-diagonalmetrics. The approach is based on the so-called anholonomic frame method withassociated nonlinear connection structure. We construct a general class of integralvarieties of Ricci ow equations and dene the constrains for the Levi-Civita con-gurations. In Sec. 3, we apply the formalism to 4D Taub-NUT congurations andtheir off-diagonal Ricci ows. The last section is devoted to conclusions and discus-sion. In Appendix, we outline the necessary material from the geometry of nonlinearconnections and related nonholonomic deformations.

2. Nonholonomic Ricci Flows

In this section, we introduce an off-diagonal ansatz for metrics depending on threecoordinates and consider the anholonomic frame method of constructing exact solu-

tions for the system of equations dening Ricci ows of metrics subjected to non-holonomic constraints.

2.1. Geometric preliminaries

The normalized Ricci ows, 19–24 with respect to a coordinate base ∂ α = ∂/∂u α ,are dened by the equations

∂ ∂τ

gα β = − 2R α β +2r5

gα β , (1)

where R α β is the Ricci tensor of a metric gα β and corresponding Levi-Civitaconnection b and the normalizing factor r = RdV/dV is introduced in order to

b By denition, this connection is torsionless and metric compatible.




preserve the volume V. In this work, we shall consider ows of generic off-diagonal4D metrics

g = gα β (u)du α du β , (2)

where

gα β =gij + N ai N bj hab N ej hae

N ei hbe hab,

with the indices of type α , β = ( i, a ), ( j,b), . . . running the values i, j , . . . , = 1 , 2and a, b , . . . , = 3 , 4 (we shall omit underlying of indices for the components of arbitrary basis or even with respect to coordinate basis if that will not result in

ambiguities) and local coordinates labeled in the form u = ( x, y ) = u α = ( x i , ya ).Applying the frame transforms

e α = e αα ∂ α and cα = e α

α du α ,

for e α cβ = δβα , where “ ” denotes the inner product and δβ

α is the Kroneckersymbol, with

e αα =

δ ii N bi (u)δ a

b

0 δ ab

and e αα =

δii − N bk (u)δk

i

0 δaa

, (3)

we represent the metric (2) in an effectively diagonalized ( n + m)-distinguishedform, see the metric (A.5) in Appendix,

g = gα (u)c α cα = gi (u)bi bi + ha (u)ba ba ,

c α = ( bi = dx i , ba = dya + N ai (u)dx i ) .(4)

Such metrics and frame transforms are considered in the geometry of nonholonomicmanifolds with associated N -connection structure dened by the set N = N bk stating a nonintegrable distribution on a 4D manifold V . In App. A, we outline thegeometry of such spaces. In this paper, we shall consider classes of solutions with

ansatz of type (4) (and (A.1) or (A.5)), whengi = gi (xk ) , ha = ha (xk , v) , N 3i = wi (xk , v) , N 4i = n i (xk , v) , (5)

for y3 = v being the so-called “anisotropic” coordinate. Such metrics are verygeneral off-diagonal ones, with the coefficients depending on two and three coordi-nates but not depending on the coordinate y4 .

To consider ows of metrics related both to the Einstein and string gravity (inthe last case there is a nontrivial antisymmetric torsion eld), it is convenient towork with the so-called canonical distinguished connection (in brief, d-connection)

D = Γγ

αβ which is metric compatible but with nontrivial torsion (see formulas(A.15) and related discussions). Imposing certain restrictions on the coefficients N bk ,see (A.17), (A.18) and (A.19), we can satisfy the conditions that the coefficients of the canonical d-connection of the Levi-Civita = Γα

βγ are dened by the samenontrivial values Γ

γαβ = Γγ

αβ with respect to N -adapted basis (A.7) and (A.8).




The Ricci ow equation (1) can be written for the Ricci tensor of the canonicald-connection (A.13) and metric (4), as it was considered in Ref. 18,

∂ ∂τ

gij = − 2R ij + 2 λg ij − hcd∂

∂τ (N ci N dj ) , (6)

∂ ∂τ

hab = − 2S ab + 2 λh ab , Rαβ = 0 and gαβ = 0 for α = β , (7)

where λ = r/ 5, y3 = v and τ can be, for instance, the timelike coordinate, τ = t,or any parameter or extra dimension coordinate. Equations (6) and (7) are justthe nonholonomic frame transform with the matrices (3) of Eq. (1). The aim of this section is to show how the anholonomic frame method developed in Ref. 18

(for Ricci ows) and in Refs. 13–17 (for off-diagonal exact solutions) can be usedfor constructing exact solutions of the system of Ricci ow equations (6) and (7)describing nonholonomic deformations of 3D and 4D Taub-NUT solutions.

2.2. Ricci ow equations for off-diagonal metric ansatz

We consider a primary (pseudo-)Riemannian metric

g = g1(xk , v,y 4)(dx1 )2 + g2(xk , v,y 4)(dx2 )2

+ˇh3(x

k

, v,y4

)(ˇb

3

)2

+ˇh4(x

k

, v,y4

)(ˇb

4

)2

,b3 = dv + wi (xk , v,y 4)dx i ,

b4 = dy4 + n i (xk , v,y 4)dx i ,

(8)

which is a particular case of metric (A.1) with N -connection coefficients N 3i =wi (xk , v,y 4) and N 4i = n i (xk , v,y 4) for y3 = v considered to be a nonholonomicallyconstrained coordinate. The metric may be an exact solution of the Einstein equa-tions, or any conformal transform of such one (which is not an exact solution). c Ananholonomic transform, N → N and g = ( g, h) → g = ( g, h), can be dened by

formulas (A.6),gi = ηi (xk , v,y 4)gi , ha = ηa (xk , v,y 4)ha , N ai = ηa

i (xk , v,y 4)N ai

when the polarizations ηα and ηai are chosen in a form that the “target” metric g

has coefficients of type (5), i.e. it is parametrized in the form

g = g1(xk )(dx1 )2 + g2(xk )(dx2)2

+ h3(xk , v)(b3)2 + h4(xk , v)(b4)2 ,

b3 = dv + wi (xk , v)dx i ,

b4 = dy4 + n i (xk , v)dx i ,

(9)

c In this work, we shall consider the primary metrics to be related to certain Taub-NUT-likesolutions, or some of their conformal transforms and/or trivial embedding/compactication inextra/lower dimension spaces.




dening an exact solution for the nonholonomic Ricci ow equations (6) and (7),when τ is one of the coordinates xk , or v.

The nontrivial components of the Ricci tensor R αβ (A.13) (see details of asimilar calculus in Ref. 15) are

R11 = R2

2 =1

2g1g2

g•1 g•

22g1

+(g•

2 )2

2g2− g••

2 +g1g22g1

+(g1)2

2g2− g1 , (10)

S 33 = S 44 =1

2h3h4− h4 +

(h4 )2

2h4+

h4h32h4

, (11)

R3i = −1

2h4

(wi β + α i ) , (12)

R4i = −h4

2h3(n i + γn i ) , (13)

where

α i = ∂ i h4 − h4∂ i ln |h3h4 | ,

β = h4 − h4 ln |h3h4 | ,

γ =3h42h4

−h3h3

, for h3 = 0, h4 = 0 ,

(14)

dened by h3 and h4 as solutions of equations (7). In the above presented formulas,it was convenient to write the partial derivatives in the form a • = ∂a/∂x 1 , a =∂a/∂x 2 and a = ∂a/∂v.

We consider a general method of constructing solutions of the Ricci ow equa-tions related to the so-called Einstein spaces with nonhomogeneously polarizedcosmological constant, when

R ij = λ [h ](xk )δi

j , S ab = λ [v ](xk , v)δab ,

Rαβ

= 0 and gαβ

= 0 for α = β ,

when λ [h ] and λ [v ] are induced by certain string gravity ansatz (A.27), or mattereld contributions. In particular case, we can x λ [h ] = λ [v ] and consider off-diagonalmetrics of the usual Einstein spaces with cosmological constant.

The nonholonomic Ricci ow equations (6) and (7) for the Einstein spaces withnonhomogeneous cosmological constant dened by ansatz of type (9) transforminto the following system of partial differential equations consisting of two subsetsof equations: the rst subset of equations consists of those generated by the Einsteinequations for the off-diagonal metric,

g•1 g•

22g1

+(g•

2 )2

2g2− g••

2 +g1g22g1

+(g1)2

2g2− g1 = 2 g1g2λ [h ](xk ) , (15)

− h4 +(h4 )2

2h4+

h4h32h4

= 2 h3h4λ [v ](xk , v) , (16)




wi β + α i = 0 , (17)

n i + γn i = 0 . (18)The second subset of equations is formed just by those describing ows of thediagonal, gij = diag[ g1, g2] and hab = diag[ h3 , h4], and off-diagonal, wi and n i ,metric coefficients,

∂ ∂τ

gij = 2 λ [h ](xk )gij − h3∂

∂τ (wi wj ) − h4

∂ ∂τ

(n i n j ) , (19)

∂ ∂τ

ha = 2 λ [v ](xk , v)ha . (20)

The aim of the next section is to show how we can integrate Eqs. (15)–(20) in aquite general form.

2.3. Integral varieties of solutions of Ricci ow equations

We emphasize that the system of equations (15)–(18) was derived and solved ingeneral form for a number of 4D and 5D metric ansatz of type (9), or (A.5), inRefs. 13–17 for various models of gravity theory. The idea of work 18 was to use theformer method and some integral varieties of those solutions in order to subject

the metric and N -connection coefficients additionally to the conditions (19) and(20) and generate Ricci ows of off-diagonal metrics. Here, we briey outline themethod of constructing such general solutions.

We begin with Eq. (15) for the metric coefficients gi (xk ) on a 2D subspace.By a corresponding coordinate transform x i → x i (x i ), such metrics always can bediagonalized and represented in conformally at form,

gi (xk )(dx i )2 = eψ (x i )1(dx 1 )2 + 2(dx 2 )2 ,

where the values i = ± 1 depend on chosen signature. Equation (15) transforms

into1ψ•• + 2ψ = 2 λ [h ](xk ) . (21)

Such equations and their equivalent 2D coordinate transform can be written inthree alternative ways convenient for different types of nonholonomic deformationsof metrics. For instance, we can prescribe that g1 = g2 and write the equation forψ = ln |g1 | = ln |g2 |. Alternatively, we can suppose that g1 = 0 for a given g1(x1 )(or g•

2 = 0 , for a prescribed g2(x2 )) and get from (15) equation

g•1 g•

2

2g1 +

(g•2 )2

2g2 − g••

2 = 2 g1g2λ [h ](xk

)(or

g1g22g1

+(g1)2

2g2− g1 = 2 g1g2λ [h ](xk ) ,




in the inverse case). In general, we can prescribe that, for instance, g1(xk ) is denedby any solution of the 2D Laplace/D’Alambert/solitonic equation and try to deneg2(xk ) constrained to be satised one of the above equations related to (15). Weconclude that such 2D equations always can be solved in explicit or nonexplicitform.

Equation (16) relates two nontrivial v-coefficients of the metric coefficientsh3(xk , v) and h4(xk , v) depending on three coordinates but with partial deriva-tives only on the third (anisotropic) coordinate. As a matter of principle, we canx h3 (or, inversely, h4) to describe any physically interesting situation being, forinstance, a solution of the 3D solitonic, or pp-wave equation, and then we can tryto dene h4 (inversely, h3) in order to get a solution of (16). Here we note that itis possible to solve such equations for any λ [v ](xk , v), in general form, if h4 = 0 (forh4 = 0 , there are nontrivial solutions only if λ [v ] = 0) . Introducing the function

φ(x i , v) = lnh4

|h3h4 |, (22)

we write that equation in the form

|h3h4 | − 1(eφ ) = − 2λ [v ] . (23)

Using (22), we express |h3h4 | as a function of φ and h4 and obtain

|h4 | = − (eφ ) / 4λ [v ] , (24)

which can be integrated in general form,

h4 = h4[0] (x i ) −14 dv

[e2φ (x i ,v ) ]λ [v ](x i , v)

, (25)

where h4[0] (x i ) is the integration function. Having dened h4 and using again (22),

we can express h3 via h4 and φ,|h3 | = 4 e− 2φ (x i ,v ) ( |h4 |) 2 . (26)

The conclusion is that prescribing any two functions φ(x i , v) and λ [v ](x i , v), we canalways nd the corresponding metric coefficients h3 and h4 solving (16). Following(26), it is convenient to represent such solutions in the form

h4 = 4[b(x i , v) − b0(x i )]2 ,

h3 = 4 3e− 2φ (x i ,v ) [b (x i , v)]2 ,

where a = ± 1 depending on xed signature, b0(x i ) and φ(x i , v) can be arbitraryfunctions and b(x i , v) is any function when b is related to φ and λ [v ] as stated bythe formula (24). Finally, we note that if λ [v ] = 0 , we can relate h3 and h4 solving(23) as (eφ ) = 0 .




For any couples h3 and h4 related by (16), we can compute the values α i , β andγ (14). This allows us to dene the off-diagonal metric ( N -connection) coefficientswi solving (17) as algebraic equation,

wi = − α i /β = − ∂ i φ/φ . (27)

We emphasize, that for the vacuum Einstein equations one can be solutions of (16)resulting in α i = β = 0 . In such cases, wi can be arbitrary functions on variables(x i , v) with nite values for derivatives in the limits α i , β → 0 eliminating the “ill-dened” situation wi → 0/ 0. For the Ricci ow equations with nonzero values of λ [v ], such difficulties do not arise. The second subset of N -connection (off-diagonal

metric) coefficients n i can be computed by integrating two times on variable v in(18), for given values h3 and h4 . One obtains

n k = n k [1](x i ) + n k [2](x i )n k (x i , v) , (28)

where

n k (x i , v) = h3( |h4 |)− 3 dv , h 4 = 0 ;

=

h3 dv , h4 = 0 ;

= ( |h4 |)− 3 dv , h3 = 0 ,

and nk [1](x i ) and n k [2](x i ) are integration functions.We conclude that any solution ( h3 , h4) of Eq. (16) with h4 = 0 and nonvanishing

λ [v ] generates the solutions (27) and (28), respectively, of Eqs. (17) and (18). Suchsolutions (of the Einstein equations) are dened by the mentioned classes of inte-gration functions and prescribed values for b(x i , v) and ψ(x i ). Further restrictionson (g1 , g2) and ( h3 , h4) are necessary in order to satisfy Eqs. (19) and (20) relatingows of the metric and N -connection coefficients in a compatible manner. It is notpossible to solve in a quite general form such equations, but in the next section weshall give certain examples of such solutions dening ows of the Taub-NUT-likemetrics.

2.4. Extracting solutions for the Levi-Civita connection

The method outlined in the previous section allows us to construct integral varietiesfor the Ricci ow equations (15)–(20) derived for the canonical d-connection with

nontrivial torsion, see formulas (A.15) and (A.11). We can restrict such integralvarieties (constraining the off-diagonal metric, equivalently, N -connection coeffi-cients wi and n i and related integration functions) in order to generate solutionsfor the Levi-Civita connection. The condition Γα

βγ = Γγαβ (i.e. the coefficients of the

Levi-Civita connection are equal to the coefficients of the canonical d-connection,




both classes of coefficients being computed with respect to the N -adapted bases(A.7) and (A.8)) holds true if Eqs. (A.17)–(A.19) are satised. d

Introducing the coefficients for the ansatz (9), we can see that the constrain(A.18) is trivially satised and Eq. (A.19) is written in the form

∂h 3

∂x k − wk h3 − 2wk h3 = 0 , (29)

∂h 4

∂x k − wk h4 = 0 , (30)

nk h4 = 0 . (31)

The relations (29) and (30) are equivalent for the general solutions h3 , see (26), h4 ,see (25) and wi , see (27), generated by a function φ(x i , v) (22) if φ → φ− ln 2, when

φ = ln |h4 | − ln |h3 |

and

wk = ( h4 )− 1 ∂h 4

∂x k = − (φ )− 1 ∂φ∂x k ,

where φ = const is possible only for the vacuum Einstein solutions. The condition(31) for h4 = 0 constrains n k = 0 which holds true if we put the integrationfunction n k [2] = 0 in (28), when n k = nk [1](x i ). These values of wk and n k have to

be constrained one more again in order to solve Eq. (A.17), which for our ansatzare of type

w1 − w•2 + w2w1 − w1w2 = 0 , (32)

n1 − n •2 = 0 , (33)

stating integrable (pseudo-)Riemannian foliations. We have to take such integra-tion functions when (33) is satised from the very beginning for some two functionsnk [1](x i ) depending on two variables. In a particular case, we can consider anyparametrization of type wk = wk (x i )q(v) for some functions wk (x i ) and q(v) den-ing a class of solutions of (32).

The nal conclusion in this section is that taking any solution of Eqs. (16)–(18), we can restrict the integral varieties to integration functions satisfying theconditions (32) and (33) allowing us to extract torsionless congurations for theLevi-Civita connection.

3. Nonholonomic Ricci Flows and 4D Taub-NUT Spaces

The techniques elaborated in previous section can be applied in order to generate

Ricci ow solutions for various classes of 4D metrics. In this section, we examinesuch congurations derived from a primary Taub-NUT metric.

d We emphasize that connections on manifolds are not dened as tensor objects. If the coefficientsof two different connections are equal with respect to one frame, they can be very different withrespect to other frames.




We begin with the primary quadratic element

ds2 = F − 1 dr 2 + ( r 2 + n2)d 2 − F (r )[dt − 2nw( )d ]2 + ( r 2 + n2)a( )d 2 , (34)for the so-called topological Taub-NUT–AdS/dS space–times 37,38,12 with NUTcharge n. There are three possibilities:

F (r ) =r 4 + ( εl2 + n2)r 2 − 2µrl 2 + εn 2(l2 − 3n2) + (1 − | ε|)n2

l2(n2 + r 2 )

for ε = 1, 0, − 1 dening, respectively

U(1) brations over S 2 ;U(1) brations over T 2 ;U(1) brations over H 2 ;

fora( ) = sin 2 , w ( ) = cos ,a( ) = 1 , w( ) = ,a( ) = sinh 2 , w ( ) = cosh .

The ansatz (34) for ε = 1, 0, − 1 but n = 0 recovers correspondingly the spherical,toroidal and hyperbolic Schwarzschild–AdS/dS solutions of 4D Einstein equationswith cosmological constant λ = − 3/l 2 and mass parameter µ. For our furtherpurposes, it is convenient to consider a coordinate transform

(r, ,t, ) → (r, ,p ( ,t , ), ) ,

with a new timelike coordinate p when

dt − 2nw( )d = dp − 2nw( )d

and t → p are substituted in (34). This is possible if

t → p = t − ν − 1( , )dξ( , ) ,

with

dξ = − ν ( , )d( p − t) = ∂ ξ d + ∂ ξ d

when

d( p − t) = 2 nw( )(d − d ) .

The last formulas state that the functions ν ( , ) and ξ( , ) should be taken tosolve the equations

∂ ξ = − 2nw( )ν and ∂ ξ = 2 nw( )ν .

The solutions of such equations can be generated by any

ξ = ef ( − ) and ν =1

2nw( )df dx

ef ( − ) ,

for x = − .




The primary ansatz (34) can be written in a form similar to (8)

g = g1(xk , v,y 4)(dx1 )2 + g2(xk , v,y 4)(dx2 )2

+ h3(xk , v,y 4)( b3)2 + h4(xk , v,y 4)( b4)2 ,

b3 = dv + wi (xk , v,y 4)dx i ,

b4 = dy4 + n i (xk , v,y 4)dx i ,

(35)

following the parametrizations

x1 = r , x 2 = , y 3 = v = p , y4 = ,

g1(r ) = F − 1(r ) , g2(r ) = ( r 2 + n2) ,

h3(r ) = − F (r ) , h4 (r, ) = ( r 2 + n2)a( ) ,

w1( ) = − 2nw( ) , w2 = 0 , n i = 0 .

An anholonomic transform N → N and g = ( g, h) → g = ( g, h) can be denedby formulas of type (A.6),

g1 = η1 (r, )g1(r ) , g2 = η2 (r, )g2(r ) ,

h3 = η3 (r, ,p )h3(r ) , h4 = η4 (r, ,p )h4(r, ) ,

w1 = η31 (r, ,p )w1 ( ) , w2 = w2 (r, ,p ) ,

n1 = n1(r, ,p ) , n2 = n2(r, ,p ) ,

(36)

resulting in the “target” metric ansatz

g = g1(r, )(dr )2 + g2(r, )(d )2

+ h3(r, ,p )(b3)2 + h4(r, ,p )(b4 )2 ,

b3 = dp + w1(r, ,p )dr + w2(r, ,p )d ,

b4 = d + n1(r, ,p )dr + n2(r, ,p )d .

(37)

Our aim is to state the coefficients when this off-diagonal metric ansatz denesolutions of the nonholonomic Ricci ow equations (15)–(20) for τ = p.

We construct a family of exact solutions of the Einstein equations with polarized

cosmological constants following the same steps used for deriving formulas (21),(22), (26), (24) and (28). By a corresponding 2D coordinate transform x i → x i (r, ),such metrics always can be diagonalized and represented in conformally at form,

g1(r, )(dr )2 + g2(r, )(d )2 = eψ (x i ) [ 1(dx 1)2 + 2(dx 2)2 ] ,




where the values i = ± 1 depend on chosen signature and ψ(x i ) is a solution of

1ψ•• + 2ψ = 2 λ [h ](x˜i ) .

For other metric coefficients, one obtains the relation

φ(r, ,p ) = lnh4

|h3h4 |,

for

(eφ ) = − 2λ [v ](r, ,p ) |h3h4 | ,

|h3 | = 4 e− 2φ ( r, ,p ) (

|h4 |) 2 ,

|h4 | = − (eφ ) / 4λ [v ] .

It is convenient to represent such solutions in the form, see (26),

h4 = 4[b(r, ,p ) − b0(r, )]2 , h3 = 4 3e− 2φ ( r, ,p ) [b (r, ,p )]2 , (38)

where a = ± 1 depend on xed signature, b0(r, ) and φ(r, ,p ) can be arbitraryfunctions and b(r, ,p ) is any function when b is related to φ and λ [v ] as stated bythe formula (24).

The N -connection coefficients are of type

n k = n k [1](r, ) + n k [2](r, )n k (r, ,p ) ,

where

nk (r, ,p ) = h3( |h4 |)− 3dp ,

and nk [1](r, ) and nk [2](r, ) are integration functions and h4 = 0 .The above constructed coefficients for the metric and N -connection depend on

arbitrary integration functions. One have to constrain such integral varieties in order

to construct Ricci ow solutions. Let us consider possible solutions of the equation(19) for n i = 0 as a possible solution of (33) (necessary for the Levi-Civita congu-rations). One obtains a matrix equation for matrices ˜ g(r, ) = [2λ [h ](r, )gij (r, )]and w(r, ,p ) = [ wi (r, ,p )wj (r, ,p )]

g(r, ) = h3(r, ,p )∂

∂pw(r, ,p ) . (39)

This equation is compatible for such 2D systems of coordinates when ˜ g is notdiagonal (when it is more easy to contract a solution (21) for the diagonalized case)because w is also not diagonal. For 2D subspaces, the coordinate and frame trans-forms are equivalent but such congurations should be correspondingly adapted tothe nonholonomic structure dened by ˜ w(r, ,p ) which is possible for a general 2Dcoordinate system. We can consider the transforms

gij = eii (xk (r, ))e j

j (xk (r, ))gi j (xk )




and

wi (xk ) = eii (xk (r, ))wi ((r, ,p )) ,

associated to a coordinate transform ( r, ) → xk (r, ) with gi j (xk ) dening, ingeneral, a symmetric but nondiagonal (2 × 2)-dimensional matrix. We can integrateEq. (39) in explicit form by separation of variables in φ, b, h3 and wi , when

φ = φ(x i )φ( p) , h3 = h3(x i )h3 ( p) ,

wi = wi (xk )q( p) , for wi = − ∂ i ln |φ(xk )| , q = ( ∂ p φ( p))− 1 ,

where separation of variables for h3 is related to a similar separation of variablesb = b(x i )b( p) as follows from (38). One obtains the matrix equation

g(xk ) = α 0 h3(x i )w0 (xk ) ,

where the matrix w0 has components ( wi wk ) and constant α 0 = 0 is chosen fromany prescribed relation

h3( p) = α 0∂ p[∂ p φ( p)]− 2 . (40)

We conclude that any given functionsˆφ(x

k

),ˇφ( p) and

ˆh3(x

i

) and constant α 0we can generate solutions of the Ricci ow equation (19) for n i = 0 with themetric coefficients parametrized in the same form as for the solution of the Einsteinequations (15)–(18). In a particular case, we can take φ( p) to be a periodic orsolitonic type function.

The last step in constructing ow solutions is to solve Eq. (20) for the ansatz(37) redened for coordinates xk = xk (r, ),

∂ ∂p

ha = 2 λ [v ](xk , p)ha .

This equation is compatible if h4 = ς (xk )h3 for any prescribed function ς (xk ).We can satisfy this condition by corresponding parametrizations of function φ =φ(x i )φ( p) and/or b = b(x i )b( p), see (38). As a result, we can compute the effectivecosmological constant for such Ricci ows,

λ [v ](xk , p) = ∂ p ln |h3(xk , p)| ,

which for solutions of type (40) is dened by a polarization running in time,

λ [v ]( p) = α 0∂ 2 p [∂ p φ( p)]− 2 .

In this case, we can identify α 0 with a cosmological constant ( λ = − 3/l 2 , for primaryTaub-NUT congurations, or any λ = λ 2

H / 4, for string congurations, see formula(A.28)) if we choose such φ( p) then ∂ 2 p [∂ p φ( p)]− 2 → 1 for p → 0.




Putting together the coefficients of metric and N -connection, one obtains

g = α 0 h3(x i ) ∂ i ln |φ(xk )|∂ j ln |φ(xk )|dx i dx j + ∂ p[∂ p φ( p)]− 2

× dp − (∂ p φ( p))− 1(dx i ∂ i ln |φ(xk )|) 2 + ς (xk )(d )2 . (41)

This metric ansatz depends on certain type of arbitrary integration and genera-tion functions h3(x i ), φ(xk ), ς (xk ) and φ( p) and on a constant α 0 which canbe identied with the primary cosmological constant. It was derived by consider-ing nonholonomic deformations of some classes of 4D Taub-NUT solutions (35) byconsidering polarization functions (36) deforming the coefficients of the primarymetrics into the target ones for corresponding Ricci ows. The target metric (41)model 4D Einstein spaces with “horizontally” polarized, λ [h ](xk ) and “vertically”running, λ [v ]( p), cosmological constant managed by the Ricci ow solutions.

Finally, we conclude that if the primary 4D topological Taub-NUT–AdS/dSspace–times have the structure of U(1) brations over 2D hypersurfaces (sphere,torus or hyperboloid) than their nonholonomic deformations to Ricci ow solutionswith effectively polarized/running cosmological constant denes the generalized 4DEinstein spaces as certain foliations on the corresponding 2D hypersurfaces. Thisholds true if the nonholonomic structures are chosen to be integrable and for theLevi-Civita connection. In more general cases, with nontrivial torsion, for instance,induced from string gravity, we deal with “nonintegrable” foliated structures, i.e.with nonholonomic Riemann–Cartan manifolds provided with effective nonlinearconnection structure induced by off-diagonal metric terms.

4. Outlook and Discussion

In summary, we have developed the method of anholonomic frames in order to con-struct exact solutions describing Ricci ows of 4D Taub-NUT-like metrics. Towardthis end, we applied a program of study and applications to physics which is based

on the geometry of nonholonomic/foliated spaces with associated nonlinear con-nection structure induced by generic off-diagonal metric terms. The premise of thismethodology is that one was possible to generate exact nonholonomic solutions forthree-, four- and ve-dimensional space–times (in brief, 3D, 4D, 5D) in the Ein-stein and low/extra dimension gravity with cosmological constant (possibly inducedby some ansatz for the antisymmetric torsion in string gravity, or other models of gravity and effective matter eld interactions). 13–17 The validity of this approach inconstructing Ricci ow solutions was substantiated by generating certain examplesof Ricci ow of solitonic pp-waves. 18

In this paper, we elaborated the geometric background for generalized Einsteinspaces with effectively polarized (anisotropically on some space coordinates andrunning on timelike coordinate) cosmological “constants” arising naturally if weconsider generic off-diagonal gravitational (vacuum and with nontrivial mattereld interactions or extra dimension corrections) and nonholonomic frame effects.




Such space–times are distinguished by corresponding nontrivial nonholonomic(foliated, if the integrability conditions are satised) structures (examined in dif-ferent approaches to Lagrange–Finsler geometry, 33 semi-Riemannian and Finslerfoliations 36 and, for instance, in modern gravity and noncommutative geometry 15 ).In searching for physical applications of such geometric methods, we addressedto the geometry and physics of Taub-NUT space–times 1–7 (see also more recentdevelopments in Refs. 8–12) being interested in an analysis of the Ricci ows of Taub-NUT spaces.

There are a number of mathematical results and certain applications in modernphysics related to Ricci ow geometry. 19–24 Nevertheless, possible applications togravity theories are connected to quite cumbersome approximated methods in con-structing solutions of the Ricci ow equations. Perhaps the rst attempt to con-struct explicit exact solutions was performed following a linearization approach 27

allowing to generate exact solutions for lower dimensions. Another class of Ricciow solutions was related to solitonic pp-waves in 5D gravity. 18 The program of constructing exact Ricci ow solutions can be naturally extending to an analysis of such ows of some physically valuable metrics describing exact solutions in gravity.

The case of ows of the so-called Taub-NUT–AdS/dS space–times presents aspecial interest. They describe a number of very interesting physical situations withnontrivial cosmological constant. Certain classes of nonholonomic deformations of such solutions by generic off-diagonal metric terms and associated anholonomicframe structures result in another classes of exact solutions of the Einstein equationsdening the so-called locally anisotropic Taub-NUT space–times, see a detailedstudy in Refs. 16 and 17. One of the important result of those investigations wasthat the occurrence of effective polarizations on spacelike coordinates and running of cosmological constant can be considered in a manner resulting in exactly integrablesystems of partial equations (to which the Einstein equations transform for verygeneral ansatz in 3–5D gravity).

The main idea of this work is to prove that effectively induced nonhomogeneous

Einstein spaces may describe Ricci ows of Taub-NUT-like metrics, for certainparametrizations of polarizations of the metric coefficients and of the cosmologicalconstant. We considered nonholonomic Ricci ows of 4D Taub-NUT spaces pri-marily dened as brated structures on 2D spherical/toroidal/hyperbolic hyper-surfaces. Such 4D off-diagonal ows can also be effectively modeled by families of nonhomogeneous Einstein spaces with corresponding nonholonomic frame struc-ture (which transform into a foliated structure for the Levi-Civita congurations).Here, we note that the integral varieties of the Ricci ow solutions depend onvarious general classes of generating functions and integration functions depending

on three, two, one variables and on integration constants (for 3D congurations,such functions depend on two and one variables). This is a general property of the systems of partial equations to which the Ricci ow and/or Einstein equationsreduce for very general off-diagonal metric and non-Riemannian linear connectionansatz.




The ansatz for usual (nondeformed) Taub-NUT spaces transform the Einsteinequations into certain systems of nonlinear ordinary differential equations on onevariable. From the very beginning such solutions were constrained to depend only onintegration constants (like the mass parameter, NUT constant which were denedfrom certain physical considerations). It is a very difficult technical task to constructexact off-diagonal solutions depending both holonomically and anholonomically ontwo, three variables and dening 4–5D space–times. The anholonomic frame methodoffers a number of such possibilities but it results in a more sophisticated conceptualproblem for possible geometrical and physical meaning/interpretations of variousclasses of integration functions and constants. In the view of such considerations, wecan argue that by imposing certain constraints on classes of integration functionswe select certain new or prescribed physical situations when the solutions are clas-sied by new nonlinear symmetries (noncommutative or Lie algebra generalizationsto Lie algebroid congurations) and self-consistent embedding in solitonic and/or pp-wave backgrounds, with anisotropic polarization of constants, deformation of horizons and so on, see detailed discussions in Refs. 13–18. The approach appearto be promising in constructing exact solutions for the Ricci ows of physicallyimportant metrics and connections when the generation and integration functionsare constrained, for instance, to dene Levi-Civita congurations being derivedeffectively by nonhomogeneous cosmological constants.

In conclusion, we emphasize that the nonholonomic Ricci ow solutions forTaub-NUT-like metrics continue to have a number of issues when they are viewedfrom the perspective of black hole ows and cosmological solutions, generalizationsto extra dimensions non-Riemann theories of gravity. We shall also analyze non-holonomic Ricci ows in string gravity in our further works.

Acknowledgments

S. Vacaru is grateful to D. Singleton, E. Gaburov and D. Gont ¸a for former col-laboration and discussions. He thanks the Fields Institute for hosting volunteer

research work. M. Visinescu has been supported in part by the MEC-CEEX Pro-gram, Romania.

Appendix. The Geometry of N -Connections andAnholonomic Deformations

We outline the geometry of anholonomic deformations of geometric structures on aRiemann–Cartan manifold. The geometric constructions will be related to Ricciows on such spaces and possible reductions to the Einstein spaces. For inte-grable frame structures and Levi-Civita connections such spaces transform into

usual (pseudo-)Riemannian brations.e

e The geometry of brations is considered in detail in Ref. 36 following a different class of linearconnections on nonholonomic manifolds not imposing the conditions that those connections aresolutions of the Einstein, or Ricci ow, equations. In this work, we develop for Ricci ows theapproach outlined for exact solutions in Einstein and string gravity, for instance, in Ref. 15.




A.1. Nonholonomic transforms of vielbeins and metrics

We consider a ( n + m)-dimensional manifold (space–time) V , n ≥ 2, m ≥ 1,f

of arbitrary signature enabled with a “primary” metric structure g = g N hdistinguished in the form

g = gi (u)(dx i )2 + ha (u)( ba )2 , ba = dya + N ai (u)dx i . (A.1)

The local coordinates are parametrized u = ( x, y ) = u α = ( x i , ya ), for the indicesof type i, j , k , . . . = 1 , 2, . . . , n (in brief, horizontal, or h-indices/components) anda, b, c , . . . = n + 1, n + 2 , . . . , n + m (vertical, or v-indices/components). g Theoff-diagonal coefficients N ai (u) in (A.1) state, in general, a nonintegrable ( n + m)-

splitting N in any point u V and dene a class of “ N -adapted” local bases(frames, equivalently, vielbeins) e = ( e, e), when

e α = ei =∂

∂x i − N ai (u)∂

∂y a , ea =∂

∂ya , (A.2)

and local dual bases (coframes) b = ( b, b), when

b α = ( bj = dx i , bb = dyb + N bi (u)dx i ) , (A.3)

for b e = I , i.e. e α b β = δβ

α, where the inner product is denoted by “ ” and the

Kronecker symbol is denoted by δβα . The nonintegrability of the frame structure and

corresponding h- and v-splitting in (A.2) results in the nonholonomy (equivalently,anholonomy) relations

e α e β − e β e α = w γαβ e γ ,

with nontrivial anholonomy coefficients

w aji = − w a

ij = Ωaij ej (N ai ) − ei (N aj ) ,

wbia = − w

bai = ea (

ˇN

bj ) .

(A.4)

A metric g = g N h parametrized in the form

g = gi (u)(bi )2 + ga (u)(ba ) ,

ba = dya + N ai (u)dx i(A.5)

is a nonholonomic transform (deformation), preserving the ( n + m)-splitting, of themetric g = g N h if the coefficients of (A.1) and (A.5) are related by formulas

gi = ηi (u)gi , ha = ηa (u)ˇha and N

a

i = ηa

i (u)ˇ

N a

i , (A.6)f In this paper, we shall consider 4D constructions and possible reductions to three dimensions,when n = 2 and m = 2 , or m = 1.g The Einstein’s summation rule on “up–low” indices will be applied if the contrary case will benot emphasized.




(the summation rule is not considered for the indices of “polarizations” ηα = ( ηi , ηa )and ηa

i in (A.6)). Under anholonomic deformations, for nontrivial values ηai (u), the

nonholonomic frames (A.2) and (A.3) transform correspondingly into

e α = ei =∂

∂x i − N ai (u)∂

∂y a , ea =∂

∂ya (A.7)

and

b α = bj = dx i , bc = dyc + N ci (u)dx i (A.8)

with the anholonomy coefficients w γαβ dened by N ai substituted in formulas (A.4).

We adopt the convention to use “bold” symbols for any geometric object adapted/dened with respect to N -elongated bases and corresponding N -connectionstructures.

A set of coefficients N = N ai states a N -connection structure on a manifold Vif it denes a locally nonintegrable (nonholonomic) distribution T V | u = hV | u N

vV | u in any point u V which can be globalized to a Whitney sum

T V = h V N vV . (A.9)

We say conventionally that an N -connection decomposes the tangent space T Vinto certain horizontal ( h), h V , and vertical ( v), vV , subspaces. With respect to

an “N -adapted” base (A.7), any vector eld X splits into its h- and v-components,X = hX + vX = X i ei + X a ea

with X i X bi and X a X ba . A similar decomposition holds for a co-vector(1-form) X ,

X = hX + vX = X i bi + X a ba .

It should be noted that the “interior product” “ ” is dened by the metric structurebut the “ h- and v-splitting” are stated by the N -connection coefficients N ai , which in

this work are related to generic off-diagonal metric coefficients dened with respectto a usual coordinate basis. h

The N -connection curvature Ω of an N -connection N is by denition just theNeijenhuis tensor

Ω (X , Y ) [vX,vY ] + v[X , Y ] − v[vX, Y ] − v[X , vY ] ,

where, for instance, [ X , Y ] denotes the commutator of vector elds X and Y on T V .The coefficients Ω

aij of an “N -curvature” Ω stated with respect to the bases (A.2)

and (A.3) are computed following the rst formula in (A.4). We can diagonalize

the metric (A.5) by certain coordinate transforms if all w γαβ vanish, i.e. the N -connection structure became trivial (integrable) with Ω = 0 and ea (N bj ) = 0 . Thesubclass of linear connections is selected as a particular case by parametrizations

h For simplicity, we shall omit (inverse) hats on symbols if this does not result in confusion.




of type N ba (x, y ) = Abai (xk )ya (for instance, in the Kaluza–Klein gravity, the values

Abai (xk ) are associated to the gauge elds after extra dimension compactications

on ya ).An anholonomic transform N → N and g = ( g, h) → g = ( g, h), dened

by formulas (A.6), deforms correspondingly the nonholonomic frame (A.2) andmetric structures (A.1). In this paper, we consider maps (transforms) of spaces(manifolds) provided with nonlinear connection ( N -connection) structure 15,33 wheninvariant conventional h- and v-splitting. A manifold V is called N -anholonomic if it is provided with a preferred anholonomic frame structure induced by the genericoff-diagonal coefficients of a metric. 13–15

A.2. Torsions and curvatures of d -connections

A linear connection (1-form) Γ γα = Γ γ

αβ b β on V denes an operator of covariantderivation,

D = D α = ( D i , D a ) .

The coefficients

Γ γαβ (D α e β ) b γ = ( L i

jk , L abk , C ijc , C abc )

can be computed in N -adapted form (i.e. with h- and v-splitting) with respect tothe local bases (A.2) and (A.3) following the formulas

L ijk (D k ej ) bi , La

bk (D k eb) ba ,

C ijc (D c ej ) bi , C abc (D c eb) ba .

Following the terminology from Refs. 33 and 15, we call Γ γαβ to be N -distinguished

(equivalently, a d-connection) if it preserves the ( n + m)-splitting N , i.e. the de-composition

D X α D α = X i D i + X a D a ,

holds for any vector eld X = X i ei + X a ea , under parallel transports on V .i

The torsion T α of a d-connection D is dened in standard form

T α Db α = b α + Γ γα b β . (A.10)

There are ve types of N -adapted components of T αβγ computed with respect to(A.7) and (A.8),

T ijk = L ijk − L i

kj , T abc = C abc − C acb ,

T ija

= C ija

, T abi

=∂N a

i

∂ya− La

bi,

T aji = e j N ai − e i N aj = Ω aij ,

(A.11)

i The geometrical objects dened with respect to N -adapted bases are called respectively, d -tensors,d-vectors, d-connections.




which for some physical models can be related to a complete antisymmetric ten-sor H αβγ = e [α Bβγ ] in string gravity 34,35 or to certain torsion elds in (non-)commutative gauge gravity. 15

The curvature

R αβ DΓ α

β = dΓ αβ − Γ γ

β Γ αγ ,

of a d-connection D , splits into six types of N -adapted components with respect to(A.7) and (A.8),

R αβγ δ = R i

hjk , R abjk , P i

hja , P cbja , S i jbc , S abdc ,

where

R ihjk = ek L i

hj − ej L ihk + Lm

hj L imk − Lm

hk L imj − C i ha Ωa

kj ,

R abjk = ek La

bj − ej Labk + Lc

bj Lack − Lc

bk Lacj − C abc Ωc

kj ,

R ijka = ea L i

jk − D k C i ja + C i jb T bka ,

R cbka = ea Lc

bk − D k C cba + C cbd T cka ,

R ijbc = ec C i jb − ebC i jc + C hjb C i hc − C hjc C i hb ,

R abcd = ed C abc − ec C abd + C ebc C aed − C ebd C aec .

(A.12)

Contracting, respectively, the components of (A.12), R αβ R τ αβτ , one com-

putes the h- and v-components of the Ricci d-tensor (there are four N -adaptedcomponents)

R ij Rkijk , R ia − R k

ika , Rai R baib , S ab R c

abc . (A.13)

The scalar curvature is dened by contracting the Ricci d-tensor with the inversemetric g αβ ,

←

R g αβ R αβ = gij R ij + hab S ab . (A.14)

There are two types of preferred linear connections uniquely determined by ageneric off-diagonal metric structure with n + m splitting, see g = g N h (A.5).

(i) The Levi-Civita connection = Γαβγ is by denition torsionless, T = 0 ,

and satises the metric compatibility condition , g = 0 (we shall use a left lowlabel “ | ” in order to emphasize that some geometric objects are constructed just for the Levi-Civita connection).

(ii) The canonical d-connection Γγαβ = ( L i

jk , Labk , C ijc , C abc ) is also metric com-

patible, i.e. Dg = 0 , but the torsion vanishes only on h- and v-subspaces,i.e. T ijk = 0 and T abc = 0 , for certain nontrivial values of T ija , T abi , T aji . In thispaper, we shall work only with the canonical d-connection. For simplicity, weshall omit hats on symbols and write, for simplicity, L i

jk instead of L ijk , T ija

instead of T ija and so on . . . but preserve the general symbols D and Γγ

αβ .




By a straightforward calculus with respect to N -adapted frames (A.7) and (A.8),one can verify that the requested properties for D are satised if

L ijk =

12

gir (ek gjr + ej gkr − er gjk ) ,

Labk = eb(N ak ) +

12

hac (ek hbc − hdc ebN dk − hdb ec N dk ) ,

C ijc =12

gik ec gjk ,

C abc =12

had (ec hbd + ec hcd − ed hbc ) .

(A.15)

We note that these formulas are computed for the components of the metric g =g N h (A.5) but in a similar form, using symbols with “inverse hats” we cancompute the components D of g = g N h (A.1) with respect to (A.2) and (A.3).

The Levi-Civita linear connection = Γαβγ , uniquely dened by the con-

ditions T = 0 and g = 0 , is not adapted to the distribution (A.9) and itsnonholonomic deformations. Let us parametrize the coefficients in the form

Γαβγ = L i

jk , Lajk , L i

bk , Labk , C ijb , C ajb , C ibc , C abc ,

where with respect to N -adapted bases (A.7) and (A.8)

e k (e j ) = L ijk e i + La

jk ea , e k (eb) = L ibk e i + La

bk ea ,

e b (e j ) = C ijb e i + C ajb ea , e c (eb) = C ibc e i + C abc ea .

A straightforward calculus j shows that the coefficients of the Levi-Civita connectioncan be expressed in the form

L ijk = L i

jk , Lajk = − C ijb gik hab −

12

Ωajk ,

L ibk =

1

2Ωc

jk hcb gji −1

2(δi

j δhk − gjk gih )C jhb ,

Labk = La

bk +12

(δac δb

d + hcd hab )[Lcbk − eb(N ck )] ,

C ikb = C ikb +12

Ωajk hcb gji +

12

(δij δh

k − gjk gih )C jhb ,

C ajb = −12

(δac δd

b − hcb had )[Lcdj − ed (N cj )] , C abc = C abc ,

C iab = −gij

2[Lc

aj − ea (N cj )]hcb + [ Lcbj − eb(N cj )]hca ,

(A.16)

where Ωajk are computed as in the rs formula in (A.4) but for the coefficients N cj .

j Such results were originally considered by R. Miron and M. Anastasiei for vector bundles pro-vided with N -connection and metric structures, see Ref. 33. Similar proofs hold true for anynonholonomic manifold provided with a prescribed N -connection structures.




For our purposes, it is important to state the conditions when both the Levi-Civita connection and the canonical d-connection may be dened by the same setof coefficients with respect to a xed frame of reference. Following formulas (A.15)and (A.16), we conclude that one holds the component equality Γα

βγ = Γγαβ if

Ωcjk = 0 (A.17)

(there are satised the integrability conditions and our manifold admits a foliationstructure),

C ikb = C ikb = 0 (A.18)

and

Lcaj − ea (N cj ) = 0

which, following the second formula in (A.15), is equivalent to

e k hbc − hdc ebN dk − hdb ec N dk = 0 . (A.19)

We conclude this section with the remark that if the conditions (A.17)–(A.19)hold true for the metric (A.5), the torsion coefficients (A.11) vanish. This results inrespective equalities of the coefficients of the Riemann, Ricci and Einstein tensors.

A.3. Gravity on N -anholonomic manifolds and foliations

Contracting with the inverse to the d-metric (A.5) in V , we can introduce the scalarcurvature of a d-connection D ,

←

R g αβ R αβ R + S , (A.20)

where R gij R ij and S hab S ab and compute the Einstein tensor

G αβ R αβ −12

g αβ←

R . (A.21)

In the vacuum case, G αβ = 0 , which mean that all Ricci d-tensors (A.13) vanish.The Einstein equations for the canonical d-connection Γ γ

αβ (A.15),

R αβ −12

g αβ←

R = κΥ αβ , (A.22)

are dened for a general source of matter elds and, for instance, possible stringcorrections, Υ αβ . It should be emphasized that there is a nonholonomically inducedtorsion T γ

αβ with d-torsions computed by introducing consequently the coefficientsof d-metric (A.5) into (A.15) and then into formulas (A.11). The gravitational eld

equations (A.22) can be decomposed into h- and v-components following formulas(A.13) and (A.20),

R ij −12

gij (R + S ) = Υ ij , S ab −12

hab (R + S ) = Υ ab ,

1P ai = Υ ai , − 2P ia = Υ ia .(A.23)




The vacuum equations, in terms of the Ricci tensor R αβ = gαγ R γβ , are

R ij = 0 , S ab = 0 , 1P ai = 0 , 2P ia = 0 . (A.24)

If the conditions (A.17), (A.18) and (A.19) are satised, Eqs. (A.23) and (A.24)are equivalent to those derived for the Levi-Civita connection. In such cases, thespace–time is modeled as foliated manifold with generic off-diagonal metric (A.5)if the anholonomy coefficients w γ

αβ , dened by N ai substituted in formulas (A.4),are not zero.

In string gravity the nontrivial torsion components (A.11) and source κΥ αβ canbe related to certain effective interactions with the strength (torsion)

H µνρ = e µ B νρ + e ρ Bµν + e ν Bρµ ,

of an antisymmetric eld B νρ , when

Rµν = −14

H νρµ H νλρ (A.25)

and

D λ H λµν = 0 , (A.26)

see details on string gravity, for instance, in Refs. 34 and 35. The conditions (A.25)and (A.26) are satised by the ansatz

H µνρ = Z µνρ + H µνρ = λ [H ] |gαβ | ενλρ , (A.27)

where ενλρ is completely antisymmetric and the distortion (from the Levi-Civitaconnection) and

Z µαβ cµ = e β T α − e α T β +12

(e α e β T γ )c γ ,

is dened by the torsion tensor (A.10) with coefficients (A.11). We emphasize thatour H -eld ansatz is different from those already used in string gravity when H µνρ =λ [H ] |gαβ |ενλρ . In our approach, we dene H µνρ and Z µνρ from the respectiveansatz for the H -eld and nonholonomically deformed metric, compute the torsiontensor for the canonical distinguished connection and, nally, dene the “deformed”H -eld as H µνρ = λ [H ] |gαβ |ενλρ − Z µνρ . Such space–times are both nonholonomicwith nontrivial torsion related to that in string gravity and with sources inducedby string corrections via an effective cosmological constant λ [H ], when

R αβ = −

λ2[H ]

4δα

β . (A.28)

To generate solutions for such equations, it is more convenient to work directly withthe canonical d-connection (A.15).




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Nonholonomic Ricci Flows and Running Cosmological Constant: I. 4D Taub-NUT Metrics

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