We prove that a bounded affine vector field on a complete Finsler manifold is
a Killing vector field. This generalizes the analogous result of Hano for
Riemannian manifolds. Even though our result is more general, the proof is
significantly simpler.Comment: 3 page

Kertesz, David Csaba

Lovas, Rezso L.

English

arXiv

Kertész, Dávid Csaba

Lovas, Rezső László

University of Debrecen Electronic Archive

A generalization and short proof of a theorem of Hano on affine vector fields

Tokyo University of Science Repository for Academic Resources / 東京理科大学学術リポジトリ

SUT Journal of MathematicsVol. 53, No. 2 (2017), 83–87A generalization and short proof of a theorem ofHano on affine vector fieldsDa´vid Csaba Kerte´sz and Rezso˝ L. Lovas∗(Received December 9, 2016; Revised November 20, 2017)Abstract. We prove that a bounded affine vector field on a complete Finslermanifold is a Killing vector field. This generalizes the analogous result of Hanofor Riemannian manifolds [3]. Even though our result is more general, the proofis significantly simpler.AMS 2010 Mathematics Subject Classification. 53B40,53C60.Key words and phrases. affine vector field, Killing vector field, Finsler manifold.§1. IntroductionYano showed that affine vector fields on a compact orientable Riemannianmanifold are Killing vector fields [7]; the proof was based on integral formulas.Hano found a generalization: bounded affine vector fields on a complete Rie-mannian manifold are Killing vector fields. The proof relied on the de Rhamdecomposition, and special properties of irreducible Riemannian manifolds. Asimilar proof can be found in [4]. We show that Hano’s result is true for themuch more general Finsler manifolds, using only the Euler–Lagrange equation.§2. Definitions and prerequisitesThroughout, M is a second countable and smooth Hausdorff manifold; thetangent bundle is τ : TM → M , and we denote by T˚M the tangent manifoldwith the zero vectors removed. If ϕ : M → N is a smooth mapping betweenmanifolds, ϕ∗ : TM → TN stands for its derivative.∗Both authors were supported by the Hungarian Scientific Research Fund (OTKA) GrantK-1116518384 D. CS. KERTE´SZ AND R. L. LOVASWe are going to work on the tangent manifold, where we use two kinds oflifts of vector fields on the base manifold. The vertical lift Xv of a vector fieldX ∈ X(M) is the velocity field of the global flow(t, v) ∈ R× TM 7−→ v + tX(τ(v)) ∈ TMon TM . If ϕX : DX ⊂ R ×M → M is the maximal local flow of X ∈ X(M)andD˜X := {(t, v) ∈ R× TM | (t, τ(v)) ∈ DX},then(t, v) ∈ D˜X 7−→ (ϕXt )∗(v) ∈ TMis a local flow on TM , whose velocity field is called the complete lift of X,denoted by Xc. The Liouville vector field C on TM is the velocity field of theflow of positive dilations:(t, v) ∈ R× TM 7−→ etv ∈ TM.It is clear that a smooth function f on T˚M is k+-homogeneous (k ∈ Z) if andonly if Cf = kf .A continuous function F on TM is a Finsler function for M if it is smoothon T˚M , 1+-homogeneous, F ↾ T˚M > 0, and for any p ∈ M and u ∈ T˚pM ,the symmetric bilinear form (E ↾ TpM)′′(u) is non-degenerate (hence positivedefinite), where E := 12F2. A Finsler manifold is a manifold together with aFinsler function.If (M,F ) is a Finsler manifold, then there exists a unique second-ordervector field S ∈ X(T˚M) such that a curve γ in M is a geodesic of (M,F ) ifand only if S ◦ γ˙ = γ¨. This vector field S is usually called the canonical sprayor geodesic spray of (M,F ). Another characterization of S is that(2.1) S(XvE)−XcE = 0 for all X ∈ X(M).This form of the Euler–Lagrange equation is due to Crampin (see, e.g., [2,p. 348] or [5, p. 16]). It can be derived directly from the elementary form(∂E∂yi◦ γ˙)′ − ∂E∂xi◦ γ˙ = 0 using the local formulae for Xv and Xc.A Finsler manifold is said to be forward complete if the domains of itsmaximal geodesics are not bounded from above, and complete if the domainof its maximal geodesics is R. For many equivalent characterizations of com-pleteness, see [1, §6.6].A vector field X on a Finsler manifold (M,F ) is affine if its flow pre-serves geodesics, and it is a Killing vector field if its flow preserves the Finslerfunction, i.e., F ◦ (ϕXt )∗ = F for all possible t ∈ R. Both properties can be ex-pressed in terms of the complete lift of X: X is affine if and only if [Xc, S] = 0,and X is a Killing vector field if and only if XcE = 0.A GENERALIZATION AND SHORT PROOF OF A THEOREM OF HANO 85§3. Proof of the resultThe key of our argument is the following simple observation. It is in fact adisguised special case of Exercise 5.4.3 from [1], but we give a short directproof.Lemma 1. If X is an affine vector field on a Finsler manifold (M,F ) and γis a geodesic, then for all t and t0 in the domain of γ we haveXvE(γ˙(t)) = XvE(γ˙(t0)) + (t− t0)XcE(γ˙(t0)).Proof. Since X is affine, we have [Xc, S] = 0. Geodesics have constant speed,hence SE = 0. From these we get0 = [Xc, S]E = Xc(SE)− S(XcE) = −S(XcE).Since γ is a geodesic, S ◦ γ˙ = γ¨, and we have(XvE ◦ γ˙)′ = S(XvE) ◦ γ˙ (2.1)= XcE ◦ γ˙,(XvE ◦ γ˙)′′ = (XcE ◦ γ˙)′ = S(XcE) ◦ γ˙ = 0.Therefore XvE ◦ γ˙ is an affine function, and our claim follows.Theorem 2. Let (M,F ) be a Finsler manifold, X an affine vector field, andsuppose that one of the following conditions holds:(1) F ◦X is bounded, and (M,F ) is complete;(2) F ◦X and F ◦ (−X) are bounded, and (M,F ) is forward complete.Then X is a Killing vector field.Proof. First we prove that XvE is bounded from above on the set U(TM) :=F−1({1}) if (1) holds, and it is bounded from above and from below if (2)holds. For any v ∈ U(TM), setting p := τ(v), we haveXvE(v) = F (v)XvF (v) = XvF (v) = (F ↾ TpM)′(v)(X(p)) ≤ F (X(p)),where in the last step we used the fundamental inequality (see [1, p. 7] or [6,Proposition 9.1.37]). In a similar way, we obtainXvE(v) = (F ↾ TpM)′(v)(X(p)) = −(F ↾ TpM)′(v)(−X(p)) ≥ −F (−X(p)).Over U(TM) these two inequalities give −F ◦ (−X) ◦ τ ≤ XvE ≤ F ◦X ◦ τ ,from which it follows that XvE has the desired boundedness property.86 D. CS. KERTE´SZ AND R. L. LOVASNow we show that XcE = 0, and hence X is a Killing vector field. Itsuffices to prove it on U(TM), because XcE is 2+-homogeneous. Indeed, theflows of Xc and C clearly commute, hence [Xc, C] = 0, and we have(3.1) C(XcE) = [C,Xc]E +Xc(CE) = 2XcE.So fix v ∈ U(TM) and let γ be the maximal geodesic with γ˙(0) = v. ThenLemma 1 givesXvE(γ˙(t)) = XvE(γ˙(0)) + tXcE(γ˙(0)) = XvE(v) + tXcE(v)for any real number t in case (1) and for any positive real number t in case(2). Geodesics have constant speed, hence γ˙ remains inside U(TM), and theleft-hand side of the above formula has to be bounded from above in case (1),and it has to be bounded from above and below in case (2), which is possibleonly if XcE(v) = 0. Thus XcE = 0 on U(TM). This together with (3.1)implies XcE = 0, that is, X is a Killing vector field.As a corollary we haveTheorem 3 (Hano). Let (M, g) be a complete Riemannian manifold, and Xan affine vector field on M such that the function g(X,X) is bounded. ThenX is a Killing vector field.The proof is immediate if we apply Theorem 2 to the Finsler function givenby F (v) :=√g(v, v), v ∈ TM . Since compact Finsler manifolds are complete,we also haveTheorem 4. An affine vector field on a compact Finsler manifold is a Killingvector field.AcknowledgmentsThe authors are grateful to Jo´zsef Szilasi and Bernadett Aradi for their sug-gestions that improved the paper.References[1] D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann–Finsler geometry,Springer, 2000.[2] M. Crampin and F. A. E. Pirani, Applicable differential geometry, vol. 59, Cam-bridge University Press, Cambridge, 1986.A GENERALIZATION AND SHORT PROOF OF A THEOREM OF HANO 87[3] J.-i. Hano, On affine transformations of a Riemannian manifold, Nagoya Math.J., 9 (1955), 99–109.[4] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, JohnWiley & Sons, Inc., New York, 1996.[5] J. Szilasi, R. L. Lovas, and D. Cs. Kerte´sz, Several ways to a Berwald manifold– and some steps beyond, Extracta Mathematicae, 26 (2011), 89–130.[6] , Connections, Sprays and Finsler Structures, World Scientific, 2014.[7] K. Yano, On harmonic and Killing vector fields, Ann. of Math. (2), 55 (1952),38–45.Da´vid Csaba Kerte´szInstitute of Mathematics, University of DebrecenH-4002 Debrecen, P.O. Box. 400, HungaryE-mail : kerteszd@science.unideb.huRezso˝ L. LovasInstitute of Mathematics, University of DebrecenH-4002 Debrecen, P.O. Box. 400, HungaryE-mail : lovas@science.unideb.hu

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arXiv:1606.08631v1  [math.DG]  28 Jun 2016A generalization and short proof of a theorem of Hanoon affine vector fieldsD. Cs. Kertész R. L. Lovas∗November 29, 2021AbstractWe prove that a bounded affine vector field on a complete Finsler manifold is aKilling vector field. This generalizes the analogous result of Hano for Riemannianmanifolds [3]. Even though our result is more general, the proof is significantlysimpler.Keywords: affine vector field, Killing field, Finsler manifold.2010 Mathematics Subject Classification: 53B40, 53C60.1 IntroductionYano showed that affine vector fields on a compact orientable Riemannian manifold areKilling fields [7]; the proof was based on integral formulas. Hano found a generalization:bounded affine vector fields on a complete Riemannian manifold are Killing fields. Theproof relied on the de Rham decomposition, and special properties of irreducible Rieman-nian manifolds. A similar proof can be found in [4]. We show that Hano’s result is truefor the much more general Finsler manifolds, using only the Euler–Lagrange equation.2 Definitions and prerequisitesThroughout, M is a second countable and smooth Hausdorff manifold; τ : TM → M isthe tangent bundle, τ̊ : T̊M → M is the slit tangent bundle of M . If ϕ : M → N is asmooth mapping between manifolds, ϕ∗ : TM → TN stands for its derivative.We are going to work on the tangent manifold, where we use two kinds of lifts ofvector fields on the base manifold. The vertical lift Xv of a vector field X ∈ X(M) is thevelocity field of the global flow(t, v) ∈ R× TM 7−→ v + tX(τ(v)) ∈ TMon TM . If ϕX : DX ⊂ R×M → M is the maximal local flow of X ∈ X(M) andD̃X := {(t, v) ∈ R× TM | (t, τ(v)) ∈ DX},then(t, v) ∈ D̃X 7−→ (ϕXt )∗(v) ∈ TMis a local flow on TM , whose velocity field is called the complete lift of X , denoted by Xc.The Liouville vector field C on TM is the velocity field of the flow of positive dilations:(t, v) ∈ R× TM 7−→ etv ∈ TM.It is clear that a smooth function f on T̊M is k+-homogeneous (k ∈ Z) if and only ifCf = kf .∗Both authors were supported by the Hungarian Scientific Research Fund (OTKA) Grant K-1116511A continuous function F on TM is a Finsler function for M if it is smooth on T̊M ,1+-homogeneous, F ↾ T̊M > 0, and for any p ∈ M and u ∈ T̊pM , the symmetric bilinearform (E ↾ TpM)′′(u) is non-degenerate (hence positive definite), where E := 12F 2. AFinsler manifold is a manifold together with a Finsler function.If (M,F ) is a Finsler manifold, then there exists a unique second-order vector fieldS ∈ X(T̊M) such that a curve γ in M is a geodesic of (M,F ) if and only if S ◦ γ̇ = γ̈.Another characterization of S is thatS(XvE)−XcE = 0 for all X ∈ X(M). (1)This form of the Euler–Lagrange equation is due to Crampin (see, e.g., [2, p. 348] or [5,p. 16]). It can be derived directly from the elementary form(∂E∂yi◦ γ̇)′− ∂E∂xi◦ γ̇ = 0 usingthe local formulae for Xv and Xc.A Finsler manifold is said to be forward complete if the domains of its maximalgeodesics are not bounded from above, and complete if the domain of its maximal geodesicsis R. For many equivalent characterizations of completeness, see [1, §6.6].A vector field X on a Finsler manifold (M,F ) is affine if its flow preserves geodesics,and it is a Killing field if its flow preserves the Finsler function, i.e., F ◦ (ϕXt )∗ = F forall possible t ∈ R. Both properties can be expressed in terms of the complete lift of X :X is affine if and only if [Xc, S] = 0, and X is a Killing field if and only if XcE = 0.3 Proof of the resultThe key of our argument is the following simple observation. It is in fact a disguisedspecial case of Exercise 5.4.3 from [1].Lemma 1. If X is an affine vector field on a Finsler manifold (M,F ) and γ is a geodesic,then for all t and t0 in the domain of γ we haveXvE(γ̇(t)) = XvE(γ̇(t0)) + (t− t0)XcE(γ̇(t0)).Proof. Since X is affine, we have [Xc, S] = 0. Geodesics have constant speed, henceSE = 0. From these we get0 = [Xc, S]E = Xc(SE)− S(XcE) = −S(XcE).Since γ is a geodesic, S ◦ γ̇ = γ̈, and we have(XvE ◦ γ̇)′ = S(XvE) ◦ γ̇(1)= XcE ◦ γ̇,(XvE ◦ γ̇)′′ = (XcE ◦ γ̇)′ = S(XcE) ◦ γ̇ = 0.Therefore XvE ◦ γ̇ is an affine function, and our claim follows.Theorem 2. Let (M,F ) be a Finsler manifold, X an affine vector field, and suppose thatone of the following conditions holds:(1) F ◦X is bounded, and (M,F ) is complete;(2) F ◦X and F ◦ (−X) are bounded, and (M,F ) is forward complete.Then X is a Killing vector field.2Proof. First we prove that XvE is bounded from above on U(TM) := F−1({1}) if (1)holds, and it is bounded from above and from below if (2) holds. For any v ∈ U(TM),setting p := τ(v), we haveXvE(v) = F (v)XvF (v) = XvF (v) = (F ↾ TpM)′(v)(X(p)) ≤ F (X(p)),where in the last step we used the fundamental inequality (see [1, p. 7] or [6, Proposi-tion 9.1.37]). In a similar way, we obtainXvE(v) = (F ↾ TpM)′(v)(X(p)) = −(F ↾ TpM)′(v)(−X(p)) ≥ −F (−X(p)).Over U(TM) these two inequalities give −F ◦ (−X) ◦ τ ≤ XvE ≤ F ◦X ◦ τ , from whichour first assertion follows.Now we show that XcE = 0, and hence X is a Killing field. It suffices to prove iton U(TM), because XcE is 2+-homogeneous. Indeed, the flows of Xc and C clearlycommute, hence [Xc, C] = 0, and we have C(XcE) = [C,Xc]E +Xc(CE) = 2XcE.So fix v ∈ U(TM) and let γ be the maximal geodesic with γ̇(0) = v. Then Lemma 1givesXvE(γ̇(t)) = XvE(γ̇(0)) + tXcE(γ̇(0)) = XvE(v) + tXcE(v)for any real number t in case (1) and for any positive real number t in case (2). Geodesicshave constant speed, hence γ̇ remains inside U(TM), and the left-hand side of the aboveformula has to be bounded from above in case (1), and it has to be bounded from aboveand below in case (2), which is possible only if XcE(v) = 0.As an immediate corollary we haveTheorem 3 (Hano). A bounded affine vector field on a complete Riemannian manifoldis a Killing field.Since compact Finsler manifolds are complete, we also haveTheorem 4. An affine vector field on a compact Finsler manifold is a Killing vector field.AcknowledgementsThe authors are grateful to József Szilasi and Bernadett Aradi for their suggestions thatimproved the paper.References[1] D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann–Finsler geometry,Springer, 2000.[2] M. Crampin and F. A. E. Pirani, Applicable differential geometry, vol. 59, CambridgeUniversity Press, Cambridge, 1986.[3] J.-i. Hano, On affine transformations of a Riemannian manifold, Nagoya Math. J., 9(1955), 99–109.[4] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, John Wiley& Sons, Inc., New York, 1996.[5] J. Szilasi, R. L. Lovas, and D. Cs. Kertész, Several ways to a Berwald manifold – andsome steps beyond, Extracta Mathematicae, 26 (2011), 89–130.[6] , Connections, Sprays and Finsler Structures, World Scientific, 2014.[7] K. Yano, On harmonic and Killing vector fields, Ann. of Math. (2), 55 (1952), 38–45..3

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A generalization and short proof of a theorem of Hano on affine vector fields

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