2 The space of Riemannian metrics

H. Ghahremani-Gol, A. Razavi

Abstract.R.Hamilton defined Ricci flow as a weak parabolic partial dif- ferential equation, in spite of weakness he could prove the existence and uniqueness in the short time, while later DeTurck found a shorter proof.

On the other hand the space of Riemannian metrics on a compact man- ifold had been proved to be an infinite dimensional manifold which is a projective limit of Banach manifolds. In this paper we consider the Ricci flow as an integral curve of certain vector fields on the manifold of Rie- mannian metrics and in spite of being infinite dimensional, we prove the existence and uniqueness for the short time, and moreover we find further results on the behavior of these curves.

M.S.C. 2010: 53C44, 58D17, 58B25.

Key words: Ricci flow; space of Riemannian metrics; projective limit; Banach man- ifold.

1 Introduction

The Ricci flow was first defined by Hamilton in the early 1980 [14]. Following works of Eells and Sampson[8], he introduced an evolution equation for a family of Riemannian metrics as follows: ( _∂

∂tg(t) =−2Rc(g(t)) g(0) =g0,

where Rc(g(t)) denotes the Ricci curvature of the metric g(t), and by rescaling the space and time we obtain its cousin, the normalized Ricci flow, as:





∂

∂tg(t) =−2Rc(g(t)) +²_n

R

MRdµ R

Mdµ g(t) g(0) =g0,

whereRdenotes the scalar curvature of the metricg(t) anddµis the volume element ofg(t). The Ricci flow is an evolution equation, considered as a partial differential equation.

Balkan Journal of Geometry and Its Applications, Vol. 18, No. 2, 2013, pp. 20-30.∗

°c Balkan Society of Geometers, Geometry Balkan Press 2013.

A fundamental problem for any system of partial differential equation is the ex- istence and uniqueness of solution for the short time. Since Ricci flow is a weak parabolic partial differential equation, standard parabolic theory does not work for it. Hamilton proved the existence and uniqueness of solution for the Ricci flow in the short time by using Nash and Moser implicit function theorem [13, 14]. Later D.M.DeTurck provided an easier proof using linearizing of differential operators [6].

Ebin studied the space of Riemannian metrics on a compact manifoldM in his paper from geometric point of view[9]. He enlarged this space to a space of a certain type of infinite dimensional manifolds. The space of Riemannian metrics on a compact, oriented, smooth n-manifold M is an infinite dimensional manifold. It is an open subset of a Fr´echet space, and is a Fr´echet manifold. Since some basic theorems of differential geometry such as implicit function theorem and theory of ordinary dif- ferential equation do not hold in the infinite dimensional manifolds such as Fr´eechet space, authors have focussed on the special type of Fr´echet manifolds namely the projective limits. Geometry of those Fr´echet manifolds which can be obtained as pro- jective limit Banach manifolds has been studied widely([1, 7, 11, 17, 18]). The space of Riemannian metrics is a type of projective limit manifolds. In this paper we use this viewpoint and give a new approach for the proof of short time existence theorem for the Ricci flow, moreover we find the properties of the Ricci flow as a curve.

2 The space of Riemannian metrics

LetM be a compact, oriented, smoothn-manifold, without boundary. Consider the collectionM et(M) of all smooth Riemannian metrics onM. In factM et(M) is the subset of all sections inS²T^∗M of symmetric rank-2 covariant tensor fields which are positive definite on eachT_p^∗Mforp∈M, moreoverM et(M) is an open convex positive cone in Γ(S²T^∗M). For convenience, we will abbreviateM et(M) asM. The space of Γ(S²T^∗M) is an infinite-dimensional Fr´echet space [13], thereforeMis also infinite- dimensional Fr´echet manifold. A Fr´echet space is a complete Hausdorff metrizable locally convex topological vector space. A Fr´echet manifold is a Hausdorff topological space with an atlas of coordinate charts taking their values in Fr´echet spaces, such that the coordinate transition functions are all smooth maps between Fr´echet spaces.

Since Mis an open subset of the vector space Γ(S²T^∗M), the tangent space TgM for anyg∈Mis Γ(S²T^∗M) itself. Geometry ofMat first has been studied by Ebin [9]. Later Freed and Groisser gave a decomposition of this space toV ol(M) andMµ

then using this decomposition they defined a Riemannian metric onM, and obtained the Levi-Civita connection and its geodesics [10]. There are enormous papers about different aspects of the space of Riemannian metrics, see [3, 4, 9, 10, 21, 12].

2.1 Met(M)

In this section we describe the decomposition ofMdefined by Freed and Groisser[10], which we use in the next sections.

For anyg ∈Man L² inner product on tensor fields is induced. For anyA, B ∈ Γ(T^∗MN

T^∗M), we set

hA, Big= Z

M

trg(AB^t)µ(g),

where in local coordinates {xⁱ}, A = AijdxⁱN

dx^j (and similarly for B, g), B^t = BjidxⁱN

dx^j (the ”transpose” ofB),{g^ij}is the inverse matrix of{gij},trg(AC) = Aijg^jkCklg^li, and µ(g) is the volume formp

det(gij)dx¹∧...∧dxⁿ. The restriction of this quadratic form to symmetric the symmetric tensor fieldsAandB is positive definite. Thus there is a metric onM.

Let V ol(M)⊂Ωⁿ(M) be the space of volume forms on M, consistent with the orientation. Forα∈ Ωⁿ(M) and ν ∈ V ol(M), let (α/ν) be the function satisfying α = (α/ν)ν. Let p : M → V ol(M) be the projection carrying g to µ(g), and let Mν =M etν(M) =p⁻¹(ν) for anyν∈V ol(M).

Each volume formµdetermines a splitting

i_µ:V ol(M)×M_µ −→M, such that

(2.1) (ν, h)7→(ν/µ(h))^2/nh.

SinceV ol(M) is an open subset of vector space Ωⁿ(M), the tangent bundle ofV ol(M) is canonically isomorphic to V ol(M)×Ωⁿ(M). A vector field β over any subset U ⊂V ol(M) may therefore be naturally identified with a functionβ :U →Ωⁿ(M), and we implicitly make this identification henceforth.

The tangent space to Mµ athis the set of h−traceless symmetric tensor fields;

that is,{A∈Γ(S²T^∗M)| h^ijAij ≡0}. Then for anyg= (ν, h)∈M∼=V ol(M)×Mµ, a tangent vector ofT_gMcan be considered to be of the formα+Awhereα∈Ωⁿ(M)∼= TνV ol(M) andA∈ThMµ.

The Koszul’s formula of Levi-Civita connection, applied to constant vector fields B, C, E onM∼=V ol(M)×Mµ, quickly leads to

∇BC|g=−1

2(Bg⁻¹C+Cg⁻¹B) +1

4{(trg(C))B+ (trg(B))C−trg(BC)g}.

3 M as a projective limit manifold

In this section we give a brief definition of a certain type of infinite dimensional manifolds for which the space of all Riemannian metrics is an example.

It is well known (see for instance [11]) that every Fr´echet spaceF can be identified with the limit of a projective system{Eⁱ;ρ^ji}i,j∈Nof Banach spaces,F ∼= lim←−Eⁱ. This means that the mappingsρ^ji:E^j−→Eⁱ(j≥i) are smooth and satisfy the following conditions for every (i, j, k) such thatj≥i≥k:

ρ^ikρ^ji=ρ^jk.

Definition 3.1. Let{Mⁱ;φ^ji}i,j∈Nbe a projective system of smooth manifolds mod- eled on the Banach spaces{Eⁱ}i∈N respectively. We assume that:

(1) The models{Eⁱ}i∈N form a projective system with connecting morphisms{ρ^ji: E^j −→Eⁱ;j≥i}and limit the Fr´echet space F=lim←−Eⁱ.

(2) For any element x = (xⁱ)i∈N ∈ M = lim←−Mⁱ there exists a family of charts {(Uⁱ, ψⁱ)}i∈N ofMⁱ-s, such that the limitslim←−Uⁱ, lim←−ψⁱ can be defined and the sets lim←−Uⁱ, lim←−ψⁱ(lim←−Uⁱ) are open inM,F respectively.

Then the limitM =lim←−Mⁱ is called a PLB-manifold.

A PLB-manifold M is a Fr´echet manifold modelled on F. The corresponding local structure is fully determined by the charts (lim←−Uⁱ, lim←−ψⁱ). The differentiability of mappings involved can be either this of J. A. Leslie ([16]) or that of A.Kriegl- P.Michor ([15]). The tangent bundleT M ofM has also a Fr´echet manifold structure with model the Fr´echet spaceF×F which is isomorphic to PLB-manifoldlim←−T Mⁱ. Proposition 3.1. [11] The tangent bundles{T Mⁱ}i∈N form a projective system with limit set-theoretically isomorphic toT M :T M 'lim←−T Mⁱ.

Omori introduced inverse limit Hilbert manifolds and inverse limit Hilbert groups, which are a special type of projective limit Banach manifolds, those where connecting morphisms of factors are the natural embedding [17, 18].

The space M is an inverse limit Hilbert manifold. In order to obtain some of results obtained by Palais in [19, 20] about space of sections of the vector bundles, we consider the bundlesT_q^pM of (p, q)-tensors as vector bundlesE overM. Then the space of sections Γ(E) = Γ(T_q^p) is the space of all smooth tensor fields of type (p, q) onM. The metricg onM determines an inner product on the bundlesT_q^pM in the usual way. Therefore, an inner product in the space Γ(E) of tensor fields of type (p, q) is defined. IfT andU are tensor fields of type (p, q), then

(3.1) hT, Uig=

Z

M

g(x)(T(x), U(x))µ(g)(x), where

g(x)(T(x), U(x)) =g^i1k1...g^iqkqgj1l1...gjplpT_i^j₁¹_...i^...j_q^pU_k^l1₁^...l_...k^p_q.

TheC^k-norm is defined in the space Γ(E) as follows: for a nonnegative integerkand a tensor fieldT of type (p, q), set

(3.2) |T|k=

Xk

i=0

supx∈Mk∇⁽ⁱ⁾T(x)k,

where∇⁽ⁱ⁾=∇ ◦...◦ ∇is theith power of the covariant derivative andk∇⁽ⁱ⁾T(x)k= h∇⁽ⁱ⁾T,∇⁽ⁱ⁾Ti^1/2g is the tensor norm at a point x ∈ M. We denote by C^k(E) the completion of the space Γ(E) with respect to the topology defined by the norm|T|k. The Banach space C^k(E) consists of tensor fields of class C^k. However, in many problems, Hilbert spaces are more convenient, then we define on the space Γ(E) inner products stronger than (3.1). Letsbe a nonnegative integer andT andU be tensor fields of type (p, q). We set

(3.3) hT, Uig,s= Xs

i=0

h∇⁽ⁱ⁾T,∇⁽ⁱ⁾Uig= Xs

i=0

Z

M

g(∇⁽ⁱ⁾T,∇⁽ⁱ⁾U)µ(g),

whereh∇⁽ⁱ⁾T,∇⁽ⁱ⁾Uigis the inner product (3.1). We denote byH^s(E) the completion of the space Γ(E) with respect to the topology defined by inner product(3.3). The spaceH^s(E) is called the space of Sobolev smoothness classH^s; it is a Hilbert space.

We denote byk.ksthe norm of this space. In particular, fors= 0, the spaceH⁰(E) is the completion of the space Γ(T_q^pM) with respect to inner space (3.1).(For detail, see ([19], Chap. IX).) Forl≥s, we haveH^l(E)⊂H^s(E); this embedding is continuous.

Obviously,C^k(E)⊂H^k(E). The inverse embedding is stated by the following Sobolev embedding theorem (for the proof, see ([19], Chap. X)).

Theorem 3.2. [19] If s ≥ n/2 + 1 +k, then H^s(E) ⊂ C^k(E) and the embedding mappingH^s(E)→C^k(E) is completely continuous.

Therefore, fors≥n/2+1+k, we can assume that a tensor fieldT of Sobolev class H^sis differentiable of classC^k. Further restrictions on s are related with smoothness conditions for tensor fields of the spaceH^s(E).

Theorem 3.3. [20] Let E and F be vector bundles over M and f : E → F be a C^∞-mapping preserving fibers. Ifs≥n/2 + 1, then the mappingφ:H^s(E)→H^s(F) defined by the formulaφ(α) =f◦αis a mapping of class C^∞.

Now we see that the space M is a projective (inverse) limit Hilbert manifold.

Assume S₂^s=H^s(S2M) is the Hilbert space of symmetric 2-forms of class H^s, s >

n/2. Let C⁰M be the space of continuous Riemannian metrics. For s > n/2, let M^s=H^s(S²M)T

C⁰M. Since H^s ⊂C⁰(S₂(M)) and the embedding is continuous, we have thatM^sis open inS₂^s=H^s(S2M) and is an open, convex, positive cone. In particular, the spaceM^sis a smooth Hilbert manifold. The system {M,M^s} forms a strong ILH-manifold (inverse limit Hilbert manifold)[21].

3.1 Integral curves

In an infinite dimensional manifold a given vector filed need not have integral curves locally, and if there exist they need not be unique for a given initial value. This is due to the fact that inverse function theorem, the implicit function theorem don’t hold for all of the infinite dimensional manifolds.

We have the following theorem about existence and uniqueness of integral curves for those Fr´echet manifolds which can be obtained as a projective limits of Banach manifolds.[1]

Theorem 3.4. [1] Let ξ be a vector field on the Fr´echet manifold M = lim←−Mⁱ which can be considered as the projective limit of vector fields {ξⁱ}i∈N of {Mⁱ}i∈N

respectively. Then ξ admits locally a unique integral curve θ, satisfying an initial condition of the formθ(0) =xforx∈M.

4 Ricci flow and space of Riemannian metrics

Hamilton introduced in [14] the Ricci flow as follows:

Definition 4.1. Let M be a manifold with an initial metric g0, a Ricci flow is a family of Riemannian metrics{g(t)}onM satisfying the P DE:

( _∂

∂tg(t) =−2Rc(g(t)) g(0) =g0,

whereRic(g(t)) denotes the Ricci curvature of the metricg(t).

The traceless part of the Ricci tensor on a Riemannian n-manifold (M, g) is the tensorEij :=Rij−_n¹Rgij, whereRij is coordinate expression of Ricci curvature and Ris the scalar curvature defined as the trace of the Ricci tensor, i.e. R:=g^ijRij. A metricg is called Einstein if the traceless part of the Ricci tensor is identically 0.

The various geometric quantities evolve when the metric evolves [2]. Suppose that gij(t) is a time-dependent Riemannian metric, and _∂t^∂gij(t) =hij(t).Then the volume element as a geometric quantities evolves according to the following equation:

∂

∂tµ(g(t)) = 1

2trghµ(g(t)).

4.1 Short-Time existence

An important foundational step in the study of any system of evolutionary partial differential equations is to show short-time existence and uniqueness. As the PDE of Ricci flow is not strictly parabolic, we can’t deduce directly the existence and uniqueness of short time solution for Ricci flow. Hamilton using Nash-Moser implicit function theorem proved short-time existence theorem for Ricci flow[14]. DeTurck later gave a more direct proof by modifying the flow by a time-dependent change of variables to make it parabolic [6].

SinceMis a projective limit Banach manifold, we can find another approach for the proof of the short time existence theorem of Ricci flow equation.

Theorem 4.1. If (M, g0) is a compact Riemannian manifold, there exists a unique solutiong(t), defined for timet∈[0, ε), to the Ricci flow such thatg(0) =g0for some ε >0.

Proof. LetX be the vector filed onMsuch thatg−→ −2Rc(g). SinceMis a type of projective limit Banach manifold, the vector filedV can be considered as projective limit. Using theorem (3.4) the Ricci flow curve can be considered as the integral curve of this vector filed and its existence can be proved as follows.

By proposition (3.1) the vector field X ∈ TM can be considered as X = lim←−Xⁱ. We show that X admit locally a unique integral curve α with initial condition as α(0) =g0 forg0 ∈M. Since for i∈N, Xⁱ is a vector field on Hilbert manifoldMⁱ, there exist a unique integral curveαⁱ such that

(ψⁱ_k◦αⁱ)⁰ = Ψ^2,i_k (Xⁱ(αⁱ(t))); k∈I

and αⁱ(0) = g₀ⁱ = ϕⁱ(g₀) where ϕⁱ : M → Mⁱ is the canonical projection and {(π⁻¹(U_kⁱ),Ψk)}k∈I is the corresponding trivialization for TMⁱ, Ψ^2,i_k is the projec- tion of Ψⁱ_k onto its second factor. Now α=lim←−αⁱ is integral curve for X. We must showφ^ji◦α^j =αⁱ forj≥i. It is sufficient to showφ^ji◦α^j is also an integral curve ofXⁱ. Then we have

(ψ_kⁱ ◦(φ^ji◦α^j)⁰ = ρ^ji◦(ψ_k^j◦α^j))⁰(t) =ρ^ji(ψ_k^j◦α^j)⁰(t)

= ρ^ji(Ψ^2,j_k (X^j(α^j(t)))) = Ψ^2,j_k (T φ^ji(X^j(α^j(t)))

= Ψ^2,j_k (X^j(φ^ji◦α^j(t)))

and alsoφ^ji◦α^j(g^j₀) =g₀ⁱ. Therefore by uniqueness of integral curve φ^ji◦α^j =αⁱ. This shows{αⁱ}i∈Nis a projective system of curves andα=lim←−αⁱ exists. αsatisfies

the conditions for theorem because

(ψ◦α)⁰(t) = ((ψ_kⁱ ◦αⁱ)⁰(t))i∈N= (Ψ^2,i_k (Xⁱ(αⁱ(t))))i∈N= Ψ²_k(X(α(t))).

Uniqueness ofαfollows from uniqueness ofαⁱ. ¤

4.2 Geodesics on M

Geodesics onMwith initial conditions have been calculated in [10].Freed and Groisser fixedµ∈V ol(M) and implicity identifiedMwithV ol(M)×Mµ as in (2.1) and found the geodesics. In this section we explain geodesic equation onM.

Proposition 4.2. [10] The geodesic in V ol(M) with initial position µ and initial velocityα∈Tµ(V ol(M) = Ωⁿ(M)is

µ(t) =µt= (1 +1 2(α

µ)t)µ.

Proposition 4.3. [10] The geodesic inMµ with initial position gand initial velocity A∈Tg(Mµ)is

g(t) =gt=ge^t(g−1A).

Theorem 4.4. [10] the geodesic inMwith initial position (µ, g)and initial velocity (α, A)∈Ωⁿ(M)×sym0(M, g)is

iµ(gt) = (q(t)²+r²t²)ⁿ²gexp(tan⁻¹(rt/q) r g⁻¹A),

where q(t) = 1 + ¹₂(α/µ)t, r = ¹₄(ntr((g⁻¹A)²))¹² and iµ is defined as in previous section. Ifr= 0after replacing the exponential term by1, then the change in volume form ofg(t)is given by the formula

µ(g(t)) = (q(t)²+r²t²)µ.

Sketch of proof. Let {g(t)} be a geodesic and B =g⁰ = ^dg_dt then the above formula of Levi-Civita connection has been applied to the geodesic equation ∇g⁰g⁰ = 0 to obtain (µt, ht). Finally using splitting mapiµ forM∼=V ol(M)×Mµ, geodesics are

obtained, you can see [10] for more details. ¤

Theorem 4.5. The velocity vector of geodesics onMis as follows:

α⁰(t) = (α(1 +1 2

α

µt) + 2r²tµ, A

q²+r²t²exp(tan⁻¹(^rt_q) r g⁻¹A)) also forr= 0

α⁰(t) = (α(1 + 1 2

α µt),0).

Proof. According to M ∼= V ol(M)×Mµ, a geodesic is of the form α(t) = (µt, ht) where

µt=µ(g(t)) = (q(t)²+r²t²)µ

and

ht=gexp

Ãtan⁻¹(^rt_q) r g⁻¹A

! . Therefore

d

dtµt= (2q(t)´q(t) + 2r²t)µ= µ

2 µ

1 +1 2 α µt

¶1 2

α µ + 2r²t

¶ µ=

µ 1 +1

2 α µt

¶

α+ 2r²tµ.

As well, we get d

dtht = gexp

Ãtan⁻¹(^rt_q r g⁻¹A

!

g⁻¹A×1 r

d dt(^rt_q) 1 + (^rt_q)²

= gexp

Ãtan⁻¹(^rt_q) r g⁻¹A

!

g⁻¹A×1

r× q²

q²+r²t² ×rq−¹₂^α_µtr q²

= exp

Ãtan⁻¹(^rt_q) r g⁻¹A

!

A× q−¹₂^α_µt q²+r²t²

= A

q²+r²t²exp

Ãtan⁻¹(^rt_q) r g⁻¹A

!

thus forr6= 0 we have α⁰(t) =

à α

µ 1 +1

2 α µt

¶

+ 2r²tµ, A

q²+r²t²exp

Ãtan⁻¹(^rt_q) r g⁻¹A

!!

,

and also forr= 0 according toα(t) = (q(t)²µ, g) the velocity vector of geodesic is as follows:

α⁰(t) = (α(1 + 1 2

α µt),0),

which concludes the proof. ¤

Remark 4.2. By Definition 4.1, a Ricci flow equation can be considered as a curve onM. We call these curves asRicci flow curve.

We will study the behavior of them on the manifoldM. In special cases we obtain the following result:

Theorem 4.6. A Ricci flow curve starting from an Einstein metric g0 is not a geodesic.

Proof. Let g0 be an Einstein metric: Ric(g0) = λg0, where λ is a constant. Then g(t) = (1−2λt)g0.Then by Theorem 4.4 this curve is not the geodesic. ¤ Moreover using Theorem 4.5, we see that velocity vector of geodesic on M is different from velocity vector of Ricci flow curve

Proposition 4.7. IfM is a compact manifold with strictly positive scalar curvature R0 then the space of Riemannian metrics on the compact manifold M is foliated by the Ricci flow.

Proof. Since M is a compact manifold with strictly positive scalar curvature R0, Ricci flow equation is a non-vanishing vector field onMalso solution of Ricci flow are integral curves on theM. It is obvious that this integral curve foliate theM. ¤

4.3 Ricci solitons

Ricci solitons are special solution of Ricci flow such that everyg(t) is the formg(t) = σ(t)ϕ^∗_t(g0) whereσ(t) andϕ(t) are respectively scalar and diffeomorphisms ofM and g0 is initial metric andϕ0=Id, σ(0) = 1[5]. We call these curve, the Ricci soliton curve.

Theorem 4.8. Ricci soliton curve results in an equation which consists of an initial metric and a vector field and a scalar as follows:

−2Ricg0= 2λg0+£Vg0.

Proof. For arbitrary curveα(t) =σ(t)ϕ^∗_t(g0) according to decompositionM∼=V ol(M)×

Mµ we have:

α(t) = Ã

µ_σ(t)ϕ^∗_t(g0),

µ µ µσ(t)ϕ^∗_t(g0)

¶²

n

σ(t)ϕ^∗_t(g0)

! .

The velocity vectorα⁰(t) is as follows:

α⁰(t) =¡₁

2tr_σ(t)ϕ^∗_t(g0)(_∂t^∂σ(t)ϕ^∗_t(g0))µ_σ(t)ϕ^∗_t(g0), (_µ ^µ

σ(t)ϕ∗

t(g0))ⁿ²(_∂t^∂σ(t)ϕ^∗_t(g0)−¹_ntrσ(t)ϕ^∗_t(g0)(_∂t^∂σ(t)ϕ^∗_t(g0))σ(t)ϕ^∗_t(g0)

´

=³

1 2 1

σ(t)(ϕ^∗)⁻¹_t (g0)(^dσ(t)_dt ϕ^∗_t(g0) +σ(t)^∂_∂ϕ^∗_t(g0))µ_σ(t)ϕ^∗_t(g0), (_µ ^µ

σ(t)ϕ∗

t(g0))ⁿ²(^dσ(t)_dt ϕ^∗_t(g0) +σ(t)^∂_∂ϕ^∗_t(g0)−¹_nσ(t)¹ (ϕ^∗)⁻¹_t (g0)(^dσ(t)_dt ϕ^∗_t(g0) +σ(t)^∂_∂ϕ^∗_t(g0))σ(t)ϕ^∗_t(g0)¢

and att= 0 we have (4.1)

α⁰(0) = (1

2g₀⁻¹(σ⁰(0)g0+£Vg0)µg0,( µ µg0

)²ⁿ(σ⁰(0)g0+£Vg0−1

n(g₀⁻¹(σ⁰(0)g0+£Vg0)g0)), where £ denotes the Lie derivative, V is the time-dependent vector field such that V(ϕt(p)) = _dt^dϕt(p)) for anyp∈M. On the other hand forα(t) as a Ricci flow curve according to (3.4), the velocity vectorα⁰(t) is as follows:

α⁰(t) = (−Rµ_σ(t)ϕ^∗_t(g0),−2(_µ ^µ

σ(t)ϕ∗

t(g0))ⁿ²[Ric(σ(t)ϕ^∗_t(g0))−^R_nσ(t)ϕ^∗_t(g0)]

= (−_σ(t)¹ ϕ^∗_t(R0)σ(t)ⁿ²µϕ^∗_t(g0),−2_σ(t)¹ (_µ ^µ

σ(t)ϕ∗ t(g0))ⁿ²·

·[ϕ^∗_t(Ricg0)−_n¹_σ(t)¹ ϕ^∗_t(R0)σ(t)ϕ^∗_t(g0)])

= (−ϕ^∗_t(R0)σ(t)ⁿ²⁻¹µϕ^∗_t(g0),_σ(t)⁻²(_µ ^µ

σ(t)ϕ∗

t(g0))ⁿ²[ϕ^∗_t(Ricg0)−^ϕ∗t(R_n ⁰⁾ϕ^∗_t(g0)]);

att= 0 we have

(4.2) α⁰(0) =

µ

−R0µg0,−2( µ µg0

)²ⁿ[Ricg0−R0

n g0]

¶ . By comparing (4.1) and (4.2) have:

( −R0= ¹₂g⁻¹₀ (σ⁰(0)g0+£Vg0),

−2(Ricg0−^R_n⁰g0) =σ⁰(0)g0+£Vg0−_n¹g₀⁻¹(σ⁰(0)g0+£Vg0)g0, Therefore−2Ricg0=σ⁰(0)g0+£Vg0,

Letσ⁰(0) = 2λ; then−2Ricg0= 2λg0+£Vg0. ¤

Remark 4.3. The above theorem coincides with the well-known result on Ricci soliton - see for example [5].

The Ricci flow curve has certain properties which will be included in another forthcoming paper.

Acknowledgements. The authors wish to appreciate Prof. D. Ebin and Prof.

H. Omori for giving helpful hints. The second author also would like to appreciate Prof. H. P. Kunzle for introducing him ”Ricci flow” together with excellent ideas.

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Authors’ address:

Hajar Ghahremani-Gol and Asadollah Razavi (corresponding author) Department of Mathematics and Computer Science,

Amirkabir University of Technology, 424 Hafez Ave., Tehran, 1591634311, Iran.

E-mail: [email protected] , [email protected]