arXiv:1807.04142v1 [math.DG] 11 Jul 2018
Ricci flow on Finsler surfaces B. Bidabad∗ and M.K. Sedaghat
Abstract Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to the original Ricci flow by pulling back the solution of the Ricci-DeTurck flow using appropriate diffeomorphisms. At the end, we illustrate this argument by some examples.
Keywords: Surface Ricci flow, Finsler surface, Ricci-DeTurck, parabolic differential equation, Berwald frame. AMS subject classification: 53C60, 53C44
Introduction The Ricci flow is a geometric evolution equation for the metric tensor on a general Riemannian manifold. The normalized Ricci flow has the property that all its fixed points are Einstein metrics. In his celebrated paper [10], Hamilton showed that in 3-manifolds, the positive Ricci curvature condition on the initial metric implies that the Ricci flow exists for all time and converges to a Riemannian metric of constant curvature. This phenomenon has been shown later for other types of curvature conditions in other dimensions by several authors. ∗
The corresponding author,
[email protected]
1
The study of Ricci flow on surfaces is far simpler than its counterparts in higher dimensional cases. Hence one can obtain much more detailed and comprehensive results. On surfaces, the Ricci flow solutions remain within a conformal class and clearly coincide with that of the Yamabe flow on surfaces. In [11] Hamilton proved that for a compact oriented Riemannian surface (M, g), if M is not diffeomorphic to the 2-sphere S2 , then any metric g converges to a constant curvature metric under the Ricci flow and if M is diffeomorphic to S2 , then any metric g with positive Gaussian curvature on S2 converges to a metric of constant curvature under the flow. Later, Chow in [8] removed the positive Gaussian curvature assumption in Hamilton’s theorem and proved that for evolution of any metric on S2 , under Hamilton’s Ricci flow, the Gaussian curvature becomes positive in finite time and concluded that under the flow any metric g on a Riemannian surface converges to a metric of constant curvature. Thus for compact surfaces, Ricci flow provides a new proof of the uniformization theorem. Much is also known in the complete case. There are also many interesting subtleties in setting up this flow in the incomplete cases. Moreover, surface Ricci flow has started making impacts on practice fields and tackling fundamental engineering problems. In Finsler geometry as a natural generalization of Riemannian geometry, the problem of constructing the Finslerian Ricci flow raises a number of new conceptual and fundamental issues in regards to the compatibility of geometrical and physical objects and their optimal configurations. A fundamental step in the study of any system of evolutionary partial differential equations is to show the short time existence and uniqueness of solutions. Recently, an evolution of a family of Finsler metrics along Finsler Ricci flow has been studied by the first named author in several joint works and it has been shown that such flows exist in short time and converge to a limit metric; for instance, see [5]. In the present work, we study the Ricci flow on the closed Finsler surfaces and prove the short-time existence and uniqueness of solutions for the Ricci flow. Intuitively, since the Ricci flow system of equations is only weakly parabolic, its short-time existence and uniqueness do not follow from the standard theory of parabolic equations. Following the procedure described 2
by D. DeTurck in Riemannian space [9], we have introduced the Finslerian Ricci-DeTurck flow on Finsler surfaces by Eq. (34) and prove existence and uniqueness of short-time solutions. More precisely, we prove: Theorem 1. Let M be a compact Finsler surface. Given any initial Finsler structure F0 , there exists a real number T > 0 and a smooth one-parameter family of Finsler structures F˜ (t), t ∈ [0, T ), such that F˜ (t) is a unique solution to the Finslerian Ricci-DeTurck flow with F˜ (0) = F0 . Next, a solution to the original Ricci flow is found by pulling back the solution to the Ricci-DeTurck flow via appropriate diffeomorphisms. This leads to Theorem 2. Let M be a compact Finsler surface. Given any initial Finsler structure F0 , there exists a real number T > 0 and a smooth one-parameter family of Finsler structures F (t), t ∈ [0, T ), such that F (t) is a unique solution to the Finslerian Ricci flow and F (0) = F0 .
1 1.1
Preliminaries and notations Chern connection; A global approach
Let M be a real smooth surface and denote by T M the tangent bundle of tangent vectors, by π : T M0 −→ M the fiber bundle of non-zero tangent vectors and by π ∗ T M −→ T M0 the pullback tangent bundle. Let F be a Finsler structure on T M0 and g the related Finslerian metric. A Finsler manifold is denoted here by the pair (M, F ). Any point of T M0 is denoted by z = (x, y), where x = πz ∈ M and y ∈ Tx M. Let us denote by T T M0 , the tangent bundle of T M0 and by ρ, the canonical linear mapping ρ : T T M0 −→ π ∗ T M, where, ρ = π∗ . For all z ∈ T M0 , Vz T M is the set of all vertical vectors at z, that is, the set of vectors which are tangent to the fiber through z. Consider the decomposition T T M0 = HT M ⊕ V T M, which permits us to uniquely represent a vector ˆ ∈ X (T M0 ) as the sum of the horizontal and vertical parts namely, field X ˆ = HX ˆ + V X. ˆ The corresponding basis is denoted here by { δ i , ∂ i }, X δx ∂y 3
where,
δ δxi
:=
j ∂ − Nij ∂y∂ j , Nij = 21 ∂G ∂xi ∂y i 2 1 ih ∂ 2 F 2 j g ( ∂yh ∂xj y − ∂F ). We 4 ∂xh
and Gi are the spray coefficients de-
denote the formal Christoffel symbols fined by Gi = 1 i by γjk = 2 g ih (∂j ghk + ∂k gjh − ∂h gjk ) where, ∂k = ∂x∂ k . The dual bases are denoted by {dxi , δy i } where, δy i := dy i +Nji dxj . Let us denote a global representation of the Chern connection by ∇ : X (T M0 )×Γ(π ∗ T M) −→ Γ(π ∗ T M). ˆ = ∇ ˆy Consider the linear mapping µ : T T M0 −→ π ∗ T M, defined by µ(X) X ∂ i ∗ ˆ where, X ∈ T T M0 and y = y ∂xi is the canonical section of π T M. The connection 1-forms of Chern connection in these bases are given by i ωj = Γijk dxk where, Γijk = 12 g ih (δj ghk + δk gjh − δh gjk ) and δk = δxδk . In the sequel, all the vector fields on T M0 are decorated with a hat and denoted by ˆ Yˆ , Zˆ and the corresponding sections of π ∗ T M by X = ρ(X), ˆ Y = ρ(Yˆ ) X, ˆ respectively unless otherwise specified. The torsion freeness and Z = ρ(Z), and almost metric compatibility of the Chern connection are given by ˆ Yˆ ], ∇Xˆ Y − ∇Yˆ X = ρ[X, ˆ X, Y ), (∇ ˆ g)(X, Y ) = 2C(µ(Z), Z
(1) (2) ∂g
respectively, where C is the Cartan tensor with the components Cijk = ∂yijk . In a local coordinates on T M the Chern horizontal and vertical covariant derivatives of an arbitrary (1, 2) tensor field S on π ∗ T M with the components i (Sjk (x, y)) on T M are denoted by i i i i s i ∇l Sjk := δl Sjk − Ssk Γsjl − Sjs Γskl + Sjk Γsl , i i ˙ l S := ∂˙l S , ∇ jk jk
where, ∇l := ∇
δ δxl
˙ l := ∇ and ∇
∂ ∂y l
. Horizontal metric compatibility of the
Chern connection is given in local coordinates by ∇l gjk = 0, see [3, p. 45]. The local Chern hh-curvature tensor is given by Rji kl = δk Γijl − δl Γijk + Γihk Γhjl − Γihl Γhjk ,
(3)
see [3, p. 52]. The reduced hh-curvature tensor is a connection free tensor field which is also referred to as the Riemann curvature by certain authors. In a local coordinates on T M, the components of the reduced hh-curvature tensor are given by Rik := F12 y j Rji km y m, which are entirely expressed in 4
terms of x and y derivatives of spray coefficients Gi as follows Rik :=
2 i 1 ∂Gi ∂ 2 Gi j ∂Gi ∂Gj j ∂ G (2 − y + 2G − ), F 2 ∂xk ∂xj ∂y k ∂y j ∂y k ∂y j ∂y k
(4)
see [3, p. 66].
1.2
Lie derivatives of Finsler metrics
The Lie derivative of an arbitrary Finslerian (0, 2) tensor field T = Tjk (x, y)dxj ⊗ dxk on ⊗2 π ∗ T M with respect to an arbitrary vector field Vˆ on T M0 is given by ˆ Y ) − T (X, ρ[Vˆ , Yˆ ]), (LVˆ T )(X, Y ) = Vˆ (T (X, Y )) − T (ρ[Vˆ , X], ˆ = X, ρ(Yˆ ) = Y and X, ˆ Yˆ ∈ Tz T M0 , see [4]. The Lie derivative where, ρ(X) of Finsler metric g with respect to the arbitrary vector field Vˆ on T M0 is given by ˆ Y ) − g(X, ρ[Vˆ , Yˆ ]). (LVˆ g)(X, Y ) = Vˆ (g(X, Y )) − g(ρ[Vˆ , X], By means of the torsion freeness of Chern connection defined by (1), Lie derivative of the Finsler metric g can be rewritten as (LVˆ g)(X, Y ) = Vˆ (g(X, Y )) − g(∇Vˆ X − ∇Xˆ V, Y ) − g(X, ∇Vˆ Y − ∇Yˆ V ) = Vˆ (g(X, Y )) − g(∇ ˆ X, Y ) + g(∇ ˆ V, Y ) X
V
− g(X, ∇Vˆ Y ) + g(X, ∇Yˆ V ).
(5)
By the almost g-compatibility of Chern connection defined by (2), we have 2C(µ(Vˆ ), X, Y ) = (∇Vˆ g)(X, Y ) = Vˆ (g(X, Y )) − g(∇Vˆ X, Y ) − g(X, ∇Vˆ Y ), Therefore, Vˆ (g(X, Y )) = 2C(µ(Vˆ ), X, Y ) + g(∇Vˆ X, Y ) + g(X, ∇Vˆ Y ).
(6)
Plugging the equation (6) in (5) we obtain (LVˆ g)(X, Y ) = 2C(µ(Vˆ ), X, Y ) + g(∇Xˆ V, Y ) + g(X, ∇Yˆ V ). 5
(7)
Replacing X and Y by the canonical section y = y i ∂x∂ i in (7) we obtain (LVˆ g)(y, y) = 2C(µ(Vˆ ), y, y) + g(∇yˆ V, y) + g(y, ∇yˆ V ), ˆ = y i δxδ i . Using C(µ(Vˆ ), y, y) = 0, see [3, p. 23], and the symmetric where, y property of g(∇yˆ V, y) one arrives at (LVˆ g)(y, y) = 2g(y, ∇yˆ V ),
(8)
where, V = v i ∂x∂ i is a section of π ∗ T M. In the local coordinates, (8) can be written as y i y j LVˆ gij = 2y i y j gik ∇j v k . Using ∇j gik = 0, we obtain y i y j LVˆ gij = 2y i y j ∇j vi ,
(9)
where, vi = gik v k .
1.3
The Berwald frame and a geometrical setup on SM
Let (M, F ) be a Finsler surface and SM the quotient of T M0 under the following equivalence relation: (x, y) ∼ (x, y˜) if and only if y, y˜ are positive multiples of each other. In other words, SM is the bundle of all directions or rays, and is called the (projective) sphere bundle. The local coordinates (x1 , x2 ) on M induce the global coordinates (y 1, y 2 ) on each fiber Tx M, through the expansion y = y i ∂x∂ i . Therefore (xi ; y i ) is a coordinate system on SM, where the coordinates y i are regarded as homogeneous coordinates in the projective space. Using the canonical projection p : SM −→ M, one can pull the tangent bundle T M back to p∗ T M which is a vector bundle with the fiber dimension 2 over the 3-manifold SM. The vector bundle p∗ T M has a global i section l := Fy(y) ∂x∂ i and a natural Riemannian metric which we here denote by g := gij (x, y)dxi ⊗ dxj . One can complete l into a positively oriented g-orthonormal frame {e1 , e2 } for p∗ T M, with e2 := l, by setting Fy1 ∂ Fy2 ∂ −√ , e1 = √ 1 g ∂x g ∂x2 y1 ∂ y2 ∂ e2 = + , F ∂x1 F ∂x2 6
p √ ∂F where, g := det(gij ) and Fyi abbreviates the partial derivative ∂y i . In 2-dimensional case, {e1 , e2 } is a globally defined g-orthonormal frame field for p∗ T M called a Berwald frame. The natural dual of l is the Hilbert form defined by ω := Fyi dxi , which is a global section of p∗ T ∗ M. The coframe corresponding to {e1 , e2 } is defined here by {ω 1, ω 2 }, where √ g 2 1 1 ω = (y dx − y 1dx2 ) = v11 dx1 + v21 dx2 , F (10) ω 2 = Fy1 dx1 + Fy2 dx2 = v12 dx1 + v22 dx2 . The sphere bundle SM ⊂ T M is a 3-dimensional Riemannian manifold equipped with the induced Sasaki metric ω1 ⊗ ω1 + ω2 ⊗ ω2 + ω3 ⊗ ω3, where,
√
g 2 δy 1 δy 2 δy 1 δy 2 (y − y1 ) = v11 + v21 . (11) F F F F F The collection {ω 1 , ω 2, ω 3 } is a globally defined orthonormal frame for T ∗ (SM). Its natural dual frame is given by {ˆ e1 , eˆ2 , eˆ3 }, where 3
ω :=
Fy1 δ Fy2 δ δ δ −√ = u11 1 + u21 2 , eˆ1 = √ 1 2 g δx g δx δx δx 2 1 y δ y δ 1 δ 2 δ + = u + u , eˆ2 = 2 2 F δx1 F δx2 δx1 δx2 Fy2 Fy1 ∂ ∂ ∂ ∂ eˆ3 = √ F 1 − √ F 2 = F u11 1 + F u21 2 . g ∂y g ∂y ∂y ∂y
(12) (13) (14)
These three vector fields on SM form a global orthonormal frame for T (SM). The first two are horizontal while the third one is vertical. The objects ω 1 , ω 2, ω 3 and eˆ1 , eˆ2 , eˆ3 are defined in terms of objects that live on the slit tangent bundle T M0 . But they are invariant under positive rescaling in y. Therefore they give bonafide objects on the sphere bundle SM, see [3, p. 92-94].
1.4
The integrability condition for Finsler metrics
Here, we first recall that all the Riemannian metrics on the fibers of the pulled-back bundle do not come from a Finsler structure. Hence, not ev7
ery arbitrary symmetric (0, 2)-tensor gij (x, y) arises from a Finsler structure F (x, y). Intuitively, in order to make sure gij (x, y) are components of a Finsler structure, the essential integrability criterion is the total symmetry of (gij )yk on all three indices i, j, k. In fact, gij (x, y) arises from a Finsler structure F (x, y) if and only if (gij )yk is totally symmetric in its three indices, see [2, p. 56]. Symmetry of (gij )yk on all three indices i, j, k is known in the literature as integrability condition. Moreover, we have to make sure the integrability criterion is satisfied in every step along the Ricci flow. To this end we consider a general evolution equation given by ∂ g(t) = ω(t), g(0) := g0 , (15) ∂t where, ω(t) := ω(t, x, y) is a family of symmetric (0, 2)-tensors on π ∗ T M, zero-homogenous with respect to y. The following Lemma establishes the integrability condition, see also [5, p. 749]. Lemma 1.1. Let g(t) be a solution to the evolution equation (15). There is a family of Finsler structures F (t) on T M such that, 1 ∂ 2 F (t) . 2 ∂y i ∂y j
gij (t) =
(16)
Proof. Let M be a compact differential manifold, F (t) a family of smooth 1-parameter Finsler structures on T M0 and g(t) the Hessian matrix of F (t) which defines a scalar product on π ∗ T M for every t. Let g(t) be a solution to the evolution equation (15). We have Z t g(t) = g(0) + ω(τ )dτ, ∀τ ∈ [0, t). (17) 0
We show that the metric g(t) satisfies the integrability condition, or equivalently there is a Finsler structure F (t) on T M0 satisfying (16). For this purpose, we multiply gij by y i and y j in (17), Z t i j i j y y gij (t) = y y gij (0) + y i y j ωij (τ )dτ. 0
By means of the initial condition y y gij (0) = F 2 (0), we get Z t i j 2 y y gij (t) = F (0) + y i y j ωij (τ )dτ. i j
0
8
(18)
1
By positive definiteness assumption of gij , we put F = (y i y j gij ) 2 . Twice vertical derivatives of (18) yields 1 1 ∂2F 2 = gkl (0) + k l 2 ∂y ∂y 2
Z
0
t
∂2 (y i y j ωij (τ ))dτ. k l ∂y ∂y
(19)
On the other hand, by straightforward calculation we have 1 ∂2 1 ∂ 2 ωij (τ ) i j ∂ωik (τ ) ∂ωil (τ ) i i j y + ωkl (τ ), (y y ω (τ )) = yy = − ij 2 ∂y k ∂y l 2 ∂y k ∂y l ∂y l ∂y k (20) for all τ ∈ [0, t). Using (15) we obtain 1 ∂ 2 ωij (τ ) i j y y = 0, 2 ∂y k ∂y l
∂ωik (τ ) = 0, ∂y l
∂ωil (τ ) = 0. ∂y k
Therefore, (20) is reduced to 1 ∂2 (y i y j ωij (τ )) = ωkl (τ ), 2 ∂y k ∂y l
(21)
for all τ ∈ [0, t). Finally, replacing (21) in (19) we get 1 ∂2F 2 = gkl (0) + 2 ∂y k ∂y l
Z
i
ωkl (τ )dτ = gkl .
o
Therefore, every gij (t) on the fibers of pulled-back bundle, arises from a Finsler structure. This completes the proof.
2
Semi-linear strictly parabolic equations on SM
Recall that a quasi-linear system is a system of partial differential equations where, the derivatives of principal order terms occur only linearly and coefficients may depend on derivatives of the lower order terms. It is called semi-linear if it is quasi-linear and coefficients of the principal order terms depend only on the independent variables, but not on the solution, see [12, p. 9
45]. Let M be a 2-dimensional manifold and u : M −→ R a smooth function on M. A semi-linear strictly parabolic equation is a PDE of the form ∂u ∂2u ∂u ij = a (x, t) i j + h(x, t, u, i ), ∂t ∂x ∂x ∂x
i, j = 1, 2,
where, aij and h are smooth functions on M and for some constant λ > 0 we have the parabolic assumption aij ξi ξj ≥ λ k ξ k2 ,
0 6= ξ ∈ χ(M),
that is, all eigenvalues of A = (aij )2×2 have positive signs or equivalently A is positive definite. Definition 2.1. Let M be a surface and φ : SM −→ R a smooth function on the sphere bundle SM. Consider the following semi-linear strictly parabolic equation on SM; ∂φ = GAB (x, y, t)ˆ eA eˆB φ + h(x, y, t, φ, eˆA φ), ∂t
A, B = 1, 2, 3,
where, eˆA is a local frame for the tangent bundle T SM and stand for partial derivatives on SM. Here, GAB and h are smooth functions on SM and G = (GAB ) is positive definite. More precisely, a semi-linear strictly parabolic equation on SM can be written in the form ∂φ = pab (x, y, t)ˆ ea eˆb φ + q(x, y, t)ˆ e3eˆ3 φ + ma (x, y, t)ˆ ea eˆ3 φ + lower order terms, ∂t (22) where a, b = 1, 2, and the matrix ! 1 M P 2 , G= 1 t M Q 2 3×3
is positive definite where, P = (pab )2×2 , Q = (q)1×1 , M = (ma )2×1 .
10
Lemma 2.1. Let (M, F ) be a Finsler surface and φ : T M −→ R a zerohomogeneous smooth function on the tangent bundle T M. The semi-linear differential equation ∂φ δ2φ ∂2φ = g ij i j + F 2 g ij i j + lower order terms, ∂t δx δx ∂y ∂y
i, j = 1, 2,
(23)
is a strictly parabolic equation on SM. Proof. Let us denote again by φ its restriction on SM. According to (12) and (13), replacing eˆa = uia δxδ i , we obtain eˆa φ = uia
δφ , δxi
eˆb eˆa φ = uia ujb
δφ δ2φ + eˆb (uia )( i ), i j δx δx δx
a, b = 1, 2.
Multiplying the both sides by g ab leads to δ2φ δφ + g ab eˆb (uia )( i ) i j δx δx δx 2 δφ δ φ = g ij i j + g ab eˆb (uia )( i ), δx δx δx
g ab eˆb eˆa φ = g ab uia ujb
where, gab = gij uia ujb . According to (10), by using the notations ω c := vic dxi 2φ and B c := vic g ab eˆb (uia ) one can rewrite the expression g ij δxδi δx j on SM with respect to eˆa as follows g ab eˆb eˆa φ − B c eˆc φ = g ij
δ2φ , δxi δxj
c = 1, 2.
Hence, (14) yields ∂φ , ∂y i ∂φ eˆ3 eˆ3 φ = eˆ3 (F ui1 i ) ∂y ∂2φ ∂φ ∂F ∂φ = F 2 uj1 ui1 j i + F (ˆ e3 ui1 )( i ) + F uj1 ( j )ui1 i . ∂y ∂y ∂y ∂y ∂y eˆ3 φ = F ui1
11
∂F Using the fact uj1 ∂y j = 0, see [1, p. 161], we have
eˆ3 eˆ3 φ = F 2 uj1 ui1
∂2φ ∂φ i + F (ˆ e u )( ). 3 1 ∂y j ∂y i ∂y i
Multiplying the both sides by g 11 and taking into account that g 11 ui1 uj1 = g ij − y i y j we get g 11 eˆ3 eˆ3 φ = F 2 g ij
∂2φ ∂φ + F g 11 (ˆ e3 ui1 ) i , j i ∂y ∂y ∂y i
where, g11 = gij ui1 uj1 . According to (11), we have ω 3 = vi1 δyF . Denoting 2φ D 1 := vi1 F g 11eˆ3 ui1 one can rewrite the expression F 2 g ij ∂y∂j ∂y i on SM with respect to eˆ3 as follows g 11 eˆ3 eˆ3 φ − D 1 eˆ3 φ = F 2 g ij
∂2φ . ∂y j ∂y i 2
2
φ 2 ij ∂ φ ab eˆb eˆa φ − Thus the principal order terms g ij δxδi δx j and F g ∂y i ∂y j convert to g c 11 1 B eˆc φ and g eˆ3 eˆ3 φ − D eˆ3 φ on SM. On the other hand, the order of lower order terms in (23) do not change after rewriting them in terms of the basis {ˆ e1 , eˆ2 , eˆ3 } on SM. Therefore (23) on SM is written as
∂φ = g ab eˆb eˆa φ + g 11 eˆ3 eˆ3 φ − B c eˆc φ − D 1 eˆ3 φ + lower order terms, ∂t
(24)
where, a, b, c = 1, 2. Using the fact that g is positive definite, the coefficient ! g ab 0 , G= 0 g 11 3×3
of principal order terms of (24) is positive definite on SM. Therefore, by virtue of (22) the differential equation (24) is a semi-linear strictly parabolic equation on SM.
3
A vector field on SM
Let (M, F ) and (N, F¯ ) be two Finsler surfaces with the corresponding metric tensors g and h, respectively. Let (xi , y i ) and (¯ xi , y¯i ) be local coordinate 12
systems on T M and T N, respectively. Let c be a geodesic on M. The natural lift of c on T M, namely, c˜ : t ∈ I :−→ c˜(t) = (xi (t), (
dxi )(t)) ∈ T M, dt
is a horizontal curve. That is, its tangent vector field c˜˙(t) = zontal. Consider a diffeomorphism
dxi δ , dt δxi
is hori-
ϕ : T M −→ T N,
(xi , y i ) 7→ ϕ(xi , y i ) = (ϕα (xi , y i )) = (ϕj (xi , y i ), ϕ2+j (xi , y i )),
such that c¯(t) := (ϕ ◦ c˜)(t) is a horizontal curve, where i, j = 1, 2, and α = 1, ..., 4. Throughout this section, ϕ takes horizontal curves to horizontal curves. ¯ i the coefficients of horizontal covariant Let us denote by Γijk and Γ jk derivatives of Chern connection on (M, F ) and (N, F¯ ), respectively. Then we have δ d2 x¯j δ d¯ xj ¯ δ d2 x¯j d¯ xj d¯ xk ¯ i xj δ ¯ c¯˙ d¯ ¯ c¯˙ c¯˙ = ∇ ∇ Γ ) ∇ = + = ( + . ˙ c ¯ jk dt δ¯ xj dt2 δ¯ xj dt δ¯ xj dt2 dt dt δ¯ xi (25) On the other hand δϕi dxp d¯ xi = p , dt δx dt
d2 x¯i δ 2 ϕi dxp dxq δϕi d2 xp = + p 2 . dt2 δxp δxq dt dt δx dt
Replacing the last equations in (25) leads to q i 2 h q p j k p 2 i ¯ i δϕ δϕ dx dx ) δ , ¯ c¯˙ c¯˙ = ( δ ϕ dx dx + δϕ d x + Γ ∇ jk δxp δxq dt dt δxh dt2 δxp δxq dt dt δ¯ xi
(26)
where, all the indices run over the range 1, 2. The geodesic c on M, satisfies q p d2 xh h dx dx + Γ = 0. pq dt2 dt dt
Substituting
d2 xh dt2
from the last equation in (26), leads to
p 2 i i q j k ¯ c¯˙ c¯˙ = dx dx ( δ ϕ − δϕ Γh + Γ ¯ i δϕ δϕ ) δ . ∇ jk dt dt δxp δxq δxh pq δxp δxq δ¯ xi
13
(27)
Next, let Aipq :=
2 i j k δ 2 ϕi δϕi h ¯ i δϕ δϕ + F 2 ∂ ϕ . − Γ + Γ jk δxp δxq δxh pq δxp δxq ∂y p ∂y q
Contracting Aipq with g pq leads to the following operators (Φg,h ϕ)i := g pq (
2 i j k δϕi h δ 2 ϕi 2 ∂ ϕ ¯ i δϕ δϕ ), + F − Γ + Γ pq jk δxp δxq ∂y p ∂y q δxh δxp δxq
(28)
where, (Φg,h ϕ)i = g pq Aipq and i = 1, 2. For greater indices we consider the following operator. (Φg,h ϕ)2+i := g pq (
2 2+i δ 2 ϕ2+i ∂ϕ2+i δNqk 2∂ ϕ + F + ), δxp δxq ∂y p ∂y q ∂y k δxp
(29)
where, i = 1, 2. Summarizing the above definitions we have α
(Φg,h ϕ) =
(Φg,h ϕ)i (Φg,h ϕ)2+i
α=i α = 2 + i,
where, i = 1, 2. Next, we show the operator (Φg,h ϕ)α is invariant under all diffeomorphisms on T M. Lemma 3.1. Let (M, F ) and (N, F¯ ) be two Finsler surfaces with the corresponding metric tensors g and h, respectively. If ψ is a diffeomorphism from T M to itself, then it leaves invariant the operator (Φg,h ϕ)α , that is (Φψ∗ (g),h ψ ∗ ϕ)α |(˜x,˜y) = (Φg,h ϕ)α |(x,y) ,
α = 1, .., 4,
where, x˜i = ψ ∗ xi and y˜i = ψ ∗ y i . Proof. Let (xi , y i ) and (¯ xi , y¯i ) be the two local coordinate systems on T M 14
and T N, respectively and x˜i = ψ ∗ xi and y˜i = ψ ∗ y i . For α = i, we have δ 2 ϕi ∂ 2 ϕi 2 (x, y) + F (x, y) (x, y) δxp δxq ∂y p∂y q δϕi δϕj δϕk ¯ i (¯ − k (x, y)Γkpq (x, y) + Γ x , y ¯ ) (x, y) (x, y) jk δx δxp δxq δ 2 ϕi ∂ 2 ϕi 2 pq (ψ(˜ x, y˜)) + F (ψ(˜ x, y˜)) p q (ψ(˜ x, y˜)) = g (ψ(˜ x, y˜)) δxp δxq ∂y ∂y δϕi − k (ψ(˜ x, y˜))Γkpq (ψ(˜ x, y˜)) δx δϕk δϕj ¯ i (¯ (ψ(˜ x , y ˜ )) (ψ(˜ x , y ˜ )) +Γ x , y ¯ ) jk δxp δxq δ 2 (ψ ∗ ϕ)i ∂ 2 (ψ ∗ ϕ)i ∗ 2 = (ψ ∗ g)pq (˜ x, y˜) (˜ x , y ˜ ) + (ψ F )(˜ x , y ˜ ) (˜ x, y˜) δ˜ xp δ˜ xq ∂ y˜p ∂ y˜q δ(ψ ∗ ϕ)i (˜ x, y˜)Γ(ψ ∗ g)kpq (˜ x, y˜) − k δ˜ x δ(ψ ∗ ϕ)j δ(ψ ∗ ϕ)k ¯ i (¯ x , y ¯ ) +Γ (˜ x , y ˜ ) (˜ x , y ˜ ) jk δ˜ xp δ˜ xq = (Φψ∗ (g),h ψ ∗ ϕ)i |(˜x,˜y) .
(Φg,h ϕ)i |(x,y) = g pq (x, y)
Similarly, for α = 2 + i, one can show that (Φg,h ϕ)2+i |(x,y) = (Φψ∗ (g),h ψ ∗ ϕ)2+i |(˜x,˜y) , where, i = 1, 2. This completes the proof. Remark 3.1. Let (M, F ) and (N, F¯ ) be two Finsler surfaces with the corresponding metric tensors g and h, respectively. Let ϕ : T M −→ T N, ϕ(xi , y i ) = (ϕα (xi , y i )), α = 1, .., 4, be a diffeomorphism and takes horizontal curves to horizontal curves. Given ϕ0 : T M −→ T N, we consider the following evolution equation ∂ α ϕ = (Φg,h ϕ)α , ∂t
ϕ(0) = ϕ0 .
(30)
By restricting ϕα ’s to SM and using Lemma 2.1, one can see that (30) is a strictly parabolic system. Hence, there is a unique solution for (30) in short time. 15
Corollary 3.1. Let (M, F˜ ) and (N, F¯ ) be two Finsler surfaces with corresponding metric tensors g˜ and h, respectively. Let N = M and ϕ be the identity map ϕ = Id : T M −→ T M, ϕ(xi , y i ) = (xi , y i ), then we have ˜ ipq + Γ ¯ ipq ), (Φg˜,h Id)α = (Φg˜,h Id)i = g˜pq (−Γ
α = i,
˜i , Γ ¯ i are the coefficients of horizontal covariant derivawhere, i = 1, 2 and Γ pq pq tives of Chern connection with respect to the metrics g˜ and h, respectively. Let ξ be a vector field on T M with the components ξ := (Φg˜,h Id)i
∂ ˜i + Γ ¯i ) ∂ . = g˜pq (−Γ pq pq i ∂x ∂xi
(31)
Using the fact that the difference of two connections is a tensor, ξ is a globally well-defined vector field. It can be easily verified that the components of ξ are homogeneous of degree zero on y, thus ξ can be considered as a vector field on SM.
4
Ricci-DeTurck flow and its existence and uniqueness of solution
There are several well known definitions for Ricci tensor in Finsler geometry. For instance, H. Akbar-Zadeh has considered two Ricci tensors on Finsler manifolds in his works namely, one is defined by Ricij := [ 21 F 2 Ric]yi yj where, Ric is the Ricci scalar defined by Ric := g ik Rik = Rii and Rik are defined by (4), see [3, p. 192]. Another Ricci tensor is defined by Rcij := 21 (Rij + Rji ), where Rij is the trace of hh-curvature defined by Rij = Rillj . The difference ∂R between these two Ricci tensors is the additional term 21 y k ∂yjki appeared in ∂R
the first definition. More precisely, we have Ricij − Rcij = 12 y k ∂yjki . D. Bao based on the first definition of Ricci tensor has considered the following Ricci flow in Finsler geometry, ∂ gjk (t) = −2Ricjk , ∂t 16
g(t=0) = g0 ,
(32)
where, gjk (t) is a family of Finslerian metrics defined on π ∗ T M × [0, T ). ∂ Contracting (32) with y j y k , via Euler’s theorem, leads to ∂t F 2 = −2F 2 Ric. That is, ∂ log F (t) = −Ric, F (t = 0) := F0 , (33) ∂t where, F0 is the initial Finsler structure, see [2]. Here and everywhere in the present work we consider the first Akbar-Zadeh’s definition of Ricci tensor and the related Ricci flow (33). One of the advantages of the Ricci quantity Ricij , used in the present work is its independence on the choice of Cartan, Berwald or Chern connections. Definition 4.1. Let M be a compact surface with a fixed background Finsler structure F¯ and related Finsler metric h. Assume that for all t ∈ [0, T ), F˜ (t) is a one-parameter family of Finsler structures on T M and g˜(t) is the tensor metric related to F˜ (t). We say that F˜ (t) is a solution to the Finslerian Ricci-DeTurck flow if ∂ ˜2 F (t) = −2F˜ 2 (t)Ric(˜ g (t)) − Lξ F˜ 2 (t), ∂t
(34)
where, Lξ is the Lie derivative with respect to the vector field ξ = (Φg˜(t),h Id)i ∂x∂ i on SM as mentioned earlier. The following theorem shows that the Ricci-DeTurck flow (34) is well defined and has a unique solution on a short time interval. Proof of Theorem 1. Let M be a compact surface with a fixed background Finsler structure F¯ and the related Finsler metric h. Here, all the indices run over the range 1, 2. The Ricci-DeTurck flow (34) can be written in the following form ypyq
∂ g˜pq (t) = −2F˜ 2 (t)Ric(˜ g (t)) − Lξ y p y q g˜pq (t), ∂t
where, g˜(t) is the metric tensor related to F˜ (t). Also we have Lξ (y py q g˜pq ) = y p y q Lξ g˜pq + 2y pg˜pq Lξ y q . 17
(35)
Therefore, (35) becomes ypyq
∂ g˜pq (t) = −2F˜ 2 (t)Ric(˜ g (t)) − y py q Lξ g˜pq − 2y p g˜pq Lξ y q . ∂t
(36)
By means of the Lie derivative formula (9) along ξ we have y p y q Lξ g˜pq = 2y p y q ∇p ξq ,
(37)
where, ∇p is the horizontal covariant derivative in Chern connection. Using its h-metric compatibility, ∇p ξq becomes ∇p ξq = (∇p g˜ql ξ l ) = g˜ql (∇p ξ l ). As mentioned earlier, if we denote the coefficients of horizontal covariant derivatives of Chern connection with respect to the metric tensors h and g˜ by Γ(h) and Γ(˜ g ), respectively, then by definition (31) of ξ we have ∇p ξq = g˜ql (δp ξ l + Γ(˜ g )lpw ξ w )
= g˜ql [δp (˜ g mn (Γ(h)lmn − Γ(˜ g )lmn ))] + g˜ql Γ(˜ g )lpw ξ w
= g˜ql [(δp g˜mn )(Γ(h)lmn − Γ(˜ g )lmn ) + g˜mn δp (Γ(h)lmn ) − g˜mn δp (Γ(˜ g )lmn )]
+ g˜ql Γ(˜ g )lpw ξ w 1 = g˜mn (δp δq g˜mn − δp δn g˜qm − δp δm g˜qn ) 2 1 − g˜ql g˜mn (δp g˜ls)(δn g˜sm − δs g˜mn + δm g˜ns ) 2 + g˜ql (δp g˜mn )(Γ(h)lmn − Γ(˜ g )lmn ) + g˜ql g˜mn δp (Γ(h)lmn ) + g˜ql Γ(˜ g )lpw ξ w .
Using the last equation, (37) is written y p y q Lξ g˜pq =y p y q g˜mn (δp δq g˜mn − δp δn g˜qm − δp δm g˜qn )
− y py q g˜ql g˜mn (δp g˜ls )(δn g˜sm − δs g˜mn + δm g˜ns )
+ 2y p y q g˜ql (δp g˜mn )(Γ(h)lmn − Γ(˜ g )lmn )
g )lpw ξ w . + 2y p y q g˜ql g˜mn δp (Γ(h)lmn ) + 2y p y q g˜ql Γ(˜
(38)
Also we have −2F˜ 2 Ric(˜ g ) = −2F˜ 2 Rnn = −2F˜ 2 lq Rqnnp lp , 18
(39)
where, Rqnnp are the components of hh-curvature tensor of Chern connection q and lq = yF˜ are the components of Liouville vector field. Replacing (3) in g ), yields (39) and using the definition of Γ(˜ −2F˜ 2 Ric(˜ g ) = −2F˜ 2 lq Rqnnp lp
= −2y p y q (δn Γnqp (˜ g ) − δp Γnqn (˜ g ) + Γnmn (˜ g )Γm g ) − Γnmp (˜ g )Γm g )) qp (˜ qn (˜ 1 = −2y p y q [δn ( g˜mn (δq g˜mp + δp g˜mq − δm g˜pq ))] 2 1 + 2y p y q [δp ( g˜mn (δq g˜mn + δn g˜qm − δm g˜qn ))] 2 p q n − 2y y (Γmn (˜ g )Γm g ) − Γnmp (˜ g )Γm g )). qp (˜ qn (˜
By applying the δp derivative we have −2F˜ 2 Ric(˜ g ) =y py q g˜mn (δp δq g˜mn + δn δm g˜pq − δp δm g˜qn − δn δq g˜mp ) − y p y q (δn g˜nm )(δq g˜mp + δp g˜qm − δm g˜pq )
+ y p y q (δp g˜mn )(δq g˜mn + δn g˜qm − δm g˜qn )
− 2y p y q (Γnmn (˜ g )Γm g ) − Γnmp (˜ g )Γm g )). qp (˜ qn (˜
(40)
Substituting (38) and (40) in (36), we obtain ypyq
∂ g˜pq (t) =y p y q g˜mn δn δm g˜pq ∂t − y p y q (δn g˜nm )(δq g˜mp + δp g˜qm − δm g˜pq )
(41)
+ y p y q (δp g˜mn )(δq g˜mn + δn g˜qm − δm g˜qn )
− 2y py q (Γnmn (˜ g )Γm g ) − Γnmp (˜ g )Γm g )) qp (˜ qn (˜
+ y p y q g˜ql g˜mn (δp g˜ls )(δn g˜sm − δs g˜mn + δm g˜ns ) − 2y py q g˜ql (δp g˜mn )(Γ(h)lmn − Γ(˜ g )lmn )
− 2y py q g˜ql g˜mn δp (Γ(h)lmn ) − 2y py q g˜ql Γ(˜ g )lpw ξ w
− 2y pg˜pq Lξ y q . Using Euler’s theorem yields y py q
∂ 2 g˜pq ∂2 = (y p y q g˜pq ) − 2˜ gnm = 0. ∂y n ∂y m ∂y n ∂y m 19
(42)
In order to get a strictly parabolic system, by virtue of (42) we add the zero 2 term F˜ 2 y py q g˜mn ∂y∂ng˜∂ypqm = 0 to the right hand side of (41). Therefore, we have y py q
∂ 2 g˜pq g˜pq (t) − g˜mn δn δm g˜pq − F˜ 2 g˜mn n m ∂t ∂y ∂y nm + (δn g˜ )(δq g˜mp + δp g˜qm − δm g˜pq )
∂
(43)
− (δp g˜mn )(δq g˜mn + δn g˜qm − δm g˜qn )
+ 2(Γnmn (˜ g )Γm g ) − Γnmp (˜ g )Γm g )) qp (˜ qn (˜
− g˜ql g˜mn (δp g˜ls )(δn g˜sm − δs g˜mn + δm g˜ns )
+ 2˜ gql (δp g˜mn )(Γ(h)lmn − Γ(˜ g )lmn ) mn
+ 2˜ gql g˜
δp (Γ(h)lmn )
p q
− 2y y
g˜ql Γ(˜ g )lpw ξ w
On the other hand, applying twice the vector field metric tensor g˜pq yields δn δm g˜pq =
lq n + 2 g˜np Lξ y = 0. F
δ δxn
on the components of
k 2 2 ∂ 2 g˜pq ∂Nm ∂˜ gpq ˜pq ˜pq k ∂ g l ∂ g − − N − N m n n m n k n k l ∂x ∂x ∂x ∂y ∂x ∂y ∂y ∂xm k 2 ∂Nm g˜pq ˜pq l k ∂ g + Nnl + N N . n m l k k ∂y ∂y ∂y ∂y l
Convecting the last equation with y p y q and using (42) we have y p y q δn δm g˜pq = y p y q
∂ 2 g˜pq . ∂xn ∂xm
Remark that, in the term y p y q g˜mn δn δm g˜pq in (41), there is no term containing 2 pq derivatives of g˜ except y p y q ∂x∂ng˜∂x m . One can rewrite (43) as follows p q
y y
∂
∂t
mn
g˜pq (t) − g˜
2 ˜pq 2 mn ∂ g ˜ δn δm g˜pq − F g˜ + lower order terms = 0. ∂y n ∂y m (44)
Recall that, M is a 2-dimensional Finsler surface and hence is isotropic. Thus, Rqnnp can be part of a symmetric quadratic form, namely, it is symmetric with respect to the indices p and q, see [1, p. 152]. Therefore, by means of symmetry of Lξ (y py q g˜pq ) with respect to the indices p and q we conclude that 20
(44) is symmetric with respect to the indices p and q. If a symmetric bilinear form vanishes on the diagonal, then by the polarization identity it vanishes identically. Therefore, from (44) we have 2 ∂ ˜pq mn 2 mn ∂ g ˜ g˜pq (t) − g˜ δn δm g˜pq − F g˜ + lower order terms = 0. n ∂t ∂y ∂y m
(45)
By restricting the metric tensor g˜ on p∗ T M and using Lemma 2.1 we can e1 , eˆ2 , eˆ3 } on SM as follows rewrite (45) in terms of the basis {ˆ ∂ g˜pq = g˜ab eˆb eˆa g˜pq + g˜11eˆ3 eˆ3 g˜pq −B c eˆc g˜pq −D 1 eˆ3 g˜pq +lower order terms, (46) ∂t where, B c := vic g˜ab eˆb (uia ) and D 1 := vi1 F˜ g˜11 eˆ3 ui1 as mentioned in Lemma 2.1 and all the indices in (46) run over the range 1, 2. By assumption M is compact and the sphere bundle SM as well. Also, the metric tensor g˜mn remains positive definite along the Ricci flow, see [5], Corollary 3.7. Since the coefficients of principal (second) order terms of (46) are positive definite, by Definition 2.1, it is a semi-linear strictly parabolic system on SM. Therefore, the standard existence and uniqueness theorem for parabolic systems on compact domains implies that, (46) has a unique solution on SM. Equation (46) is a special case of the general flow (15) and g˜(t) is a solution to it. Therefore, by means of Lemma 1.1, g˜(t) satisfies the integrability condition or equivalently, there exists a Finsler structure F˜ (t) 2F ˜ on T M such that g˜ij = 12 ∂y∂i ∂y ˜ is a Finsler metric and determines j . Hence, g a Finsler structure F˜ 2 := g˜pq y p y p which is a unique solution to the Finsler Ricci-DeTurck flow. This completes the proof of Theorem 1. ✷
5
Short time solution to the Ricci flow on Finsler surfaces
In this section, we will show that there is a one-to-one correspondence between the solutions to Ricci flow and Ricci-DeTurck flow on Finsler surfaces. Here, we recall some results which will be used in the sequel.
21
Lemma A. [7, p. 82] Let {Xt : 0 ≤ t < T ≤ ∞} be a continuous timedependent family of vector fields on a compact manifold M, then there exists a one-parameter family of diffeomorphisms {ϕt : M −→ M; 0 ≤ t < T ≤ ∞} defined on the same time interval such that ∂ ϕ (x) = Xt [ϕt (x)], ∂t t ϕ0 (x) = x, for all x ∈ M and t ∈ [0, T ). Remark 5.1. Let M be a compact Finsler surface. According to Lemma A there exists a unique one-parameter family of diffeomorphisms ϕ˜t on SM, such that ∂ ϕ (z) = ξ(ϕt (z), t), ∂t t ϕ0 = IdSM , where, z = (x, [y]) ∈ SM and t ∈ [0, T ). Remark 5.2. Let g˜pq be a solution to the Ricci-DeTurck flow and ϕt the one-parameter global group of diffeomorphisms according to the vector field ξ. Since ξ is a vector field on SM, then ϕt are homogeneous of degree zero. Zero-homogeneity of g˜pq implies that ϕ∗t (˜ gpq ) be also homogeneous of degree zero. In fact, (ϕ∗t g˜pq )(x, λy) = g˜pq (ϕt (x, λy)) = g˜pq (ϕt (x, y)) = (ϕ∗t g˜pq )(x, y). Using the fact that g˜pq is positive definite and ϕ∗t are diffeomorphisms, ϕ∗t (˜ gpq ) ∗ is also positive definite. As well ϕt (˜ gpq ) is symmetric. More intuitively, (ϕ∗t g˜)(X, Y ) = g(ϕt∗ (X), ϕt∗ (Y )) = g(ϕt∗ (Y ), ϕt∗ (X)) = (ϕ∗t g˜)(Y, X). Therefore, ϕ∗t (˜ gpq ) determines a Finsler structure as follows F 2 := gpq y˜py˜q , where, gpq := ϕ∗t (˜ gpq ) and ϕ∗t y p := y˜p. Lemma 5.1. Let ϕt be a global one parameter group of diffeomorphisms i corresponding to the vector field ξ and (γjk )g˜ and (Gi )g˜ are the Christoffel 22
symbols and spray coefficients related to the Finsler metric g˜, respectively. Then we have
∂˜ g
i where, (γjk )g˜ = g˜is 21 ( ∂xsjk −
i i ϕ∗t ((γjk )g˜) = (γjk )ϕ∗t (˜g ) ,
(47)
ϕ∗t (Gig˜) = Giϕ∗t (˜g ) ,
(48)
∂˜ gjk ∂xs
+
∂˜ gks ) ∂xj
i and Gig˜ = 12 (γjk )g˜y j y k .
Proof. Let us denote ϕ∗t xi = x˜i and ϕ∗t y i = y˜i . By definition, we have 1 ∂˜ gsj ∂˜ gjk ∂˜ gks i ϕ∗t ((γjk )g˜) = ϕ∗t (˜ g is ( k − + )) s 2 ∂x ∂x ∂xj gsj ∂˜ gjk ∂˜ gks 1 ∂˜ + )) = ϕ∗t (˜ g is )ϕ∗t ( ( k − s 2 ∂x ∂x ∂xj gsj ) ∂ϕ∗t (˜ gjk ) ∂ϕ∗t (˜ gks ) 1 ∂ϕ∗ (˜ + ) = ϕ∗t (˜ g is ) ( t k − s 2 ∂ x˜ ∂ x˜ ∂ x˜j i = (γjk )ϕ∗t (˜g) . Next, by means of (47) we have 1 i 1 i ϕ∗t (Gig˜) = ϕ∗t ( (γjk )g˜y j y k ) = ϕ∗t ((γjk )g˜)ϕ∗t y j ϕ∗t y k 2 2 1 i )ϕ∗t (˜g) y˜j y˜k = Giϕ∗t (˜g ) . = (γjk 2 This completes the proof. Lemma 5.2. Let ϕt be a global one parameter group of diffeomorphisms generating the vector field ξ and Ricg˜ the Ricci scalar related to the Finsler metric g˜, then we have ϕ∗t (Ricg˜ ) = Ricϕ∗t (˜g) . Proof. Let us consider the reduced hh-curvature tensor Rik which is expressed entirely in terms of the x and y derivatives of the spray coefficients Gig˜. (Rik )g˜ :=
∂ 2 Gig˜ ∂Gig˜ ∂Gjg˜ ∂ 2 Gig˜ 1 ∂Gig˜ (2 k − j k y j + 2Gjg˜ j k − ). ∂x ∂y ∂y ∂y ∂y j ∂y k F˜ 2 ∂x 23
Therefore, we have ϕ∗t ((Rik )g˜)
2 i ∂ 2 Gig˜ j ∂Gig˜ ∂Gjg˜ 1 ∂Gig˜ j ∂ Gg˜ = − j k y + 2Gg˜ j k − )) (2 ∂x ∂y ∂y ∂y ∂y j ∂y k F˜ 2 ∂xk ∂ 2 Gig˜ ∂Gig˜ ∂Gjg˜ ∂ 2 Gig˜ ∂Gig˜ 1 = ϕ∗t ( )ϕ∗t (2 k − j k y j + 2Gjg˜ j k − ). ∂x ∂x ∂y ∂y ∂y ∂y j ∂y k F˜ 2
ϕ∗t (
Thus, we get ϕ∗t ((Rik )g˜)
∂(ϕ∗t (Gig˜)) ∂ 2 (ϕ∗t (Gig˜)) j 1 = ∗ − y˜ (2 ∂ x˜k ∂ x˜j ∂ y˜k ϕt (F˜ 2 ) 2 ∗ i ∂(ϕ∗t (Gig˜)) ∂(ϕ∗t (Gjg˜)) j ∂ (ϕt (Gg˜ )) ∗ + 2ϕt (Gg˜) − ). ∂ y˜j ∂ y˜k ∂ y˜j ∂ y˜k
Putting i = k in this equation together with (48) implies ϕ∗t (Ricg˜ ) = Ricϕ∗t (˜g) , as we have claimed. Now we are in a position to prove the following proposition. Proposition 5.1. Fix a compact Finsler surface (M, F¯ ) with related Finsler metric tensor h. Let F˜ (t) be a family of solutions to the Ricci-DeTurck flow ∂ ˜2 F (t) = −2F˜ 2 (t)Ric(˜ g (t)) − Lξ F˜ 2 (t), ∂t
(49)
where, ξ = (Φg˜(t),h Id)i ∂x∂ i and t ∈ [0, T ). Moreover, let ϕt be a one-parameter family of diffeomorphisms satisfying ∂ ϕt (z) = ξ(ϕt (z), t), ∂t for z ∈ SM and t ∈ [0, T ). Then the Finsler structures F (t) form a solution to the Finslerian Ricci flow (33) where, F (t) is defined by F 2 (t) := gpq y˜p y˜q = ϕ∗t (F˜ 2 (t)). where, gpq := ϕ∗t (˜ gpq ) and ϕ∗t y p := y˜p. 24
Proof. In order to show F (t) form a solution to the Finslerian Ricci flow (33) ∂ we have to show ∂t (log F (t)) = −Ric. Derivation of F 2 (t) = ϕ∗t (F˜ 2 (t)) with respect to the parameter t, leads to ∂ 1 (log F (t)) = ∂t 2 The term
∂ (ϕ∗t F˜ 2 (t)) ∂t
∂ (ϕ∗t (F˜ 2 (t))) ∂t . ϕ∗t (F˜ 2 (t))
(50)
becomes
∂ ∗ ˜2 ∂ ∗ ˜2 (ϕt F (t)) = (ϕ (F (s + t))) |s=0 (51) ∂t ∂s s+t ∂ ∂ = ϕ∗t ( F˜ 2 (t)) + (ϕ∗s+t (F˜ 2 (t))) |s=0 ∂t ∂s ∂ ∗ ∗ ˜2 ∗ ∂ ˜2 = ϕt ( F (t)) + ((ϕ−1 t ◦ ϕt+s ) (ϕt (F (t)))) |s=0 ∂t ∂s ∂ ϕ∗ (F˜ 2 (t)). = ϕ∗t ( F˜ 2 (t)) + L ∂ (ϕ−1 t ◦ ϕt+s )|s=0 t ∂s ∂t On the other hand, we have ∂ ∂ −1 (ϕt ◦ ϕt+s ) |s=0 = (ϕ−1 ϕs+t ) |s=0 ) = (ϕ−1 t )∗ (( t )∗ (ξ). ∂s ∂s Hence, (51) is written ∂ ∗ ˜2 ∂ ϕ∗ (F˜ 2 (t)). (ϕt F (t)) = ϕ∗t ( F˜ 2 (t)) + L(ϕ−1 t )∗ (ξ) t ∂t ∂t Replacing the last relation in (50) and using the assumption (49) we get ∗ ∂ 2 ϕ∗ (F˜ 2 (t)) ∂ 1 ϕt ( ∂t F˜ (t)) + L(ϕ−1 t )∗ (ξ) t (log F (t)) = ∂t 2 ϕ∗t (F˜ 2 (t)) ∗ 2 2 ϕ∗ (F˜ 2 (t)) 1 ϕt (−2F˜ (t)Ric(F˜ (t)) − Lξ F˜ (t)) + L(ϕ−1 t )∗ (ξ) t = 2 ϕ∗t (F˜ 2 (t)) ∗ 2 ∗ 2 ϕ∗ (F˜ 2 (t)) 1 ϕt (−2F˜ (t)Ric(F˜ (t))) − ϕt (Lξ F˜ (t))) + L(ϕ−1 t )∗ (ξ) t = 2 ϕ∗t (F˜ 2 (t)) 1 −2ϕ∗t (F˜ 2 (t))ϕ∗t (Ric(F˜ (t))) = . 2 ϕ∗t (F˜ 2 (t))
25
By virtue of Lemma 5.2 we have ∂ (log F (t)) = −ϕ∗t (Ric(F˜ (t))) = −Ricϕ∗t (F˜ (t)) = −RicF (t) . ∂t Therefore, the Finsler structures F (t) form a solution to the Finslerian Ricci flow. Hence the proof is complete. In the next step we assume that there is a solution to the Finslerian Ricci flow based on which we construct a solution to the Ricci-DeTurck flow in Finsler space. Proposition 5.2. Fix a compact Finsler surface (M, F¯ ) with the related Finsler metric tensor h. Let F (t), t ∈ [0, T ), be a family of solutions to the Ricci flow and ϕt a one-parameter family of diffeomorphisms on SM evolving under the following flow, ∂ ϕt = Φg(t),h ϕt . ∂t Then the Finsler structures F˜ (t) defined by F 2 (t) = ϕ∗t (F˜ 2 (t)) form a solution to the following Ricci-DeTurck flow ∂ ˜2 F (t) = −2F˜ 2 (t)Ric(˜ g (t)) − Lξ F˜ 2 (t), ∂t where, ξ = (Φg˜(t),h Id)i ∂x∂ i and g˜(t) is the metric tensor related to F˜ (t). Furthermore, for all z ∈ SM and t ∈ [0, T ) we have ∂ ϕt (z) = ξ(ϕt (z), t). ∂t Proof. Using Lemma 3.1 we have ∂ ϕt = Φg(t),h ϕt = Φϕ∗t (˜g(t)),h ϕt = Φϕ∗t (˜g (t)),h Id ◦ ϕt ∂t = Φϕ∗t (˜g(t)),h ϕ∗t Id = Φg˜(t),h Id = ξ, 26
for all z ∈ SM and t ∈ [0, T ). Using F 2 (t) = ϕ∗t (F˜ 2 (t)) leads to 1 ∂ (log F (t)) = ∂t 2 =
1 2
=
1 2
∂ (ϕ∗t (F˜ 2 (t))) ∂t ϕ∗t (F˜ 2 (t)) ∂ ˜2 F (t)) + L(ϕ−1 ϕ∗ (F˜ 2 (t)) ϕ∗t ( ∂t t )∗ (ξ) t ϕ∗t (F˜ 2 (t)) ∂ ˜2 F (t) + Lξ F˜ 2 (t)) ϕ∗t ( ∂t . ϕ∗t (F˜ 2 (t))
(52)
By assumption, F (t) form a solution to the Finslerian Ricci flow (33) 0=
∂ (log F (t)) + RicF (t) . ∂t
(53)
Thus by means of (52), (53) and Lemma 5.2 we have 0=
∂ ˜2 ϕ∗t ( ∂t F (t) + Lξ F˜ 2 (t)) + 2RicF (t) ϕ∗t (F˜ 2 (t))
=
∂ ˜2 F (t) + Lξ F˜ 2 (t)) ϕ∗t ( ∂t + 2Ricϕ∗t (F˜ (t)) ϕ∗t (F˜ 2 (t))
=
∂ ˜2 F (t) + Lξ F˜ 2 (t)) ϕ∗t ( ∂t + 2ϕ∗t (RicF˜ (t) ) ∗ ˜2 ϕt (F (t))
∂ ˜2 F (t) + Lξ F˜ 2 (t)) + 2ϕ∗t (F˜ 2 (t))ϕ∗t (RicF˜ (t) ) ϕ∗t ( ∂t = ϕ∗t (F˜ 2 (t)) ∂ ˜2 ϕ∗t ( ∂t F (t) + Lξ F˜ 2 (t) + 2F˜ 2 (t)RicF˜ (t) ) = . ϕ∗t (F˜ 2 (t)) ∂ ˜2 F (t) + Lξ F˜ 2 (t) + 2F˜ 2(t)RicF˜ (t) ) = 0. This implies Therefore, ϕ∗t ( ∂t
∂ ˜2 F (t) = −2F˜ 2 (t)RicF˜ (t) − Lξ F˜ 2 (t). ∂t Therefore, F˜ (t) is a solution to the Ricci-DeTurck flow, as we have claimed. Proof of Theorem 2. In order to check the existence statement, recall that by means of Theorem 1, there exists a solution F˜ (t) to the Finslerian 27
Ricci-DeTurck flow (34) which is defined on some time interval [0, T ) and satisfies F˜ (0) = F0 . Let ϕt be the solution of the ODE ∂ ϕt (z) = (Φg˜(t),h Id)(ϕt (z), t) = ξ(ϕt(z), t), ∂t with the initial condition ϕ0 (z) = z, for z ∈ SM and t ∈ [0, T ). By Proposition 5.1, the Finsler structures F 2 (t) = ϕ∗t (F˜ 2 (t)) form a solution to the Finslerian Ricci flow (33) with F (0) = F0 . This completes the existence statement. For uniqueness statement assume that F1 (t) and F2 (t) are both solutions to the Finslerian Ricci flow defined on some time interval [0, T ) and satisfy F1 (0) = F2 (0). We claim F1 (t) = F2 (t) for all t ∈ [0, T ). In order to prove this fact, we argue by contradiction. Suppose that F1 (t) 6= F2 (t) for some t ∈ [0, T ). Let’s consider a real number τ ∈ [0, T ) where τ = inf{t ∈ [0, T ) : F1 (t) 6= F2 (t)}. Clearly, F1 (τ ) = F2 (τ ). Let ϕ1t be a solution of the flow ∂ 1 ϕt = Φg1 (t),h ϕ1t , ∂t 1 with initial condition ϕτ = Id and ϕ2t a solution of the flow
∂ 2 ϕ = Φg2 (t),h ϕ2t , ∂t t with initial condition ϕ2τ = Id. It follows from the standard theory of parabolic differential equations that ϕ1t and ϕ2t are defined on some time interval [τ, τ + ǫ), where, ǫ is a positive real number. Moreover, if we choose ǫ > 0 small enough, then ϕ1t and ϕ2t are diffeomorphisms for all t ∈ [τ, τ + ǫ). For each t ∈ [τ, τ + ǫ) we define two Finsler structures F˜1 (t) and F˜2 (t) by (F1 (t))2 = (ϕ1t )∗ (F˜1 (t))2 and (F2 (t))2 = (ϕ2t )∗ (F˜2 (t))2 . It follows from Proposition 5.2 that F˜1 (t) and F˜2 (t) are solutions of the Finslerian Ricci-DeTurck flow. Since F˜1 (τ ) = F˜2 (τ ), the uniqueness statement in Theorem 1 implies that F˜1 (t) = F˜2 (t) for all t ∈ [τ, τ + ǫ). For each t ∈ [τ, τ + ǫ), we define a vector field ξ on SM by ξ = Φg˜1 (t),h Id = Φg˜2 (t),h Id. By Proposition 5.2, we have ∂ 1 ϕ (z) = ξ(ϕ1t (z), t), ∂t t 28
and ∂ 2 ϕt (z) = ξ(ϕ2t (z), t), ∂t for z ∈ SM and t ∈ [τ, τ + ǫ). Since ϕ1τ = ϕ2τ = Id, it follows that ϕ1t = ϕ2t for all t ∈ [τ, τ + ǫ). Putting these facts together, we conclude that (F1 (t))2 = (ϕ1t )∗ (F˜1 (t))2 = (ϕ2t )∗ (F˜2 (t))2 = (F2 (t))2 , for all t ∈ [τ, τ + ǫ). Therefore, F1 (t) = F2 (t) for all t ∈ [τ, τ + ǫ). This contradicts the definition of τ . Thus the uniqueness holds well. This completes the proof of Theorem 2. ✷ Example 5.1. Let (M, F0 ) be a compact Finsler surface. We are going to obtain a solution to the Ricci flow (33). It’s well known in dimension two, a Finsler metric is of isotropic Ricci scalar (or Einstein) if and only if it is of isotropic flag curvature. Therefore, F0 is an Einstein metric and we have RicF0 = K where, K = K(x) is a scalar function on M. Consider a family of scalars τ (t) defined by τ (t) := 1 − 2Kt > 0. Define a smooth one-parameter family of Finsler structures on M by F 2 (t) := τ (t)F02 . Thus we have log(F (t)) =
1 log(τ (t)F02 ). 2
Derivative with respect to t yields ∂ K RicF0 log(F (t)) = − =− . ∂t τ (t) τ (t)
(54)
1 RicF0 = On the other hand, by straight forward computations we have τ (t) Ricτ (t) 12 F , for more details see [6, p. 926]. Replacing the last relation in 0 (54) leads to ∂ log(F (t)) = −Ricτ (t) 12 F = −RicF (t) . 0 ∂t Hence, F (t) is a solution to the Ricci flow equation (33).
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Example 5.2. Let F (t) be a family of Finsler structures on the sphere S2 defined by F 2 (t) = aij (t)y i y j where, aij (t) is a well known Riemannian metric on IR2 , called the Rosenau metric aij (t) =
8 sinh(−t) δij , 1 + 2 cosh(−t)|x|2 + |x|4
t ∈ (−∞, 0), x ∈ IR2 .
It is well known that aij extends to a metric on S2 . The related Finsler metric tensor of F (t) is 1 gij (t) := ( F 2 )yi yj = aij (t). 2 By straight forward computations, R(a(t)) the scalar curvature of the Riemannian metric aij (t) is R(a(t)) =
cosh(−t) 2 sinh(−t)|x|2 − . sinh(−t) 1 + 2 cosh(−t)|x|2 + |x|4
The Ricci tensor of gij and aij coincides. Hence 1 Ricij (g(t)) = Ricij (a(t)) = R(a(t))aij (t). 2 1
From F (t) = (aij (t)y i y j ) 2 , we have log(F (t)) =
1 log(aij (t)y i y j ). 2
Derivative with respect to t leads to ∂ 1∂ log F (t) = (aij (t))li lj , ∂t 2 ∂t
(55)
where, ∂ 16 sinh2 (−t)|x|2 −8 cosh(−t) δij . (aij (t)) = + ∂t 1 + 2 cosh(−t)|x|2 + |x|4 (1 + 2 cosh(−t)|x|2 + |x|4 )2
On the other hand, we have
1 Ric(g(t)) = li lj Ricij (g(t)) = li lj R(a(t))aij (t) 2 1 16 sinh2 (−t)|x|2 8 cosh(−t) δij . = li lj − 2 1 + 2 cosh(−t)|x|2 + |x|4 (1 + 2 cosh(−t)|x|2 + |x|4 )2 30
Comparing the last equation and (55) we have ∂ log F (t) = −Ric(g(t)). ∂t Consequently, F(t) form a solution to the Finsler Ricci flow (33) on S2 . Acknowledgment The authors would like to thank Prof. David Bao for his valuable comments. This work is partially supported by Iran National Science Foundation (INSF), under the grant 95002579.
References [1] H. Akbar-Zadeh, Initiation to global Finslerian geometry, Vol. 68. Elsevier Science, 2006. [2] D. Bao, On two curvature-driven problems in Riemann-Finsler geometry: In memory of Makoto Matsumoto, Advanced studies in pure mathematics, Vol. 48, Mathematical Society, Japan, Tokyo, (2007), 19-71. [3] D. Bao, S. Chern, Z. Shen, An introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, Vol. 200, Springer, 2000. [4] B. Bidabad, P. Joharinad, Conformal vector fields on Finsler spaces, Differential Geometry and its Applications, 31, (2013), 33–40. [5] B. Bidabad, M. Yar Ahmadi, Convergence of Finslerian metrics under Ricci flow, Sci. China. Math, Vol. 59, (2016), 741-750. [6] B. Bidabad, M. Yar Ahmadi, On quasi-Einstein Finler spaces, Bull. Iranian Math. Soc., Vol. 40, No. 4, (2014), 921-930. [7] B. Chow, D. Knopf, The Ricci Flow: An Introduction, Mathematical Survays and Monographs, Vol. 110, AMS, Providence, RI, 2004. 31
[8] B. Chow, The Ricci flow on the 2-sphere, J. Differential Geom. 33, No. 2, (1991), 325-334. [9] D. M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18(1) (1983), 157–162. [10] R. S. Hamilton, Three manifolds with positive Ricci curvature, J. Differential Geom. 17, (1982), 255-306. [11] R. S. Hamilton, The Ricci flow on surfaces, Math. and General Relativity, Contemporary Math. 71, (1988), 237-262. [12] M. Renardy, R. C. Rogers, An introduction to partial differential equations, Texts in applied mathematics, second edition, Springer 2004. Behroz Bidabad,
[email protected] Maral Khadem Sedaghat, m
[email protected] Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Hafez Ave., 15914 Tehran, Iran.
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