Fractional Nonholonomic Ricci Flows: Sergiu I. Vacaru
Fractional Nonholonomic Ricci Flows: Sergiu I. Vacaru
Fractional Nonholonomic Ricci Flows: Sergiu I. Vacaru
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Fractional Nonholonomic Ricci Flows
Sergiu I. Vacaru
University Al. I. Cuza Iasi, Science Department,
54 Lascar Catargi street, Iasi, Romania, 700107
April 5, 2010
Abstract
We formulate the fractional Ricci ow theory for (pseudo) Rie-
mannian geometries enabled with nonholonomic distributions dening
fractional integrodierential structures, for noninteger dimensions.
There are constructed fractional analogs of Perelmans functionals and
derived the corresponding fractional evolution (Hamiltons) equations.
We apply in fractional calculus the nonlinear connection formalism
originally elaborated in Finsler geometry and generalizations and re-
cently applied to classical and quantum gravity theories. There are
also analyzed the fractional operators for the entropy and fundamen-
tal thermodynamic values.
Keywords: fractional Ricci ows, nonholonomic manifolds, non-
linear connections.
2000 MSC: 26A33, 53C44, 53C99, 83E99
PACS: 45.10Hj, 05.30.Pr, 02.90.+p, 04.90.+e, 02.30Xx
Contents
1 Introduction 2
1.1 Basic concepts and ideas . . . . . . . . . . . . . . . . . . . . . 2
1.2 Remarks on notations and proofs . . . . . . . . . . . . . . . . 4
2 Nonholonomic Manifolds with Fractional Distributions 6
2.1 Fractional (co) tangent bundles . . . . . . . . . . . . . . . . . 6
2.2 Fundamental geometric objects on fractional manifolds . . . . 8
sergiu.vacaru@uaic.ro, Sergiu.Vacaru@gmail.com
1
2.2.1 Nconnections for fractional nonholonomic manifolds . 8
2.2.2 Nadapted fractional metrics . . . . . . . . . . . . . . 10
2.2.3 Distinguished fractional connections . . . . . . . . . . 11
2.2.4 The fractional canonical dconnection and LeviCivita
connection . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Perelman Type Fractional Functionals 15
3.1 On (non) holonomic Ricci ows . . . . . . . . . . . . . . . . . 15
3.2 Fractional functionals for nonholonomic Ricci ows . . . . . . 16
4 Fractional Hamiltons Evolution Equations 18
4.1 Main Theorems on fractional Ricci ows . . . . . . . . . . . . 19
4.2 Statistical Analogy for Fractional Ricci Flows . . . . . . . . . 22
A Fractional IntegroDierential Calculus on R
n
23
A.1 RiemannLiouville and Caputo fractional derivatives . . . . . 24
A.1.1 Left and right fractional RL derivatives . . . . . . . . 24
A.1.2 Fractional Caputo derivatives . . . . . . . . . . . . . . 25
A.2 Vector operations and noninteger dierential forms . . . . . 25
A.2.1 Fractional integral . . . . . . . . . . . . . . . . . . . . 25
A.2.2 Denition of fractional vector operations . . . . . . . . 26
A.2.3 Fractional dierential forms . . . . . . . . . . . . . . . 27
1 Introduction
The purpose of this paper is to generalize the Ricci ow theory [1, 2, 3, 4]
(see [5, 6, 7] for reviews of results and methods) to fractional evolution of
geometries of noninteger dimensions. The most important achievement of
this theory was the proof of W. Thurstons Geometrization Conjecture by
Grisha Perelman [2, 3, 4]. The main results on Ricci ow evolution were
proved, in the bulk, for (pseudo) Riemannian and K ahler geometries. We
show that similar results follow for geometries with noninteger dimensions
when a fractional dierential and integral calculus is corresponding encoded
into nonholonomic frame structures and adapted geometric objects.
1.1 Basic concepts and ideas
In a series of works (see, for instance, [8, 9, 10] and references therein), we
proved that nonholonomic constraints on Ricci ow evolution may transform
(pseudo) Riemannian metrics and LeviCivita connections into (in general)
2
nonsymmetric metrics and (for instance) LagrangeFinsler type linear con-
nections
1
. Such geometries can be modeled by nonholonomic distributions
and frames with associated nonlinear connection (Nconnection) structures
on (pseudo) Riemannian spaces and/or generalizations.
2
If nonholonomic distributions on a manifold M contain corresponding
integrodierential relations, we can model fractional geometries (for spaces
with derivatives and integrals of noninteger order). The rst example of
derivative of order = 1/2 has been described by Leibnitz in 1695, see
historical remarks in [16]. The theory of fractional calculus with derivatives
and integrals of noninteger order goes back to Leibniz, Louville, Grunwald,
Letnikov and Riemann [17, 18, 19, 20]. Derivatives and integrals of frac-
tional order, and fractional integrodierential equations, have found many
applications in physics (for example, see monographs [21, 22] and papers
[23, 24, 25, 26]).
We consider that the question if analogous of Thurston (in particular,
Poincare) Conjecture can be formulated (and may be proven ?) for some
spaces with fractional dimension is of fundamental importance in modern
mathematics and physics. As a rst step, the goal of this paper is to for-
mulate a fractional version of the HamiltonPerelman theory of Ricci ows.
On Perelmans functionals, we shall follow the methods elaborated in Sec-
tions 1-5 of Ref. [2] but generalized for fractional nonholonomic manifolds
by developing certain constructions from our works on nonholonomic Ricci
ow evolution [8, 9].
We dene a fractional nonholonomic manifold (equivalently, space)
V
to be given by a quadruple (V,
N,
d,
I on V. For such
fractional nonholonomic geometries (including as particular cases, for in-
stance, (pseudo
3
) Riemannian and Finsler geometries), we shall develop the
approach to encode a corresponding fractional integrodierential calculus
adapted to nonholonomic distributions. Perhaps, the simplest way is to fol-
1
see reviews [11, 12] and monograph [13], and references therein, on modern develop-
ments and applications in modern physics of the geometry of nonholonomic manifolds and
LagrangeFinsler spaces [14, 15]
2
There are used also equivalent terms like anholonomic and nonintegrable manifolds.
A nonholonomic manifold is dened by a pair (V, N), where V is a manifold and N is a
nonintegrable distribution on V.
3
mathematicians use the term semi
3
low locally a fractional vector calculus with combined RiemannLiouville
and Caputo derivatives as in [24] resulting in a selfconsistent fractional
generalization of integral operations (and corresponding fractional Gausss,
Stokes, Greens etc integral theorems which, in our approach, are crucially
important for constructing fractional Perelmans functionals).
4
The article is organized as follows: In section 2, we provide a brief in-
troduction into the geometry of fractional nonholonomic manifolds. Grisha
Perelmans functional approach to Ricci ow theory is generalized for frac-
tional nonholonomic manifolds in section 3. We derive the fractional non-
holonomic evolution equations in section 4. A statistical interpretation of
fractional nonholonomic spaces and Ricci ows is proposed. Formulas for
fractional dierential and integral calculus are summarized in Appendix.
1.2 Remarks on notations and proofs
1. We shall elaborate for fractional nonholonomic spaces a system of
notations unifying that for the nonholonomic manifolds and bundles
[12, 8, 9] and fractional integrodierential calculus [17, 18, 19, 20, 24]
(we consider the reader to be familiar with the results of such works).
For geometric objects spaces with nonholonomic distributions, we shall
use boldface symbols like V, N etc and put an up label for fractional
generalizations, for instance, of operators
d,
, or
D, or
D. We
shall put a fractional label on the left, like
A, if that will result
in a more compact system of notations.
2. There will be also used up and low labels for some canonical geo-
metric objects/operators, for instance, for the horizontal (h) and verti-
cal (v) of a distinguished linear connection D = (
h
D,
v
D) and/or its
fractional generalization
D= (
h
D,
v
D). Splitting of space dimension
4
There were formulated various approaches to fractional dierential and integral cal-
culus; on nonholonomic manifolds, some of them can be related via nonholonomic trans-
forms/deformations of geometric structures. In this paper, we elaborate a formalism with
Caputo fractional derivative (which gives zero acting on constants) adapted to nonlin-
ear connections preserving a number of similarities with an unied covariant calculus for
(pseudo) Riemannian manifolds and LagrangeFinsler geometries [11, 12, 13].
4
dimV =n+m is considered for a nonholonomic distribution (dening
a nonlinear connection, Nconnection, structure) N : TV =hVvV,
where is the Whitney sum, dim(hV) =2n and dim(vV) =2m. In
some important particular cases, we can consider V to be a (pseudo)
Riemannian manifold enabled with a nonholonomic distribution N in-
duced by N, or V = E/TM for a vector/tangent bundle (E, , M) /
(TM, , M) on a manifold M, dimM = n, dimE = n+m / dimTM =
2n, with being a corresponding surjective projection. Indices of lo-
cal coordinates on a point u on a open chart U for an atlas {U}
covering V are split in the form u
= (x
i
, y
a
), (or in brief u = (x, y)),
where the general Greek indices , , , ... split correspondingly into h
indices i, j, k, ... = 1, 2, ..., n and vindices a, b, c, ... = n+1, n+2, ..., m.
There are possible various types of transforms of local frames and
coframes, e
= (e
j
, e
b
) and e
= (e
j
, e
b
) (in particular, coordinates
with primed, underlined indices etc), when, for instance, e
=
e
(u)e
or e
= e
(u)e
, e
= (
j
= /x
j
,
a
= /y
a
)
for the Einsteins summation rule on indices being accepted. Ge-
ometric objects on V, for instance, tensors, connections etc can be
adapted to a Nconnection structure and dened by symbols with co-
ecients on the right, running corresponding values with respect to N
adapted bases (preserving a chosen hvdecomposition), for instance,
R = {R
= {R
j
klm
, R
b
klm
, R
b
klc
, ...}}.
3. Generalizing correspondingly a fractional integrodierential calculus
from [24] to nonholonomic manifolds, we can elaborate a formal (ab-
stract) analogy with the integer case but with certain modications
of the rules of local dierentiation and mixed nonholonomic integral
dierential rules. The fractional spaces and geometric objects will
be enabled with an up label in the form
V,
D,
etc.
4. Following the abstract fractional Nadapted calculus, the proofs of the-
orems became very similar to those given in Nadapted form [12, 8, 9],
which was used for a nonholonomic generalization of the Ricci ow
theory [1, 2, 5, 6, 7]. In this paper, we sketch proofs using the above
mentioned formal analogy between Nadapted fractional and integer
geometric constructions. Proofs of the fractional integrodierential
theorems related to Ricci ow evolution became very sophisticate if
we do apply the formalism of nonholonomic distributions, do not in-
troduce nonlinear connections and do not apply certain methods from
the geometry of nonholonomic manifolds.
5
2 Nonholonomic Manifolds with Fractional Distri-
butions
The fractional dierential calculus for at spaces elaborated in [24, 25],
and outlined in Appendix A, is extended for nonholonomic manifolds. For
simplicity, such fractional manifolds can be modelled as real (pseudo) Rie-
mannian spaces enabled with nonholonomic distributions containing such
integrodierential relations when the fractional calculus on curved spaces
with nonintegrable constraints is elaborated in a form maximally similar to
integer dimensions.
2.1 Fractional (co) tangent bundles
For the integer dierential calculus, the tangent bundle TM over a
manifold can be constructed for a given local dierential structure with
standard partial derivatives
i
. Such an approach can be generalized to a
fractional case when instead of
i
the dierential structure is substituted,
for instance, by the left Caputo derivatives
1
x
i
i
of type (A.1) for every
local coordinate x
i
on a local cart X on M.
Let us review, in brief, the denition for fractional tangent bundle
TM
for (0, 1) (the symbol T is underlined in order to emphasize that we shall
associate the approach to a fractional Caputo derivative). Here we cite the
paper [28] for some similar constructions with fractional tangent spaces but
for the left fractional RL derivative. We do not follow that approach because
it is not suitable for elaborating fractional Ricci ow and gravitational mod-
els with exactly integrable evolution and, respectively, eld equations, and
a selfconsistent fractional integral calculus with simplied integral theo-
rems. For our purposes, it is more convenient to use the fractional calculus
formalism proposed in Ref. [24].
We have a fractional Caputo left contact in a point
0
x X for the
parametrized curves on M parametrized by a real parameter and
1
c,
2
c :
I M, with 0 I;
1
c(0) =
2
c(0) M if
1
x
i
i
(f
1
c)
|=0
=
1
x
i
i
(f
2
c)
|=0
holds for all analytic functions f on X. This denes a relation of
equivalence when the classes [
c]
0
x
determines the fractional left tangent
Caputo space
T
0
x
M. The corresponding fractional tangent bundle is
TM :=
0
x
T
0
x
M when the surjective projection
:
TM M acts as
[
c]
0
x
=
0
x. Such a fractional bundle space is given by a triple
_
TM,
, M
_
. For
6
simplicity, we shall write, in brief, for the total space only the symbol
TM
if that will not result in ambiguities.
Locally on M, the class [
c]
0
x
is characterized by a curve
x
i
() = x
i
(0) +
(1 +)
1
c
x
i
|=0
,
for (, ). So, the horizontal and vertical coordinates (respectively, h
and vcoordinates) on
1
(X)
T
0
x
M are
u
= (x
j
,
y
j
), where
x
i
= x
i
(0) and
y
j
=
1
(1 +)
1
c
x
i
()
|=0
.
For simplicity, we shall write instead of
u
= (x
j
,
y
j
), the local coordinates
u
= (x
j
, y
j
) both for integer and fractional tangent bundles considering
that there were chosen certain such parametrizations of local coordinate
systems by using classes of equivalence only for the left Caputo fractional
derivatives.
On
= e
(u
(1)
where the fractional local coordinate basis
=
_
j
=
1
x
j
,
b
=
1
y
b
_
(2)
is with running of indices of type j
= 1, 2, ..., n and b
= e
(u
du
, (3)
where the fractional local coordinate cobasis
du
=
_
(dx
i
, (dy
a
_
(4)
with h and vcomponents, (dx
i
and (dy
a
is inverse to e
, b
M on M. Using corresponding
denitions of fractional forms (A.4) and dubbing the for fractional dier-
entials the above constructions, we can construct the fractional cotangent
bundle
on M.
We omit details on such constructions (and possible higher order fractional
tangent/vector generalizations, fractional osculator bundles etc) in this pa-
per.
2.2 Fundamental geometric objects on fractional manifolds
Let us consider a fractional nonholonomic manifold
V dened by a
quadruple (V,
N,
d,
d is stated
by some (1) and (3) and the noninteger integral structure
I is given by rules
of type (A.2). A prime integer manifold V is of integer dimension dim
V = n + m, n 2, m 1. Local coordinates on V are labeled in the form
u = (x, y), or u
= (x
i
, y
a
), where indices i, j, ... = 1, 2, ..., n are horizontal
(h) ones and a, b, ... = 1, 2, ..., m are vertical (v) ones. For some important
examples, we have that V = TM is a tangent bundle, or V = E is a vector
bundle, on M, or V is a (semi) Riemann manifold, with prescribed local
(nonintegrable) bred structure. A nonholonomic manifold V is consid-
ered to be enabled with a nonintegrable distribution dening a nonlinear
connection as we explained in point 2 of section 1.2.
2.2.1 Nconnections for fractional nonholonomic manifolds
A nonintegrable distribution
N for
d and
I.
Denition 2.1 A nonlinear connection (Nconnection)
N is dened by a
Whitney sum of conventional h and vsubspaces, h
V and v
V,
V = h
Vv
V, (5)
8
where the fractional tangent bundle
N determined by a
N are called N
anholonomic fractional manifolds. In brief, we shall call them as fractional
spaces (geometries/manifolds). Locally, a fractional Nconnection is dened
by its coecients,
N={
N
a
i
}, stated with respect to a local coordinate ba-
sis,
N=
N
a
i
(u)(dx
i
)
a
, (6)
see formulas (2) and (4).
Nconnections are naturally considered in Finsler and Lagrange geome-
try, Einstein gravity, and various supersymmetric, noncommutative, quan-
tum generalizations in modern (super) string/brane theories and geometric
mechanics, see reviews of results in [14, 15, 11, 12, 13, 29, 30, 31].
Proposition 2.1 A Nconnection
=
_
e
j
=
j
N
a
j
a
,
e
b
=
b
_
(7)
and coframe
= [
e
j
= (dx
j
)
,
e
b
= (dy
b
)
+
N
b
k
(dx
k
)
] (8)
nonholonomic structures.
Proof. The corresponding nonholonomic integrodierential fractional
structure is induced by the left Caputo derivative (A.1) and Nconnection
coecients in (6). The nontrivial nonholonomy coecients are computed
W
a
ib
=
N
a
i
and
W
a
ij
=
a
ji
=
e
i
N
a
j
e
j
N
a
i
(where
a
ji
are
the coecients of the Nconnection curvature) for
[
e
,
e
] =
e
=
W
.
9
For simplicity, in above formulas derived for (7) and (8), we omitted under-
lying of symbols of type
e
= [
e
j
,
e
b
] even such values are determined
by fractional Caputo derivatives of type (A.1), which are underlined.
(End proof.)
2.2.2 Nadapted fractional metrics
A second fundamental geometric object on
V, a metric
g, can be de-
ned similarly to (pseudo) Riemannian spaces of integer dimension but for
a chosen fractional dierential structure.
Denition 2.2 A (fractional) metric structure
g = {
g
} is determined
on a
g =
g
(u)(du
(du
(9)
for a tensor product of fractional coordinate cobases (4).
For Nadapted constructions, it is important to introduce and prove:
Claim 2.1 Any fractional metric
g can be represented equivalently as a
distinguished metric structure (dmetric),
g = [
g
kj
,
g
cb
] , which is N
adapted to splitting (5),
g =
g
kj
(x, y)
e
k
e
j
+
g
cb
(x, y)
e
c
e
b
, (10)
where fractional Nelongated bases
e
= [
e
j
,
e
b
] are dened as in (7).
Proof. For coecients of metric (9), we consider parametrization
=
_
g
ij
=
g
ij
+
N
a
i
N
b
j
g
ab
g
ib
=
N
e
i
g
be
g
aj
=
N
e
i
g
be
g
ab
_
,
(11)
for
g
=
g
=
_
e
i
i
=
i
i
e
a
i
=
N
b
i
a
b
e
i
a
= 0 e
a
a
=
a
a
_
, e
=
_
e
i
i
=
i
i
e
b
i
=
N
b
k
k
i
e
i
a
= 0 e
a
a
=
a
a
_
,
(12)
where
i
i
is the Kronecher symbol, and dene nonholonomic frames
= e
and
e
= e
(du
,
which are Nadapted frames, respectively, of type (7) and (8). Regrouping
the coecients, we get the formula (10).
10
2.2.3 Distinguished fractional connections
Linear connections on fractional
D on
V is a
linear connection preserving under parallel transports the Whitney sum (5).
A covariant fractional calculus on nonholonomic manifolds can be devel-
oped following the formalism of fractional dierential forms. For a fractional
dconnection
, (13)
with the coecients dened with respect to (8) and (7) and parametrized
the form
=
_
L
i
jk
,
L
a
bk
,
C
i
jc
,
C
a
bc
_
.
We also consider that the absolute fractional dierential
d =
1
x
d
x
+
1
y
d
y
is a Nadapted fractional operator
d :=
e
e
dened by exterior
h- and vderivatives of type (A.3), when
1
x
d
x
:= (dx
i
)
1
x
i
=
e
j
e
j
and
1
y
d
y
:= (dy
a
)
1
x
a
=
e
b
e
b
.
Denition 2.4 The torsion of a fractional dconnection
D = {
} is
D
e
=
d
e
. (14)
Following an explicit fractional (and Nadapted) dierential form calcu-
lus with respect to (8), we prove:
Theorem 2.1 Locally, the fractional torsion
T
(14) is characterized by
its coecients (dtorsion)
T
i
jk
=
L
i
jk
L
i
kj
,
T
i
ja
=
T
i
aj
=
C
i
ja
,
T
a
ji
=
a
ji
,
T
a
bi
=
T
a
ib
=
e
b
N
a
i
L
a
bi
,
T
a
bc
=
C
a
bc
C
a
cb
. (15)
For integer , we get the same formulas as in [11, 12, 13, 14]. This
is possible if we consider on
D = {
} is
=
d
=
R
e (16)
A straightforward fractional dierential form calculus for (13) gives a
proof of
Theorem 2.2 Locally, the fractional curvature
R
(16) is characterized
by its coecients (dcurvature)
R
i
hjk
=
e
k
L
i
hj
e
j
L
i
hk
+
L
m
hj
L
i
mk
L
m
hk
L
i
mj
C
i
ha
a
kj
,
R
a
bjk
=
e
k
L
a
bj
e
j
L
a
bk
+
L
c
bj
L
a
ck
L
c
bk
L
a
cj
C
a
bc
c
kj
,
R
i
jka
=
e
a
L
i
jk
D
k
C
i
ja
+
C
i
jb
T
b
ka
, (17)
R
c
bka
=
e
a
L
c
bk
D
k
C
c
ba
+
C
c
bd
T
c
ka
,
R
i
jbc
=
e
c
C
i
jb
e
b
C
i
jc
+
C
h
jb
C
i
hc
C
h
jc
C
i
hb
,
R
a
bcd
=
e
d
C
a
bc
e
c
C
a
bd
+
C
e
bc
C
a
ed
C
e
bd
C
a
ec
.
Formulas (15) and (17) encode integrodierential nonholonomic struc-
tures modeling certain types of fractional dierential geometric models. For
integer dimensions, on vector/tangent bundles, such constructions are typ-
ical ones for LagrangeFinsler geometry [14] and various types generaliza-
tions in modern geometry and gravity [30, 13, 12, 31].
Contracting respectively the components of (17), we can prove
Proposition 2.2 The fractional Ricci tensor
Ric = {
R
}
is characterized by h- vcomponents, i.e. dtensors,
R
ij
R
k
ijk
,
R
ia
R
k
ika
,
R
ai
R
b
aib
,
R
ab
R
c
abc
. (18)
It is obvious that the fractional Ricci tensor
R
D is
s
R
g
R
=
R +
S, (19)
R =
g
ij
R
ij
,
S =
g
ab
R
ab
,
dened by a sum the h and vcomponents of (18) and contractions with the
inverse coecients to a dmetric (10).
12
Proposition 2.3 -Denition: The Einstein tensor
Ens = {
G
}
for a fractional dconnection
:=
R
1
2
s
R. (20)
Such a tensor can be used for various fractional generalizations of the
Einstein and LagrangeFinsler gravity models from [11, 12, 13, 14]. It
should be emphasized that variants of fractional Ricci and Einstein tensor
were considered in [28, 32], respectively, for generalized fractional Rimann
Finsler and Einstein spaces but with RL fractional derivatives. Techni-
cally, it is a very cumbersome task to nd solutions of such sophisticate
integrodierential equations and study possible physical implications. In
our approach, working with the left Caputo fractional derivative and by
corresponding nonholonomic transforms, we can separate the equations in
fractional equations in such a form that the resulting systems of partial dif-
ferential and integral equations can integrated exactly in very general form
similarly to the integer cases outlined for dierent models of gravity theory
in [33, 11, 12, 13] and, for nonholonomic Ricci ows and applications to
physics, in [34, 35, 35, 37, 38, 10].
2.2.4 The fractional canonical dconnection and LeviCivita con-
nection
There are an innite number of fractional dconnections
D on
V. For
applications in modern geometry and physics, a special interest present sub-
classes of such linear connections which are metric compatible with a metric
structure, i.e.
D(
g) = 0, with more special cases when
D is completely
and uniquely determined by
g and
D =
{
=
_
L
i
jk
,
L
a
bk
,
C
i
jc
,
C
a
bc
_
} which is compatible with the met-
ric structure,
D (
g) = 0, and satises the conditions
T
i
jk
= 0 and
T
a
bc
= 0.
13
Proof. It follows from explicit formulas for coecients of (10) and
L
i
jk
=
1
2
g
ir
(
e
k
g
jr
+
e
j
g
kr
e
r
g
jk
) , (21)
L
a
bk
=
e
b
(
N
a
k
) +
1
2
g
ac
_
e
k
g
bc
g
dc
e
b
N
d
k
g
db
e
c
N
d
k
_
,
C
i
jc
=
1
2
g
ik
e
c
g
jk
,
C
a
bc
=
1
2
g
ad
(
e
c
g
bd
+
e
c
g
cd
e
d
g
bc
) .
Introducing the values (21) into formulas (15) we obtain that
T
i
jk
= 0 and
T
a
bc
= 0, but
T
i
ja
,
T
a
ji
and
T
a
bi
are not zero, that the metricity conditions
are satised in component form.
On a fractional nonholonomic
V with
D = {
} instead of
. Even
D has a nontrivial d
torsion, such an object is very dierent from a similar one, for instance, in
integer EinsteinCartan gravity when additional gravitational equations
have to be introduces for the nontrivial torsion components. In our case, the
canonical
=
_
L
i
jk
,
L
a
jk
,
L
i
bk
,
L
a
bk
,
C
i
jb
,
C
a
jb
,
C
i
bc
,
C
a
bc
_
,
where
e
k
(
e
j
) =
L
i
jk
e
i
+
L
a
jk
e
a
,
e
k
(
e
b
) =
L
i
bk
e
i
+
L
a
bk
e
a
,
e
b
(
e
j
) =
C
i
jb
e
i
+
C
a
jb
e
a
,
ec
(
e
b
) =
C
i
bc
e
i
+
C
a
bc
e
a
.
Following a straightforward fractional coecient computation, we can prove
Corollary 2.1 With respect to Nadapted fractional bases (7) and (8), the
coecients of the fractional LeviCivita and canonical dconnection satisfy
14
the distorting relations
+
Z
(22)
where the explicit components of distortion tensor
Z
are computed
Z
i
jk
= 0,
Z
a
jk
=
C
i
jb
g
ik
g
ab
1
2
a
jk
,
Z
i
bk
=
1
2
c
jk
g
cb
g
ji
1
2
(
i
j
h
k
g
jk
g
ih
)
C
j
hb
,
Z
a
bk
=
1
2
(
a
c
b
d
+
g
cd
g
ab
) [
L
c
bk
e
b
(
N
c
k
)] ,
Z
i
kb
=
1
2
a
jk
g
cb
g
ji
+
1
2
(
i
j
h
k
g
jk
g
ih
)
C
j
hb
,
Z
a
jb
=
1
2
(
a
c
d
b
g
cb
g
ad
)
_
L
c
dj
e
d
(
N
c
j
)
, (23)
Z
a
bc
= 0,
Z
i
ab
=
g
ij
2
{
_
L
c
aj
e
a
(
N
c
j
)
g
cb
+
_
L
c
bj
e
b
(
N
c
j
)
g
ca
}.
We emphasize that there are not simple relations of type (22) and (23)
if the fractional integrodierential structure would be not elaborated in N
adapted form for the left Caputo derivative. For the fractional RL deriva-
tives, it is not possible to introduce Nanholonomic distributions when the
formulas would preserve a maximal similarity with the integer nonholonomic
case.
3 Perelman Type Fractional Functionals
The goal of this section is to show that there is a fractional integro
dierential calculus admitting generalizations of the HamiltonPerelman
Ricci ow evolution theory. Proofs are simplied for correspondingly de-
ned nonholonomic fractional distributions.
3.1 On (non) holonomic Ricci ows
For Riemannian spaces of integer dimension, the Grisha Perelmans fun-
damental idea was to prove that the Ricci ow is not only a gradient ow
but, introducing two Lyapunov type functionals, can be dened also as a
dynamical system on the spaces of Riemannian metrics.
15
The Ricci ow equation was postulated by R. Hamilton [1] as an evolu-
tion equation
5
g
()
= 2 R
() (24)
for a set of Riemannian metrics g
F(f) =
_
V
_
R +|f|
2
_
e
f
dV, (25)
W(f, ) =
_
V
_
(
R +|f|)
2
+f
n +m
2
_
dV,
where dV is the volume form of g, integration is taken over compact V and
F(
g,
N,
f) =
_
V
(
R +
S +|
f|
2
)e
f
d
V, (26)
W(
g,
N,
f,
) =
_
V
[
(
R +
S +|
h
D
f| +|
v
D
f|)
2
+
f
n +m
2
]
d
V, (27)
where
d
V is the volume fractional form of
g (10),
R and
S
are respectively the h- and vcomponents of the curvature scalar (19) of
D, for
= (
D
i
,
D
a
), or
D = (
h
D,
v
D),
2
=
h
D
2
+
v
D
f
2
, and
f satises
_
V
d
V = 1 for
=
(4 )
(n+m)/2
e
f
and fractional ow parameter > 0.
Proof. Formulas (25) can be rewritten for some fractional functions
f
and
f when
(
R +|
f|
2
)e
f
= (
R +
S +
h
D
2
+
v
D
f
2
)e
f
+
R +|
f|
_
2
+
f
n +m
2
)
_
=
_
_
R +
S +
h
D
v
D
_
2
+
f
n +m
2
_
+
1
,
for some
and
1
for which
_
d
V = 0 and
_
1
d
V = 0.
For proofs of the Main Results in section 4, the next lemma will be
important.
17
Lemma 3.1 The rst Nadapted fractional variations of (26) are given by
F(v
ij
, v
ab
,
h
f,
v
f) = (28)
_
V
{[v
ij
(
R
ij
+
D
i
D
j
f) + (
h
v
2
h
f)(2
h
f |
h
D
f|) +
R]
+[v
ab
(
R
ab
+
D
a
D
b
f) + (
v
v
2
v
f)
_
2
v
f |
v
D
f|
_
+
S]}e
f
d
V,
where
h
=
D
i
D
i
and
v
=
D
a
D
a
,
=
h
+
v
, and
h
v =
g
ij
v
ij
,
v
v =
g
ab
v
ab
; for hvariation
h
g
ij
= v
ij
, v-variation
v
g
ab
=
v
ab
and variations
h
f =
h
f,
v
f =
v
f.
Proof. We x a Nconnection structure
= 2
R
+
2r
5
g
, (29)
describing normalized (holonomic) Ricci ows with respect to a coordinate
base
= /u
.
6
In (29), the normalizing factor r =
_
RdV/dV is in-
troduced in order to preserve the volume V ;
R
and
R = g
are
computed for the LeviCivita connection . Then we change the geometric
6
In this integer case, we underline the indices with respect to the coordinate bases in
order to distinguish them from those dened with respect to the Nelongated local bases
(7) and (8).
18
objects (tensors, derivatives and parameter) into fractional ones, and obtain
a noninteger generalization of Hamiltons equations,
1
g
ij
= 2[
N
a
i
N
b
j
(
R
ab
g
ab
)
R
ij
+
g
ij
]
g
cd 1
(
N
c
i
N
d
j
), (30)
1
g
ab
= 2
R
ab
+ 2
g
ab
, (31)
1
(
N
e
j
g
ae
) = 2
R
ia
+ 2
N
e
j
g
ae
, (32)
where
=
r/5, with
r =
_
R
d
V/
d
V, and the metric
coecients are those for (9) parametrized by ansatz (11), with respect to a
fractional local coordinate basis (4).
A fractional dierential geometry is modelled by nonholonomic integro
dierential structures. A selfconsistent system of fractional equations has
to be Nadapted. We change in (30)(32) the corresponding values:
D and
g
ij
= 2[
N
a
i
N
b
j
(
R
ab
g
ab
)
R
ij
+
g
ij
]
g
cd 1
(
N
c
i
N
d
j
), (33)
1
g
ab
= 2
_
R
ab
g
ab
_
, (34)
R
ia
= 0 and
R
ai
= 0, (35)
where the fractional Ricci tensor coecients
R
ij
and
R
ab
are computed
with respect to coordinate coframes (4), being frame transforms (12) of the
corresponding formulas (18) dened with respect to Nadapted coframes
(8). The equations (35) constrain the nonholonomic fractional Ricci ows to
result in symmetric fractional metrics. In general, fractional geometries are
with nonholonomic integrodierential structures resulting in nonsymmetric
fractional metrics, a similar conclusion for integer dimensions was proven in
Ref. [10].
4.1 Main Theorems on fractional Ricci ows
One of the most important Perelmans contributions to the theory of
Ricci ows was that he proved that Hamiltons evolution equations, in some
19
adapted forms, can be derived from certain functionals following a varia-
tional procedure. We show that Perelmans approach can be generalized
to a fractional Nadapted formalism for evolution of geometric objects. In
explicit form, we show how equations of type (33) and (34) can be derived
by a fractional integrodierential calculus (for simplicity, we take a zero
normalized term with
= 0).
Denition 4.1 A general fractional metric
g evolving via a general frac-
tional Ricci ow is called a breather if for some
1
<
2
and > 0 the
metrics
g(
1
) and
g(
2
) dier only by a fractional dieomorphism
preserving the Whitney sum (5). The cases =, <, > 1 dene correspond-
ingly the steady, shrinking and expanding breathers.
We note that because of nonholonomic character of fractional evolution
we can model processes when, for instance, the hcomponent of metric is
steady but the vcomponent is shrinking. Clearly, the expending properties
depend on the type of calculus and connections are used for denition of
Ricci ows.
Following a Nadapted variational calculus for
F(
g,
N,
f), see
Lemma 3.1, with Laplacian
R
ij
and
S
ij
, dened by
g
ij
= 2
R
ij
,
1
g
ab
= 2
R
ab
,
1
f =
f +
R
S
and the properties that
_
V
e
f
d
V = const and
1
F(
g(),
N(),
f()) = 2
_
V
[|
R
ij
+
D
i
D
j
f|
2
+|
R
ab
+
D
a
D
b
f|
2
]e
f
d
V.
Proof. Such a proof which is very similar to those for Riemannian
spaces, originally proposed by G. Perelman [2], see also details in the Propo-
sition 1.5.3 of [5], and nonholonomic manifolds (additional remarks on the
20
canonical dconnection
D on nonholonomic manifolds of integer dimension
given in [8]). All those constructions can be reproduced in Nadapted frac-
tional form using
f() and parameter function () evolve subjected to the conditions of the
system of equations
1
g
ij
= 2
R
ij
,
1
g
ab
= 2
R
ab
,
1
f =
f +
R
S +
n +m
2
,
1
= 1,
there are satised the properties
_
V
(4 )
(n+m)/2
e
f
d
V = const and
1
W(
g(),
N(),
f(), ()) =
2
_
V
[|
R
ij
+
D
i
D
j
f
1
2
g
ij
|
2
+
|
R
ab
+
D
a
D
b
f
1
2
g
ab
|
2
](4 )
(n+m)/2
e
f
d
V.
The functional
W(
g(),
N(),
f(), ()) is nondecreasing in time
and the monotonicity is strict unless we are on a shrinking fractional gradient
soliton. This property depends on the type of fractional dconnection, or
covariant connection we use.
Corollary 4.1 The fractional evolution, for all time [0,
0
), of N
adapted frames
() =
e
(, u)
(, u) =
_
e
i
i
(, u)
N
b
i
(, u)
e
a
b
(, u)
0
e
a
a
(, u)
_
,
with
g
ij
() =
e
i
i
(, u)
e
j
j
(, u)
ij
,
21
where
ij
= diag[1, ... 1] establish the signature of
g
[0]
(u), is given by
equations
1
=
g
,
if we prescribe fractional ows for the LeviCivita connection
, and
1
=
g
R
,
if we prescribe fractional ows for the canonical dconnection
D.
We conclude that the fractional ows are characterized additionally by
fractional evolutions of Nadapted frames (12) (see a similar proof for ows
of integer dimension nonholonomic frames in [8]).
4.2 Statistical Analogy for Fractional Ricci Flows
A functional
Z = exp{
_
V
[
f +
n+m
2
]
d
V }, we
prove:
Theorem 4.3 Any family of fractional nonholonomic geometries satisfying
the fractional evolution equations for the canonical dconnection is charac-
7
Let us remember some concepts from statistical mechanics. The partition function
Z =
V
(
R +
S +
h
D
f
2
+
v
D
f
n +m
2
)
d
V,
S =
_
V
[
_
R +
S +
h
D
2
+
v
D
2
_
+
n +m
2
]
d
V,
= 2
4
_
V
[|
R
ij
+
D
i
D
j
f
1
2
g
ij
|
2
+|
R
ab
+
D
a
D
b
f
1
2
g
ab
|
2
]
d
V.
A fractional, or integer, dierential geometry dened by corresponding
fundamental geometric objects and a xed, in general, noninteger dier-
ential system is thermodynamically more convenient in dependence of the
values of the above mentioned characteristics of Ricci ow evolution.
Conclusion 4.1 Finally, we draw the conclusions:
There is a version of fractional dierential and integral calculus based
on the left Caputo derivative when the resulting models of fractional
dierential geometry are with a Nconnection adapted calculus simi-
larly to noholonomic manifolds and FinslerLagrange geometry.
A Ricci ow theory of fractional geometries can be considered as a
nonholonomic evolution model transforming standard integer metrics
and connections (for instance, in Riemann geometry) into generalized
ones on nonsymmetric/noncommutative/fractional ... spaces.
A very important property of fractional calculus theories and related
geometric and physical models is that we can work with more singu-
lar functions and eld interaction/evolution models in physics and
applied mathematics.
A Fractional IntegroDierential Calculus on R
n
We summarize the formalism for a vector fractional dierential and in-
tegral calculus elaborated on at spaces [24]. The constructions involve
23
a fundamental theorem of calculus and fractional integral Greens, Stokes
and Gausss theorems, which are important for denition, in this work, of
fractional Perelmans functionals.
A.1 RiemannLiouville and Caputo fractional derivatives
It is possible to elaborate dierent types of models of fractional geometry
using dierent types of fractional derivatives. We follow an approach when
the geometric constructions are most closed to integer calculus.
A.1.1 Left and right fractional RL derivatives
Let us consider that f(x) is a derivable function f : [
1
x,
2
x] R, for
R > 0, and denote the derivative on x as
x
= /x.
The left RiemannLiouville (RL) derivative is
1
x
x
f(x) :=
1
(s )
_
x
_
s
x
_
1
x
(x x
)
s1
f(x
)dx
,
where is the Eulers gamma function. The left fractional Liouville deriva-
tive of order , where s1 < < s, with respect to coordinate x is dened
x
f(x) := lim
1
x
1
x
x
f(x).
The right RL derivative is
x
2
x
f(x) :=
1
(s)
_
x
_
s
2
x
_
x
(x
x)
s1
f(x
)dx
f(x
k
) :=
lim
2
x x
2
x
f(x). In this work, we shall not use right derivatives.
Only the fractional Liouville derivatives dene operators satisfying the
semigroup properties on function spaces. The fractional RL derivative of a
constant C is not zero but, for instance,
1
x
x
C = C
(x
1
x)
(1)
. Complete
fractional integrodierential constructions based only on such derivatives
seem to be very cumbersome and has a number of properties which are very
dierent from similar ones for integer calculus.
24
A.1.2 Fractional Caputo derivatives
The respective left and right fractional Caputo derivatives are
1
x
x
f(x) :=
1
(s )
x
_
1
x
(x x
)
s1
_
x
_
s
f(x
)dx
, (A.1)
and
x
2
x
f(x) :=
1
(s )
2
x
_
x
(x
x)
s1
_
_
s
f(x
)dx
,
where we underline the partial derivative symbol, , in order to distinguish
the Caputo operators from the RL ones with usual . A very important
property is that for a constant C, for instance,
1
x
x
C = 0. In our approach,
we shall give priority to the fractional left Caputo derivative resulting in
constructions which are very similar to those with integer calculus.
A.2 Vector operations and noninteger dierential forms
A.2.1 Fractional integral
To formulate fractional integral (Gausss, Stokes, Greens etc) theorems
with a formal noninteger order integral
1x
I
2x
, we need a generalization
of the NewtonLeibniz formula
1x
I
2x
_
1x
x
f(x)
_
= f(
2
x)f(
1
x), for a
fractional derivative
1
x
x
. Such mutually inverse operations do not exist
for an arbitrary taken type of fractional derivative.
Let us denote by L
z
(
1
x,
2
x) the set of those Lebesgue measurable func-
tions f on [
1
x,
2
x] for which ||f||
z
= (
2
x
_
1
x
|f(x)|
z
dx)
1/z
< . We write
C
z
[
1
x,
2
x] is a space of functions, which are z times continuously dieren-
tiable on this interval.
Using the fundamental theorem of fractional calculus [24], we have that
for any realvalued function f(x) dened on a closed interval [
1
x,
2
x],
there is a function F(x) =
1
x
I
x
f(x) dened by the fractional Riemann
Liouville integral
1
x
I
x
f(x) :=
1
()
x
_
1
x
(xx
)
1
f(x
)dx
x
F(x) satises the conditions
1
x
x
_
1
x
I
x
f(x)
_
= f(x), > 0,
1
x
I
x
_
1
x
x
F(x)
_
= F(x) F(
1
x), 0 < < 1,
for all x [
1
x,
2
x]. So, the right fractional RL integral is inverse to the
right fractional Caputo derivative.
There is a corresponding fractional generalization for the Taylor formula
f(
2
x) f(
1
x) =
1
x
I
x
_
1
x
x
f(x)
_
+
s1
s
1
=0
1
s
1
!
(
2
x
1
x)
s
1
f
(s
1
)
(
1
x),
for s 1 < s, where f
(s
1
)
(x) =
1
x
x
f(x).
A.2.2 Denition of fractional vector operations
Let X be a domain of R
n
parametrized as X = {
1
x
i
x
i
2
x
i
} (or,
in brief, X = {
1
x x
2
x}, which substitute the closed one dimensional
interval [
1
x,
2
x]. Let f(x
i
) and F
k
(x
i
) be realvalued functions that have
continuous derivatives up to order k1 on X, such that the k1 derivatives
are absolutely continuous, i.e., f, F = {F
i
} AC
k
[X], see details in [19].
For a basis e
i
on X, we can dene a fractional generalization of gradient
operator = e
i
i
= e
i
/x
i
, when
1
x
= e
i
1
x
i
, for
1
x
i
:=
1
x
x
i being
the left fractional Caputo derivatives on x
i
dened by (A.1). If f(x) = f(x
i
)
is a (k 1) times continuously dierentiable scalar eld such that
k1
i
f is
absolutely continuous, we can dene the fractional gradient of f,
grad f :=
1
x
f = e
i
1
x
i
f. Let us consider that X R
n
has a at metric
ij
and its
inverse
ij
. Then we can dene F
i
=
ij
F
j
(x
k
) and construct the fractional
divergence operator
divF : =
1
x
i
F
i
(x
k
).
We can not use the Leibniz rule in a fractional generalization of the
vector calculus because for two analytic functions
1
f and
2
f we have
grad(
1
f
2
f) = (
grad
1
f)
2
f + (
grad
2
f )
1
f,
div(fF) = (
grad f , F) +f
divF.
This follows from the property that
1
x
i
(
1
f (x
j
)
2
f(x
k
)) =
(
1
x
1
f(x
j
))
2
f(x
k
) + (
1
x
2
f(x
k
))
1
f(x
j
).
26
A fractional volume integral is a triple fractional integral within a re-
gion X R
3
, for instance, of a scalar eld f(x
k
),
I(f) =
I[x
k
]f(x
k
) =
I[x
1
]
I[x
2
]
I[x
3
]f(x
k
). For = 1 and f(x, y, z), we have
I(f) =
___
X
dV f =
___
X
dxdydz f.
The fundamental fractional integral theorems with
grad,
d in the form
d := (dx
j
)
j
, where
dx
j
= (dx
j
)
(x
j
)
1
(2 )
.
For the integer calculus, we use as local coordinate co-bases/frames the
dierentials dx
j
= (dx
j
)
=1
. The fractional symbol (dx
j
)
, related to
dx
j
,
is used instead of dx
i
for elaborating a covector/dierential form calculus,
see below the formula (A.5). It is considered that for 0 < < 1 we have
dx = (dx)
1
(dx)
.
An exterior fractional dierential can be dened through the fractional
Caputo derivatives in the form
d =
n
j=1
(2 )(x
j
)
1
dx
j
0
j
.
27
Dierentials are dual to partial derivatives, and derivation is inverse to inte-
gration. For a fractional calculus, the concept of dual and inverse have
a more sophisticate relation to integration and, in result, there is a more
complex relation between forms and vectors.
The fractional integration for dierential forms on L = [
1
x,
2
x] is intro-
duced using the operator
L
I[x] :=
2
x
_
1
x
(dx)
1
()(
2
xx)
1
when, for 0 < < 1,
L
I[x]
1
x
d
x
f(x) = f(
2
x) f(
1
x). (A.2)
The fractional dierential of a function f(x) is
1
x
d
x
f(x) = [...], with
square brackets considered are dened by formula
2
x
_
1
x
(dx)
1
()(
2
x x)
1
_
(dx
1
x
x
f(x
)
_
= f(x) f(
1
x).
The exact fractional dierential 0form is a fractional dierential of the
function
1
x
d
x
f(x) := (dx)
1
x
x
f(x
)
when the equation (A.2) is considered as the fractional generalization of the
integral for a dierential 1form. So, the fractional exterior derivative
can be written
1x
d
x
:= (dx
i
)
1x
i
. (A.3)
Then, the fractional dierential 1form
with coecients {F
i
(x
k
)} is
= (dx
i
)
F
i
(x
k
) (A.4)
The exterior fractional derivatives of 1form
gives a fractional 2form,
1
x
d
x
(
) = (dx
i
)
(dx
j
)
1
x
j
F
i
(x
k
).
This rule can be proven following the relations [19] that for any type frac-
tional derivative
x
, we have
x
(
1
f
2
f) =
k=0
_
k
__
k
x
1
f
_
=k
x
_
2
f
_
,
for integer k, where
_
k
_
=
(1)
k1
(k )
(1 )(k + 1)
and
k
x
(dx)
= 0, k 1.
28
There are used also the properties:
1
x
x
(x
1
x
=
(+1)
(+1)
(x
1
x)
,
where n 1 < < n and > n, and
1
x
x
(x
1
x
)
k
= 0 for k =
0, 1, 2, ..., n1. We obtain
1
x
d
x
(x
1
x)
= (dx)
1
x
i
x
i
= (dx)
(+1),
i.e.
(dx)
=
1
( + 1)
1
x
d
x
(x
1
x)
(A.5)
and write the fractional exterior derivative (A.3) in the form
1x
d
x
:=
1
( + 1)
1x
d
x
(x
i
1
x
i
)
1x
i
.
Using this formula, the fractional dierential 1form (A.4) can be written
alternatively
=
1
( + 1)
1x
d
x
(x
i
1
x
i
)
F
i
(x).
Having a well dened exterior calculus of fractional dierential forms
on at spaces R
n
, we can generalize the constructions for a real manifold
M, dimM = n.
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