Ricci solitons on Finsler spaces, previously developed by the present
authors, are a generalization of Einstein spaces, which can be considered as a
solution to the Ricci flow on compact Finsler manifolds. In the present work it
is shown that on a Finslerian space, a forward complete shrinking Ricci soliton
is compact if and only if it is bounded. Moreover, it is proved that a compact
shrinking Finslerian Ricci soliton has finite fundamental group and hence the
first de Rham cohomology group vanishes.Comment: 9 page

Ahmadi, Mohamad Yar

Bidabad, Behroz

English

arXiv

Yar Ahmadi, Mohamad

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1023–1027Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comDifferential geometryOn compact Ricci solitons in Finsler geometrySur les solitons de Ricci compacts en géométrie finslérienneMohamad Yar Ahmadi, Behroz BidabadFaculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Irana r t i c l e i n f o a b s t r a c tArticle history:Received 23 January 2015Accepted after revision 14 September 2015Available online 21 October 2015Presented by the Editorial BoardRicci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work, it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact if and only if it is bounded. Moreover, it is proved that a compact shrinking Finslerian Ricci soliton has finite fundamental group, and hence the first de Rham cohomology group vanishes.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éLes solitons de Ricci sur les espaces de Finsler, précédemment définis et étudiés par les auteurs de la présente note, sont une généralisation des espaces d’Einstein, et peuvent être considérés comme des solutions du flot de Ricci sur les variétés finslériennes compactes. Dans ce travail, on démontre qu’un soliton de Ricci complet contractant en temps croissant sur un espace de Finsler est compact si et seulement s’il est borné. En outre, il est démontré qu’un soliton de Ricci contractant compact donne lieu à un groupe fondamental de type fini et donc que le premier groupe de cohomologie s’annule.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionThe Ricci flow in Riemannian geometry was introduced by R.S. Hamilton in 1982, cf. [9], and since then has been exten-sively studied thanks to its applications in geometry, physics and different branches of real-world problems. Quasi-Einstein metrics or Ricci solitons are considered as solutions to the Ricci flow equation and are subject of great interest in geometry and physics, specially in relation with string theory, cf. [10].Let (M, g) be a Riemannian manifold, a triple (M, g, X) is said to be a quasi-Einstein metric or Ricci soliton if g satisfies the equation2 Ric +LX g = 2λg, (1)E-mail addresses: m.yarahmadi@aut.ac.ir (M. Yar Ahmadi), bidabad@aut.ac.ir (B. Bidabad).http://dx.doi.org/10.1016/j.crma.2015.09.0121631-073X/© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1024 M. Yar Ahmadi, B. Bidabad / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1023–1027where Ric is the Ricci tensor, X a smooth vector field on M , LX the Lie derivative along X , and λ a real constant. A Ricci soliton is said to be shrinking, steady or expanding if λ > 0, λ = 0 or λ < 0, respectively. If the vector field X is the gradient of a function f , then (M, g, X) is said to be gradient and (1) takes the familiar form:Ric + ∇∇ f = λg.On a compact Riemannian manifold, a quasi-Einstein metric is a special solution to the Ricci flow equation defined by∂∂tg(t) = −2 Ric, g(t = 0) := g0 .J. Lott has shown that the fundamental group of a closed Riemannian manifold is finite for any gradient shrinking Ricci soliton, cf. [11]. M.F. López and E.G. Río have proved that a Riemannian compact shrinking Ricci soliton has finite fun-damental group, cf. [10]. W. Wylie has shown that a Riemannian complete shrinking Ricci soliton has finite fundamental group, cf. [12].The concept of Ricci flow on Finsler manifolds is defined first by D. Bao, cf. [5], using the Ricci tensor defined by H. Akbar-Zadeh, [2]. Recently the present authors have developed the concept of Ricci solitons as a generalization of Einstein spaces and convergence of Ricci flow on Finsler spaces, cf. [7,8]. It is proved that if there is a Ricci soliton on a compact Finsler manifold then there exists a solution to the Ricci flow equation and vice-versa. Since Finslerian Ricci solitons generalize Ein-stein manifolds, it is natural to ask whether classical results like the Bonnet–Myers theorem for Finsler–Einstein manifolds of positive Ricci scalar remain valid for Finslerian Ricci solitons. In the present work, in analogy with Riemannian space, the shrinking Finslerian Ricci soliton is defined and it is shown that a forward complete shrinking Finslerian Ricci soliton (M, F , V ) is compact if and only if ‖V ‖ is bounded. Moreover, it is proved that in this case the fundamental group is finite and, as a consequence, the first de Rham cohomology group of M vanishes.2. Preliminaries and notationsLet M be a real n-dimensional differentiable manifold. We denote by T M its tangent bundle and by π : T M0 −→ M , the fiber bundle of non-zero tangent vectors. A Finsler structure on M is a function F : T M −→ [0, ∞), with the following properties:I. Regularity: F is C∞ on the entire slit tangent bundle T M0 = T M\0.II. Positive homogeneity: F (x, λy) = λF (x, y) for all λ > 0.III. Strong convexity: the n × n Hessian matrix gij = ([ 12 F 2]yi y j ) is positive definite at every point of T M0. A Finsler manifold (M, F ) is a pair consisting of a differentiable manifold M and a Finsler structure F . The formal Christoffel symbols of second kind and spray coefficients are denoted respectively byγ ijk := gis12(∂ gsj∂xk− ∂ g jk∂xs+ ∂ gks∂x j),where gij(x, y) = [ 12 F 2]yi y j , and Gi := 12 γ ijk y j yk . We consider also the reduced curvature tensor Rik , which is expressed entirely in terms of the x and y derivatives of spray coefficients Gi .Rik :=1F 2(2∂Gi∂xk− ∂2Gi∂x j∂ yky j + 2G j ∂2Gi∂ y j∂ yk− ∂Gi∂ y j∂G j∂ yk). (2)In the general Finslerian setting, one of the Ricci tensors is introduced by H. Akbar-Zadeh [1] as follows:Ric jk := [12 F2Ric]y j yk , (3)where Ric = Rii and Rik is defined by (2). Akbar-Zadeh’s definition of Einstein–Finsler space related to this Ricci tensor is obtained as a critical point of an Einstein–Hilbert functional and, (see [2] chapter IV). One of the advantages of the Ricci quantity defined here is its independence of the choice of Cartan, Berwald or Chern (Rund) connections. Based on the Akbar-Zadeh’s Ricci tensor, in analogy with Eq. (3), D. Bao has considered the following natural extension of Ricci flow in Finsler geometry, cf. [5]:∂∂tg jk = −2Ric jk, g(t = 0) := g0 .This equation leads to the following differential equation∂∂t(log F (t)) = −Ric, F (t = 0) := F0 ,where F0 is the initial Finsler structure. Let V = V i(x) ∂∂xi be a vector field on M .The Lie derivative of a Finsler metric tensor g jk is given in the following tensorial form byLV̂ g jk = ∇ j Vk + ∇k V j + 2(∇0 V l)Cljk, (4)M. Yar Ahmadi, B. Bidabad / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1023–1027 1025where V̂ is the complete lift of a vector field V on M , ∇ is the Cartan connection, ∇0 = yp∇p and ∇p = ∇ δδxp, (see [13], p. 180, and see [6]).Let M be a connected smooth manifold, then there exists a simply connected smooth manifold M̃ , called the universal covering manifold of M , and a smooth covering map p : M̃ −→ M such that it is unique up to a diffeomorphism. The complete lift of p is a map p̄ : T M̃ −→ T M is given byp̄(x̃, ỹ) = (p(x̃), ỹi ∂ p∂ x̃i) = (p(x̃), ỹi ∂ pj∂ x̃i∂∂x j),where ỹ ∈ Tx̃M̃ .3. Shrinking Finslerian Ricci solitonLet (M, F0) be a Finsler manifold and V = V i(x) ∂∂xi a vector field on M . We call the triple (M, F0, V ) a Finslerian quasi-Einstein or a Finslerian Ricci soliton if g jk , the Hessian related to the Finsler structure F0, satisfies2Ric jk +LV̂ g jk = 2λg jk, (5)where V̂ is the complete lift of V and λ ∈ R. A Finslerian Ricci soliton is said to be shrinking, steady or expanding if λ > 0, λ = 0 or λ < 0, respectively. The Finslerian Ricci soliton is said to be forward complete (resp. compact) if (M, F0) is forward complete (resp. compact). Note that according to the Hopf–Rinow’s theorem, the both notions forward complete and forward geodesically complete are equivalent. Denote by S M the sphere bundle, defined by S M := ⋃x∈MSxM where SxM := {y ∈ TxM|F (x, y) = 1}. For a vector field X = Xi(x) ∂∂xi on M , define‖X‖x = maxy∈Sx M√gij(x, y)Xi X j, (6)where x ∈ M (see [4] at p. 321). Since Sx M is compact, ‖X‖x is well defined.Theorem 1. Let (M, F0) be a forward geodesically complete Finsler manifold satisfying2 Ric jk +LV̂ g jk ≥ 2λg jk, (7)where λ > 0. Then, M is compact if and only if ‖V ‖ is bounded on M by a constant D and moreover, in such a case, diam(M) ≤πλ(D + √D2 + λ(n − 1)).Proof. Let M be a compact manifold, it is clear that ‖V ‖ is bounded on M . Conversely, let p, q be two points in M joined by a minimal geodesic γ parameterized by the arc length t , γ : [0, ∞) −→ M . Using (4) we have along γ :γ ′ jγ ′kLV̂ g jk = γ ′ jγ ′k(∇ j Vk + ∇k V j + 2(∇0 V l)Cljk). (8)Along γ , we have γ ′ jγ ′k(∇0 V l)Cljk(γ (t), γ ′(t)) = 0. Hence (8) reduces toγ ′ jγ ′kLV̂ g jk = 2γ ′ jγ ′k∇ j Vk. (9)On the other hand, by compatibility of metric with the Cartan connection, we have along the geodesic γ :γ ′ jγ ′k∇ j Vk = ∇γ ′ j δδx j(γ ′k Vk) = ∇γ̂ ′(γ ′k Vk) = ddt (γ′k Vk), (10)where γ̂ ′ = γ ′ j δδx j. Replacing (10) in (9), we have:γ ′ jγ ′kLV̂ g jk = 2ddt(γ ′k Vk). (11)By means of (7) and (11) we get:2γ ′ jγ ′kRic jk + 2 ddt (γ′k Vk) ≥ 2λγ ′ jγ ′k g jk.By the last inequality, we conclude thatγ ′ jγ ′kRic jk ≥ λγ ′ jγ ′k g jk − ddt (γ′k Vk) = λ + ddt (−γ′k Vk).On the other hand, by means of the Cauchy–Schwarz inequality, we have along the geodesic γ :1026 M. Yar Ahmadi, B. Bidabad / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1023–1027|−γ ′k Vk| = |γ ′k Vk| = |gkl(γ (t), γ ′(t))γ ′k V l|≤ |gpq(γ (t), γ ′(t))γ ′pγ ′q| 12 |grs(γ (t), γ ′(t))V r V s| 12≤ maxy∈Sγ (t)M|grs(γ (t), y)V r V s| 12 = ‖V ‖γ (t).Since ‖V ‖ is assumed to be bounded on M , there exists a positive constant D such that ‖V ‖γ (t) ≤ D and therefore along γ , |−γ ′k Vk| ≤ D . Now, the result follows from generalization of Mayers Theorem, cf. [3]. That is, M is compact and moreover it is bounded from above by diam(M) ≤ πλ(D + √D2 + λ(n − 1)). This completes the proof. Corollary 2. Let (M, F , V ) be a forward complete shrinking Finslerian Ricci soliton. Then, M is compact if and only if ‖V ‖ is bounded on M by a constant D and moreover, in this case, diam(M) ≤ πλ(D + √D2 + λ(n − 1)).Theorem 3. Let (M, F ) be a compact Finsler manifold satisfying (7). Then the fundamental group π1(M) of M is finite and its first cohomology group vanishes, i.e., H1dR(M) = 0.Proof. Let M̃ be the universal covering manifold of M with the smooth covering map p : M̃ −→ M . Pull back of complete lift of the smooth covering map p, i.e., p̄∗ F := F ◦ p̄ : T M̃ −→ [0, ∞) is a Finsler structure on M̃ . In fact, we check simply the three conditions of the Finsler structure. We have the regularity condition since F and p are C∞ , and so is p̄∗ F . Next,p̄∗ F (x, λy) = F ◦ p̄(x, λy) = F (p(x), λyi ∂ p∂xi)= λF (p(x), yi ∂ p∂xi) = λp̄∗ F (x, y).Thus the positive homogeneity is satisfied. Finally, assume that p̄∗xi = x̃i and p̄∗ yi = ỹi . For strong convexity we have:g̃i j := [12 (p̄∗ F )2] ỹi ỹ j =12∂2((p̄∗ F )2)∂ ỹi∂ ỹ j= 12∂2(p̄∗ F 2)∂ ỹi∂ ỹ j.One can easily check that∂(p̄∗ F 2)∂ ỹi= p̄∗ ∂ F2∂ yi,from whichg̃i j = [12 (p̄∗ F )2] ỹi ỹ j =12∂2(p̄∗ F 2)∂ ỹi∂ ỹ j= p̄∗[12F 2]yi y j = p̄∗gij . (12)Using the facts that [ 12 F 2]yi y j is positive definite on T M0 and p̄∗ is a local diffeomorphism (note that p is the smooth cov-ering map), p̄∗[ 12 F 2]yi y j is also positive definite on T M̃0 and hence F̃ := p̄∗ F defines a Finsler structure on T M̃0. Moreover, (M̃, F̃ ) is locally isometric to (M, F ). Let W denote the lift of V , that is, W := p∗V = (p−1)∗V . More precisely, since p is a local diffeomorphism, we can define W := p∗V = (p−1)∗V . By means of the local isometry p : (M̃, F̃ ) −→ (M, F ) and the inequality (7), we have:p̄∗(2Ric jk +LV̂ g jk) ≥ 2̄ p∗(λg jk).By linearity of p̄∗ we get:2 p̄∗Ric jk + p̄∗LV̂ g jk ≥ 2λp̄∗(g jk). (13)By means of (12), W = p∗V and commutativity of Lie derivative and the pull back p̄∗ , we obtain:p̄∗LV̂ g jk = LŴ g̃ jk. (14)On the other hand, one can easily check that R̃ic jk = p̄∗Ric jk . In fact we have:p̄∗Ric jk = p̄∗[12 F2Ric]y j yk =12p̄∗ ∂2(F 2Ric)∂ yi∂ y j= 12∂2∂ ỹi∂ ỹ j(p̄∗(F 2Ric)) = 12∂2∂ ỹi∂ ỹ j(p̄∗(F 2)p̄∗(Ric)).Since p̄∗(Ric) = R̃ic, cf. [7], and p̄∗(F 2) = F̃ 2, we getM. Yar Ahmadi, B. Bidabad / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1023–1027 1027p̄∗Ric jk = 12∂2∂ ỹi∂ ỹ j(p̄∗(F 2)p̄∗(Ric))= 12∂2∂ ỹi∂ ỹ j( F̃ 2R̃ic) = R̃ic jk. (15)Replacing (12), (14) and (15) in (13), leads to2R̃ic jk +LŴ g̃ jk ≥ 2λg̃ jk.On the other hand, we have:‖W ‖x̃ = maxỹ∈Sx̃ M̃((p̄∗gij)(x̃, ỹ)W i W j) 12 = maxỹ∈Sx̃ M̃(gij(p(x̃), ỹi ∂ p∂ x̃i)p̄∗W i p̄∗W j) 12≤ maxy∈Sx̃ M̃(gij(p(x̃), y)p̄∗W i p̄∗W j) 12 = ‖p̄∗W ‖p(x̃). (16)By compactness of M , the norm ‖p̄∗W ‖ is bounded on M and therefore, by means of (16), the norm ‖W ‖ is bounded on M̃ . It follows from Theorem 1 that (M̃, F̃ ) is compact. Thus the closed subset p−1(x) of M̃ is compact and, being discrete, is finite. By assumption, M is connected, so all of its fundamental groups π1(M, x) are isomorphic, where x denotes the base point. Since M̃ is a universal cover, π1(M, x) is bijective with p−1(x) and therefore π1(M) is finite. Thus, by a well-known result, the first cohomology group H1dR(M) = 0. This completes the proof. Corollary 4. Let (M, F , V ) be a compact shrinking Finslerian Ricci soliton. Then the fundamental group π1(M) of M is finite and therefore H1dR(M) = 0.Corollary 5. Let (M, F , V ) be a compact shrinking Finslerian Ricci soliton. Then the fundamental group π1(S M) of S M is finite and therefore H1dR(S M) = 0.Proof. Let M̃ be the universal covering manifold of M with the smooth covering map p : M̃ −→ M . It is well known that the homotopic sequence of the fiber bundle (S M̃, π̃ , M̃, Sn−1) is exact. That is that· · · −→ π1(Sn−1) −→ π1(S M̃) −→ π1(M̃) −→ · · · , (17)is exact. Since M̃ is simply connected, π1(M̃) = 0. We know that π1(Sn−1) = 0. Thus, by (17) we get π1(S M̃) = 0. One can easily check that p̄ : S M̃ −→ S M is a smooth covering map. Therefore, S M̃ is the universal covering manifold of S M . According to the proof of Theorem 3, M̃ is compact and so is S M̃ . Thus the fundamental group π1(S M) is finite and therefore H1dR(S M) = 0. References[1] H. Akbar-Zadeh, Sur les espaces de Finsler á courbures sectionnelles constantes, Acad. R. Belg. Bull. Cl. Sci. (5) 74 (1988) 281–322.[2] H. Akbar-Zadeh, Initiation to Global Finslerian Geometry, vol. 68, Elsevier Science, 2006.[3] M. Anastasiei, A generalization of Myers theorem, An. Stiint. Univ. “Al.I. Cuza” Din Iasi (S.N.) Mat. LIII (Supliment) (2007) 33–40.[4] D. Bao, S.S. Chern, Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics, vol. 200, Springer, 2000.[5] D. Bao, On two curvature-driven problems in Riemann–Finsler geometry, Adv. Stud. Pure Math. 48 (2007) 19–71.[6] B. Bidabad, P. Joharinad, Conformal vector fields on complete Finsler spaces of constant Ricci curvature, Differ. Geom. Appl. 33 (2014).[7] B. Bidabad, M. Yarahmadi, On quasi-Einstein Finler spaces, Bull. Iran. Math. Soc. 40 (4) (2014) 921–930.[8] B. Bidabad, M. Yar Ahmadi, Convergence of Finslerian metrics under Ricci flow, Sci. China Math. (2015), in press.[9] R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom. 17 (2) (1982) 255–306.[10] M.F. López, E.G. Río, A remark on compact Ricci solitons, Math. Ann. 340 (4) (2008) 893–896.[11] J. Lott, Some geometric properties of the Bakry–Émery–Ricci tensor, Comment. Math. Helv. 78 (2003) 865–883.[12] W. Wylie, Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc. 136 (5) (2008) 1803–1806.[13] K. Yano, The Theory of Lie Derivatives and Its Applications, North-Holland Publishers, 1957.

On compact Ricci solitons in Finsler geometry

Mohamad Yar Ahmadi

Behroz Bidabad

Comptes Rendus Mathématique

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On compact Ricci solitons in Finsler geometry

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