On General Solution in Topological Massive Gravity and Ricci Flows

On General Solutions in Topological Massive Gravity and Ricci Flows

On General Solutions in TopologicalMassive Gravity and Ricci Flows

Research Seminar

Sergiu I. Vacaru

Science DepartmentUniversity Al. I. Cuza, Iasi, Romania

Seminar at Department of Mathematics(host prof. Metin Gurses) Bilkent University

July 22, 2010; Ankara, Turkey

c© Sergiu I. Vacaru, July 15, 2010 1/ 24100722˙trsp˙ank˙bilkent.tex

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Subjects

I. Motivations: Topological Massive Gravity andNonholonomic Structures

II. Nonholonomic Ricci flow evolution of topologicalmassive gravity solutions

III. Nonholonomic Distributions and Massive Gravity

IV. General Solutions in Massive Gravity

a. Separation of Einstein eqs fordistinguished connections

b. Integration of (non) holonomic Einstein eqs

V. Extracting topological massive black ellipsoids

VI. Conclusions and Perspectives


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Motivations:

Concept/ coefficient form of N–connections: E. Cartan, A. Kawaguchi, C. Ehresmann ....

1. M. Gurses, Killing Vector Fields in Three Dimensions: A Method to Solve MassiveGravity Field Equations, arXiv: 1001.10392. A. Ulas Ozgur Kisised, O. Sarioglu and B. Tekin, Cotton Flow, arXiv: 0803.1603v23. David D. K. Chow, C. N. Pope and E. Sezgin, Classification of solutions in topologicallymassive gravity, CQG 27 (2010) 1050014. David D. K. Chow, C. N. Pope and E. Sezgin, Kundt spacetimes as solutions of topolog-ically massive gravity, CQG 27 (2010) 105002

5. S. Vacaru, On General Solutions in Einstein Gravity, accepted: Int. J. Geom. Meth.Mod. Phys. (IJGMMP) 8 N1 (2011); arXiv: 0909.3949v16. S. Vacaru, On General Solutions in Einstein and High Dimensional Gravity, Int. J. Theor.Phys. 49 (2010) 884-913; arXiv: 0909.3949v47. S. Vacaru, Curve Flows and Solitonic Hierarchies Generated by Einstein Metrics, ActaApplicandae Mathematicae [ACAP] 110 (2010) 73-107; arXiv: 0810.07078. S. Vacaru, Parametric Nonholonomic Frame Transforms and Exact Solutions in Gravity,Int. J. Geom. Meth. Mod. Phys. (IJGMMP) 4 (2007) 1285-1334; arXiv: 0704.3986

Topological Massive Gravity and Nonholonomic Strs

TMG equations found by Deser, Jackiw and Templeton(DJT). For nontrivial cosmological constant,

Eαβ + µ−1Cα

β = λδαβ

The Bergshoeff–Hohm–Townsend (BHT) theorypreserving parity (more sophisticate) can be also solved”almost” in general form.


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Nonholonomic Structures and Exact Solutions inTopological Massive Gravity

3-d Levi–Civita topological massive configs

Eαβ

+ µ−1C αβ

= λδαβ,

Eαβ

= Rαβ− 1

2δα

β[3]R,

C αβ =1

2√

g

(εατ γ∇τR

βγ + εβτ γ∇τR

αγ

)

for the Levi–Civita connection ∇ = pΓγ

αβ.

For metrics g,

ds2 = ± (dx1

)2+ gαβ(x2, y3, y4)duαduβ

= gαβ(uα)duαduβ

coordinates uα = (xi, ya) = (x1, x2, y3, y4) = (x1, uα);

i, j, ...k = 1, 2; a, b... = 3, 4; α, β = 2, 3, 4

4-d nonholonomic gravitational configurations

∇g = 0, torsion ∇T = 0.

Another class of linear connections gD = g Γγαβ,

gD = g∇ + gZ

with distortion/torsion gZ completely determined by g,when gDg = 0.


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Consider metrics g of type

ds2 = gi(xk)(dxi)2 + ω2(xk, yb)ha(x

k, y3)(dya + Na

j (xk, y3))2

(1)

= g1(x1, x2)(dx1)2 + g2(x

1, x2)(dx2)2 + ω2(x1, x2, y3, y4)×[h3(x

1, x2, y3)(dy3 + w1(x

1, x2, y3)dx1 + w2(x1, x2, y3)dx2)2

+h4(x1, x2, y3)

(dy4 + n1(x

1, x2, y3)dx1 + n2(x1, x2, y3)dx2)2

]

for N 3i = wi and N 4

i = ni.

Remark: Reduction 4-d →3-d

Source: Υαβ =

[vΛ(xk, y3)δi

k,hΛ(xk)δa

v

]

Theorem 1: The Einstein equations for gD,

gRαβ − 1

2gαβ

gsR = Υα

β

for the metrics g (1) transform into an exactly integrablesystem of partial differential equations

gR11 = gR2

2 = hΛ(xk) (2)gR3

3 = gR44 = vΛ(xk, y3)

Einstein gravity solutions with g∇ are extracted byconstraining gZ = 0.


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Effective 4–d topological theory

Eαβ + µ−1Cα

β = λδαβ, (3)

Gαβ = Rα

β − 1

2δα

β sR,

Cαβ =1

2√

g

(εατγDτR

βγ + εβτγDτR

αγ

)

Lemma: Cαβ(uγ) = ( hC(uγ) δi

j,vC (uγ)δa

b)

Theorem 2: The solutions of (3) can be extracted from(2) for

hΛ(xk) = λ− µ−1 hC(xk),vΛ(xk) = λ− µ−1 vC(xk).

Theorem: Nonholonomic Cotton flows are determined bysolutions:

∂χgij(χ) =

(1− 1

4δl

l

)εikj√gh

Dk( hΛ(χ)) = Cij(χ),

∂χhab(χ) =

(1− 1

4δe

e

)εacb√hg

Dc( vΛ(χ)) = Cab(χ)


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II. Black Ellipses in TMG

Nontrivial effective cosmological source

Consider a diagonal 4-d metricελg = −dξ ⊗ dξ − r2(ξ) dϑ⊗ dϑ−

r2(ξ) sin2 ϑ dϕ⊗ dϕ + λ$2(ξ) dt⊗ dt,

where coordinates and nontrivial metric coefficients

x1 = ξ, x2 = ϑ, y3 = ϕ, y4 = t,

g1 = −1, g2 = −r2(ξ), h3 = −r2(ξ) sin2 ϑ, h4 = λ$2(ξ)

forξ =

∫dr

∣∣∣∣1−2µ

r+ ε

(1

r2+

λ

34κ

2 r2

)∣∣∣∣1/2

,

λ$2(r) = 1− 2µ

r+ ε

(1

r2− λ

34κ

2 r2

),

where 4κ2 = 1/M 2

∗ is the 4–dimensional Newton’sconstant, λ = ε λ is a positive cosmological constant andµ1 is the ADM mass (review on Schwarzschild–de Sitterblack holes in (4 + n1)–dimensions, for n1 = 1, 2, ...)

For ε → 0 and µ taken to be a point mass (for astationary locally anisotropic model, µ = µ

0+ ε

µ1(ξ, ϑ, ϕ) , for µ

0= const and function µ

1(ξ, ϑ, ϕ)

taken from phenomenological considerations).


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The metric εg has a true singularity at r = 0 and theequation

1− 2µ0

r+

1

3λ 4κ

2 r2 = 0

has three solutions for not small r (when we can neglectthe term 1/r2) corresponding to three horizons.

3–d configurations are generated if we do notconsider coordinate ϑ


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Off–diagonal Configurations

We consider the ansatzλg = −eφ(ξ,ϑ) dξ ⊗ dξ − eφ(ξ,ϑ) dϑ⊗ dϑ

+h3 (ξ, ϑ, ϕ) δϕ⊗ δϕ + h4 (ξ, ϑ, ϕ) δt⊗ δt,

δϕ = dϕ + w1 (ξ, ϑ, ϕ) dξ + w2 (ξ, ϑ, ϕ) dϑ,

δt = dt + n1 (ξ, ϑ, ϕ) dξ + n2 (ξ, ϑ, ϕ) dϑ,

for h3 = η3(ξ, ϑ, ϕ)r2(ξ) sin2 ϑ, h4 = η4(ξ, ϑ, ϕ) λ$2(ξ),

where the coefficients (solutions of TMG) satisfy

φ••(ξ, ϑ) + φ′′(ξ, ϑ) = 2 hΛ(ξ, ϑ);

h3 = ± (φ∗)2

4 vΛ(ξ, ϑ)e−2 0φ(ξ,ϑ),

h4 = ∓ 1

4 vΛ(ξ, ϑ)e2(φ− 0φ(ξ,ϑ));

wi = −∂iφ/φ∗;

ni = 1ni(ξ, ϑ) + 2ni(ξ, ϑ)

∫(φ∗)2 e−2(φ− 0φ(ξ,ϑ))dϕ,

= 1ni(ξ, ϑ) + 2ni(ξ, ϑ)

∫e−4φ(h∗4)

2

h4dϕ, n∗i 6= 0;

= 1ni(ξ, ϑ), n∗i = 0;

for any nonzero ha and h∗a and (integrating) functions1ni(ξ, ϑ), 2ni(ξ, ϑ), generating function φ(ξ, ϑ, ϕ), and0φ(ξ, ϑ) to be determined from boundary cond.


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Ellipsoid de Sitter configurations and TMG

rotλ g = −eφ(ξ,ϑ) (dξ ⊗ dξ + dϑ⊗ dϑ)

−h20

[(√|q|)∗

]2

1 + ε

1

(√|q|)∗

s√

|q|

∗ δϕ⊗ δϕ

+(q + εs

)δt⊗ δt,

δϕ = dϕ + w1dξ + w2dϑ, δt = dt + n1dξ + n2dϑ.

q = 1− 2 1µ(r, ϑ, ϕ)

r, s =

q0(r)

4µ20

sin(ω0ϕ + ϕ0),

with 1µ(r, ϑ, ϕ) = µ + ε(r−2 − λ 4κ

2 r2/3)/2, chosen

to generate an anisotropic rotoid configuration for thesmaller ”horizon” (when h4 = 0), for a q

0(r),

r+ '2 1µ

1 + εq0(r)

4µ20

sin(ω0ϕ + ϕ0).

Levi–Civita: φ(r, ϕ, ϑ) = ln |h∗4/√|h3h4|| must be any

function 2e2φφ = vΛ.

N–connection coefficients: w1w2

(ln |w1

w2|)∗

= w•2 − w′

1 for

w∗i 6= 0; w•

2 − w′1 = 0 for w∗

i = 0; and take ni = 1ni(xk)

for 1n′1(xk)− 1n•2(x

k) = 0.


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Rotoids and solitonic distributions

On a N–anholonomic spacetime V defined by a rotoidd–metric rotg, we can consider a static 3–d solitonicdistribution η(ξ, ϑ, ϕ) as a solution of solitonic equation1

η•• + ε(η′ + 6η η∗ + η∗∗∗)∗ = 0, ε = ±1.

It is possible to define a nonholonomic transform from rotgto a d–metric rot

st g determining a stationary metric for arotoid in solitonic background to be reduces for 3–d TMG:

rotst g = −eψ (dξ ⊗ dξ + dϑ⊗ dϑ)

−4[(√|ηq|)∗

]2[1 + ε

1

(√|ηq|)∗

(s√|ηq|

)∗]δϕ⊗ δϕ

+η (q + εs) δt⊗ δt,

δϕ = dϕ + w1dξ + w2dϑ, δt = dt + 1n1dξ + 1n2dϑ,

N–connection coefficients are taken the same as inprevious solution.

Limit ε → 0, a nonholonomic embedding of theSchwarzschild solution into a solitonic vacuum, whichresults in a vacuum solution of the Einstein gravity definedby a stationary generic off–diagonal metric and datainduced by TMG.

1as a matter of principle, we can consider that η is a solution of any three dimensional solitonic and/ or other nonlinear wave equations


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”Fictive” Vacuum Solutions

Solutions of Rαβ = 0 by d-metrics with stationarycoefficients subjected to conditions

ψ••(ξ, ϑ) + ψ′′(ξ, ϑ) = 0;

h3 = ±e−2 0φ(h∗4)2

h4for given h4(ξ, ϑ, ϕ), φ = 0φ = const;

wi = wi(ξ, ϑ, ϕ) are any functions if vΛ = 0;

ni = 1ni(ξ, ϑ) + 2ni(ξ, ϑ)

∫(h∗4)

2 |h4|−5/2dv, n∗i 6= 0;

= 1ni(ξ, ϑ), n∗i = 0,

for h4 = η(ξ, ϑ, ϕ) [q(ξ, ϑ, ϕ) + εs(ξ, ϑ, ϕ)] . In the limitε → 0, we get a so–called Schwarzschild black holesolution mapped nonholonomically on a N–anholonomic(pseudo) Riemannian spacetime, or constrained to TMG.


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III. Nonholonom. Distr. & Gravity

Nonholonomic Manifolds/Bundles

Geometrization of nonholonomic mechanics =⇒concept of nonholonomic manifold V = (M,D)smooth & orientable M , non–integrable distribution DN–anholonomic manifold enabled with nonlinearconnection (N–connection) structure

N : TV = hV ⊕ vV, D = hV, integrable vV

Examples: 1) V (semi/pseudo) Riemannian manifold2) V = E(M), or = TM, is a vector, or tangent, bundle

Local coordinates u = (x, y), or uα = (xi, ya)

h–indices: i, j, ... = 1, 2, 3 and v–indices: a, b, ... = 4, 5

Local coefficients N = Nai (u) dxi ⊗ ∂

∂ya

Particular case: Nai (u) = Γa

bi(x)yb


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∃ N–adapted (co) frames (vielbeins)

eν =

(ei =

∂

∂xi−Na

i (u)∂

∂ya, ea =

∂

∂ya

),

eµ =(ei = dxi, ea = dya + Na

i (u)dxi).

Nonholonomy: [eα, eβ] = eαeβ − eβeα = W γαβeγ

Anholonomy coefficients W bia = ∂aN

bi and W a

ji = Ωaij,

N–connection curvature Ωaij = ej (Na

i )− ei

(Na

j

)

holonomic/ integrable case W γαβ = 0.

Distinguished Connections:A d–connection D on V is a linear connection preservingunder parallelism the N–connection splitting.

Locally, D ⇒ Γγαβ =

(Li

jk, Labk, C

ijc, C

abc

),

hD = (Lijk, L

abk) and vD = (C i

jc, Cabc)

Distinguished objects: d–objects, d–tensors

d–vectors X = hX + vX = hX + vX

N–adapted geometric constructions.

Levi–Civita connection ∇ is not N–adapted


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Torsion and Curvature d–tensors

Torsion of D =( hD, vD), by the d–tensor field

T(X,Y) + DXY −DYX− [X,Y],

T(X,Y) = hT(hX, hY), hT(hX, vY), ..., vTvX, vY)N–adapted: T = Tα

βγ = (T ijk, T

ija, T

ajk, T

bja, T

bca)

Curvature: R(X,Y) + DXDY −DYDX−D[X,Y],

R = Rαβγδ =

(Ri

hjk,Rabjk,R

ihja,R

cbja,R

ihba,R

cbea

)Ricci: Ric + Rβγ = Rα

βγα = (Rij, Ria, Rai, Rab)Distinguished metric (d–metric)

g = hg⊕Nvg = [ hg, vg]

g = gij(x, y) ei ⊗ ej + hab(x, y) ea ⊗ eb

Coordinate co–frames: g = gαβ

(u) duα ⊗ duβ

gαβ

=

[gij + Na

i N bjhab N e

j hae

N ei hbe hab

]

for N ej (u) = N e

j (x, y)

Data for generic off–diagonal metrics and N–connections

g = gαβ =[gij, hab, N

bj

]


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Levi–Civita and canonical d–connection

Levi–Civita connection ∇ = gΓγαβ,

T αβγ = 0 and ∇g =0

Canonical d–connection D = gΓγαβ

Dg =0 and hT(hX, hY ) = 0, vT(vX, vY ) = 0

gΓγαβ = gΓγ

αβ + gZγαβ

Distortion tensor gZγαβ completely defined by g

N–adapted coefficients Γγαβ =

(Li

jk, Labk, C

ijc, C

abc

),

Lijk =

1

2gir (ekgjr + ejgkr − ergjk) ,

Labk = eb(N

ak ) +

1

2hac

(ekhbc − hdc ebN

dk − hdb ecN

dk

),

C ijc =

1

2gikecgjk, Ca

bc =1

2had (echbd + echcd − edhbc) .

Nontrivial torsion Tγαβ ∼ gZγ

αβ

T ija = C i

jb, Taji = −Ωa

ji, Tcaj = Lc

aj − ea(Ncj )

If Tγαβ = 0

gΓγαβ = gΓγ

αβ

even, in general, ∇ 6= D


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IV. General Sols in Topolog. Massive Grav.

a. Separation of Einstein eqs for d–connections

Einstein eqs for gβδ rewritten equivalently using thecanonical d–connection,

R βδ − 1

2gβδ

sR = Υβδ,

Lcaj = ea(N

cj ), C i

jb = 0, Ωaji = 0,

Ricci tensor R βδ is for Γγαβ,

sR = gβδR βδ for D → ∇,also contributions of Cotton tensor.

For instance, (2+2) splitting, (uα = (xk, t, y4 = y),

ansatz with Killing symmetry ∂/∂y4,

Kg = g1(xk)dx1 ⊗ dx1 + g2(x

k)dx2 ⊗ dx2

+h3(xk, t)e3⊗e3 + h4(x

k, t)e4⊗e4

where for N 3i = wi(x

k, t), N 4i = ni(x

k, t),

e3 = dt + wi(xk, t)dxi,

e4 = dy4 + ni(xk, t)dxi


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Theorem 1 (Separation of Eqs)

The Einstein eqs for ansatz Kg and D are:

−R11 = −R2

2 =1

2g1g2(4)

[g••2 − g•1g

•2

2g1− (g•2)

2

2g2+ g′′1 −

g′1g′2

2g2− (g′1)

2

2g1

]= Υ4(x

k),

−R33 = −R4

4 =

1

2h3h4

[h∗∗4 − (h∗4)

2

2h4− h∗3h

∗4

2h3

]= Υ2(x

k, t), (5)

R3k =wk

2h4

[h∗∗4 − (h∗4)

2

2h4− h∗3h

∗4

2h3

](6)

+h∗44h4

(∂kh3

h3+

∂kh4

h4

)− ∂kh

∗4

2h4= 0,

R4k =h4

2h3n∗∗k +

(h4

h3h∗3 −

3

2h∗4

)n∗k2h3

= 0, (7)

where a• = ∂a/∂x1, a′ = ∂a/∂x2, a∗ = ∂a/∂t.


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b)Integration of (non)holonomic Einstein eq

Theorem 2 (Integral Varieties)

Eqs(4)–(7), written for h∗3,4 6= 0 and Υ2,4 6= 0,

ψ + ψ′′ = 2Υ4(xk) (8)

h∗4 = 2h3h4Υ2(xi, t)/φ∗ (9)

βwi + αi = 0 (10)

n∗∗i + γn∗i = 0 (11)

αi = h∗4∂iφ, β = h∗4 φ∗, φ = ln | h∗4√|h3h4|

|, γ =(ln |h4|3/2

|h3|)∗

General solution: g1 = g2 = eψ(xk)

h4 = 0h4(xk)± 2

∫(exp[2 φ(xk, t)])∗

Υ2dt,

h3 = ± 1

4

[√|h∗4(xi, t)|

]2

exp[−2 φ(xk, t)]

wi = −∂iφ/φ∗

nk = 1nk

(xi

)+ 2nk

(xi

) ∫[h3/(

√|h4|)3]dt

Levi–Civita (LC) conditions:

w∗i = ei ln |h4|, ekwi = eiwk, n∗i = 0, ∂ink = ∂kni


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General Commutative Solutions

Non–Killing metrics

Dependence on 4th coordinate via ω2(xj, y3 = t, y4 = y)

g = gi(xk)dxi ⊗ dxi + ω2(xj, t, y)ha(x

k, t)ea⊗ea,

e3 = dy3 + wi(xk, t)dxi, e4 = dy4 + ni(x

k, t)dxi,

ekω = ∂kω + wkω∗ + nk∂ω/∂y = 0,

ω2 = 1 results in solutions with Killing symmetry.

Sketch proof: Recompute the Ricci h–v and v–componentsand get zero distortion contribution for v– d’Allambertconditions, and/or additional sources determined by ω.

N–deformations and exact solutions

’Polarizations’ ηα and ηai , nonholonomic deformations,

g = [ gi,ha,

Nak ] → ηg = [ gi, ha, N

ak ].

Deformations of frame/metric/fundamental geometricstructures are more general than moving frame method:

ηg = ηi(xk, t) gi(x

k, t)dxi ⊗ dxi

+ ηa(xk, t) ha(x

k, t)ea⊗ea,

e3 = dt + η3i (x

k, t) wi(xk, t)dxi,

e4 = dy4 + η4i (x

k, t) ni(xk, t)dxi.


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V. Eqs for Nonhol. Ricci Flows

1. Families fundamental Einstin/ Cotton/ Lagrange/Finsler L(x, y, χ)/ F (x, y, χ), on TM, or V, real χ,

g = gi(xk, χ)dxi ⊗ dxi

+ω2(xj, t, y, χ)ha(xk, t, χ)ea⊗ea,

e3 = dy3 + wi(xk, t, χ)dxi, e4 = dy4 + ni(x

k, t, χ)dxi,

2. For the previous works, gαβ are solutions of Einsteineqs for a d–connection, Rαβ = λgαβ, for instance,

R βδ − 1

2gβδ

sR = Υβδ,

LC–cond. Lcaj = ea(N

cj ), C i

jb = 0, Ωaji = 0

3. This seminar, gαβ(χ) are solutions of∂gαβ

∂χ = −2Cαβ

4. Canonical d–connection evolution (postulates)∂

∂χgi = −2

(Rii − λgi

)− hc

∂

∂χ(N c

i )2,

∂

∂χha = −2

(Raa − λha

),

Rαβ = 0, for α 6= β, (R− λg) → µ−1C

In general, Rαβ 6= Rβα resulting in nonsymmetric metrics.


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N–connection evolution of massive gravity

Include exact solutions for Einstein (non) holonomicmanifolds into nonholonomic Ricci/ -Cotton flow eqs?

Examples with nontrivial N ci (χ):

a) ∂∂χgi = ∂

∂χha = 0, Υ2 = Υ4 = 0

constraints:∂

∂χ[hc (N c

i )2] = 0, (12)

gij and hab are d–metrics not depending on χfamilies of N 3

i = wi(xk, t, χ), N 4

i = ni(xk, t, χ) with (12)

g = gi(xk)dxi ⊗ dxi + ω2(xj, t, y, χ)ha(x

k, t)ea⊗ea,

e3 = dy3 + wi(xk, t, χ)dxi, wi(x

k, t, χ) → wi(xk, t), (15)

e4 = dy4 + ni(xk, t, χ)dxi,

Use Theorem 2, h∗3,4 6= 0, Υ2,4 → λ,

ψ + ψ′′ = 2λ (13)

h∗4 = 2h3h4λ/φ∗ (14)

βwi + αi = 0 (15)

n∗∗i (λ) + γ n∗i (λ) = 0 (16)

αi = h∗4∂iφ, β = h∗4 φ∗, φ = ln | h∗4√|h3h4|

|,

γ =

(ln|h4|3/2|h3|

)∗


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Solutions for N–con. Ricci flows

g1 = g2 = eψ(xk), wi = −∂iφ/φ∗,

h4 = 0h4(xk)± 2

λexp[2 φ(xk, t)],

h3 = ± 1

4

[√|h∗4(xi, t)|

]2

exp[−2 φ(xk, t)]

nk(χ) = 1nk(xi, χ) + 2nk(x

i, χ)

∫[h3/(

√|h4|)3]dt

ekω(χ) = ∂kω(χ) + wkω∗(χ) + nk(χ)∂ω(χ)/∂y = 0

LC–conditions: w∗i = ei ln |h4|, ekwi = eiwk,

n∗i (χ) = 0, ∂ink(χ) = ∂kni(χ)

b) We can consider nontrivial Υ2, Υ4 imposing constr.(12)

R11 = R2

2 = −Υ4(xk, χ), R3

3 = R44 = −Υ2(x

k, t, χ)∂

∂χgi = 2(Υ4 + λ)gi,

∂

∂χha = 2(Υ2 + λ)ha

ekω(χ) = ∂kω(χ) + wk(χ)ω∗(χ) + nk(χ)∂ω(χ)/∂y = 0

Conclusion: Exact solutions for Einstein–Lagrange/–Finsler, massive gravity models, both with trivial andnontrivial matter sources, can be generalized tononholonomic Ricci flows, and re–defined for Cotton flows.


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VI. Conclusions and Perspectives

1. ”Almost” any solution of Generalized Einstein eqs withsources, for gα′β′, via frame/ coordinate transform

eα = eα′α(xi, ya)eα′, gαβ = eα′

αeβ′βgα′β′,

can be expressed in a form gαβ =∣∣∣∣∣∣∣∣

g1 + ω2(w 21 h3 + ω2(n 2

1 h4) ω2(w1w2h3 + n1n2h4) ω2 w1h3 ω2 n1h4

ω2(w1w2h3 + n1n2h4) g2 + ω2(w 22 h3 + n 2

2 h4) ω2 w2h3 ω2 n2h4

ω2 w1h3 ω2 w2h3 h3 0ω2 n1h4 ω2 n2h4 0 h4

∣∣∣∣∣∣∣∣

2. Exact solutions Related to Nonholonomic Ricci/ – Cottonflows: physical applications in classical and quantumgravity.

3. Black ellipsoid/torus/solitonic/ pp–waves in massivetopological gravity (TMG)

4. Cosmological solutions in TMG

5. Imbedding of M. Gurses solutions into nonholonomicconfigurations and generalizations

6. Solitonic hierarchies in TMG

7. Algebraic classification and nonholonomic deforms TMG

8. Deformation quantization of TMG

9. Quantum Einstein–Finsler gravity and TMG


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On General Solution in Topological Massive Gravity and Ricci Flows