实射影空间和哈密顿系统的闭轨道问题
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摘要
本篇论文我们考虑实射影空间上的闭测地线问题和二阶自治哈密顿系统的最小周期解问题。
在第一部分中,基于Westerland的工作[1]和[2]我们使用Leray-Serre谱序列理论和Chas-Sullivan关于环路同调的B atalin-Vilkovisky代数结构的研究工作得到下面的结果：对于非单连通流形M=RP2n|1,记LgM为M上非可缩的环路组成的空间,则或者序列{dimHkS1(LgM；Z2)}无界,或者对于前一种情形,我们可以利用Gromoll-Meyer理论证明M上具有无穷多条不同的、非可缩的闭测地线。而如果是后一种情形,则我们可以建立关于Lg(M)上闭测地线的平均指标恒等式。注意到Sn上类似的平均指标恒等式是Bangert,Rademacher,龙以明,王嵬和段华贵等人证明球面上至少存在两条不同闭测地线的一系列工作的基石之一,因此我们可以期望利用本文得到的平均指标恒等式再结合龙以明精确指标迭代公式来证明M上至少有两条不同的、非可缩的闭测地线。
利用相同的方法,我们还证明了：对于k≥4且k为偶数,在Cohen-Jones同构下,Sk上的两种B atalin-Vilkovisky代数结构同构,即(H*(LSk；Z2),·,△)(?)(HH*(H*(Sk),H*(Sk),Z2),∪,Δ).与Menichi[3]中关于奇数维球面的结果结合得到,任意k≥3维球面上的这两种B-V代数结构同构。而k=2时,Menichi[3]证明了二者是不同构的。尽管如此,我们可以证明对任意的球面Sk,k≥2,纤维LSk→ES1×s1LSk→BS1的Leray-Serre谱序列在第三页坍塌。
在第二部分,我们研究辛道路的ω=-1的ω指标迭代公式。结合Masloy指标理论(可看作ω=1情形的ω指标理论),我们得到不可定向闭测地线Morse指标的精确迭代公式。我们希望这些公式将来可以用来研究不可定向流形上(譬如RP2n)的闭测地线问题。
在第三部分,我们考虑二阶自治哈密顿系统x+V'(x)=0,(?)x∈Rn,并证明了下面的结果：如果V(x)=V(-x)并且存在常数e>1使得00上面方程至少存在一个最小周期为T的周期解。
In this paper, we consider the problem of closed geodesics on the real projective spaces and the minimal period problem of the second order autonomous Hamiltonian systems with even po-tentials.
In the first part, based on the works of Westerland [1] and [2] we use the theory of Leray-Serre spectral sequence and the work of Chas and Sullivan on the Batalin-Vilkovisky algebraic structure of the homology of the free loop space to obtain the following result:For the non-simply connected space M=RP2n-1, let LgM denote the space consisting of the non-contractible loops on M. Then, either dimkS1(LgM; Z2) is unbounded, or For the first case, we can use the theory of Gromoll and Meyer to prove that there are infinite different non-contractible closed geodesics on LgM. While if it is the second case, we can establish a mean index identity for closed geodesics on LgM. Notice that a similar mean index identity for closed geodesics on Sn is one of the foundation stones for a series of works in search of the second geodesic on the spheres by many authors such as Bangert, Rademacher, Y. Long, W. Wang and H. Duan, so we may anticipate that combining with Long's precise iteration index formulae the mean index identity in this paper can be used to prove that there are at least two different non-contractible closed geodesics on M.
By the same way, we have also proved that under the Cohen-Jones isomorphism, the two Batalin-Vilkovisky algebras for Sk with even k≥4are isomorphic, that is,(H*(LSk;Z2),·,△)(?)(HH*(H*(Sk),H*(Sk),Z2),U,△). Combining with the results for odd k≥3in [3], we get that the above two B-V algebras are isomorphic for every k≥3.
While for S2, the above two algebras are not isomorphic due to Menichi[3]. Nevertheless, we can prove for any sphere Sk with k≥2that the Leray-Serre spectral sequence for the fibration LSk→ES1×S1LSk→BS1collapses at the third page.
In the second part, we study the co-index iteration formulae for symplectic paths with ω=-1. Combing with the Maslov index theory, which can be seen as the case of co-index with ω=1, we obtain the precise iteration formulae of Morse indices for unorientable closed geodesics. We hope that the formulaes in this paper will be used in study of the problem of close geodesies on unorientable manifolds such as RP2n.
In the third part, we consider the second order autonomous Hamiltonian system x+V'(x)=0,(?)x∈Rn, and have proved that if V(x)=V(-x) and there exists a constant θ>1such that0<θV'(x)·x≤V"(x)x-x,(?)x∈Rn\{0}, then for any T>0the above equation has at least a periodic solution with the minimal period T.
在第一部分中,基于Westerland的工作[1]和[2]我们使用Leray-Serre谱序列理论和Chas-Sullivan关于环路同调的B atalin-Vilkovisky代数结构的研究工作得到下面的结果：对于非单连通流形M=RP2n|1,记LgM为M上非可缩的环路组成的空间,则或者序列{dimHkS1(LgM；Z2)}无界,或者对于前一种情形,我们可以利用Gromoll-Meyer理论证明M上具有无穷多条不同的、非可缩的闭测地线。而如果是后一种情形,则我们可以建立关于Lg(M)上闭测地线的平均指标恒等式。注意到Sn上类似的平均指标恒等式是Bangert,Rademacher,龙以明,王嵬和段华贵等人证明球面上至少存在两条不同闭测地线的一系列工作的基石之一,因此我们可以期望利用本文得到的平均指标恒等式再结合龙以明精确指标迭代公式来证明M上至少有两条不同的、非可缩的闭测地线。
利用相同的方法,我们还证明了：对于k≥4且k为偶数,在Cohen-Jones同构下,Sk上的两种B atalin-Vilkovisky代数结构同构,即(H*(LSk；Z2),·,△)(?)(HH*(H*(Sk),H*(Sk),Z2),∪,Δ).与Menichi[3]中关于奇数维球面的结果结合得到,任意k≥3维球面上的这两种B-V代数结构同构。而k=2时,Menichi[3]证明了二者是不同构的。尽管如此,我们可以证明对任意的球面Sk,k≥2,纤维LSk→ES1×s1LSk→BS1的Leray-Serre谱序列在第三页坍塌。
在第二部分,我们研究辛道路的ω=-1的ω指标迭代公式。结合Masloy指标理论(可看作ω=1情形的ω指标理论),我们得到不可定向闭测地线Morse指标的精确迭代公式。我们希望这些公式将来可以用来研究不可定向流形上(譬如RP2n)的闭测地线问题。
在第三部分,我们考虑二阶自治哈密顿系统x+V'(x)=0,(?)x∈Rn,并证明了下面的结果：如果V(x)=V(-x)并且存在常数e>1使得0
In this paper, we consider the problem of closed geodesics on the real projective spaces and the minimal period problem of the second order autonomous Hamiltonian systems with even po-tentials.
In the first part, based on the works of Westerland [1] and [2] we use the theory of Leray-Serre spectral sequence and the work of Chas and Sullivan on the Batalin-Vilkovisky algebraic structure of the homology of the free loop space to obtain the following result:For the non-simply connected space M=RP2n-1, let LgM denote the space consisting of the non-contractible loops on M. Then, either dimkS1(LgM; Z2) is unbounded, or For the first case, we can use the theory of Gromoll and Meyer to prove that there are infinite different non-contractible closed geodesics on LgM. While if it is the second case, we can establish a mean index identity for closed geodesics on LgM. Notice that a similar mean index identity for closed geodesics on Sn is one of the foundation stones for a series of works in search of the second geodesic on the spheres by many authors such as Bangert, Rademacher, Y. Long, W. Wang and H. Duan, so we may anticipate that combining with Long's precise iteration index formulae the mean index identity in this paper can be used to prove that there are at least two different non-contractible closed geodesics on M.
By the same way, we have also proved that under the Cohen-Jones isomorphism, the two Batalin-Vilkovisky algebras for Sk with even k≥4are isomorphic, that is,(H*(LSk;Z2),·,△)(?)(HH*(H*(Sk),H*(Sk),Z2),U,△). Combining with the results for odd k≥3in [3], we get that the above two B-V algebras are isomorphic for every k≥3.
While for S2, the above two algebras are not isomorphic due to Menichi[3]. Nevertheless, we can prove for any sphere Sk with k≥2that the Leray-Serre spectral sequence for the fibration LSk→ES1×S1LSk→BS1collapses at the third page.
In the second part, we study the co-index iteration formulae for symplectic paths with ω=-1. Combing with the Maslov index theory, which can be seen as the case of co-index with ω=1, we obtain the precise iteration formulae of Morse indices for unorientable closed geodesics. We hope that the formulaes in this paper will be used in study of the problem of close geodesies on unorientable manifolds such as RP2n.
In the third part, we consider the second order autonomous Hamiltonian system x+V'(x)=0,(?)x∈Rn, and have proved that if V(x)=V(-x) and there exists a constant θ>1such that0<θV'(x)·x≤V"(x)x-x,(?)x∈Rn\{0}, then for any T>0the above equation has at least a periodic solution with the minimal period T.
引文
[1] C. Westerland, Dyer-Lashof operations in the string topology of spheres and projectivespaces, Math. Z.250(2005), no.3,711～727.
[2] C. Westerland, String topology of spheres and projective spaces, Algebr. Geom. Topol.7(2007),309～325.
[3] L. Menichi, String topology for spheres, Comment. Math. Helv.84(2009), no.1,135～157.
[4] M. Morse, The calculus of variations in the large, vol.18, New York, Colloquium Publ.,1934.
[5] D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J.Differential Geom.3(1969),493～510.
[6] R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure.Appl. Math.(1956), vol.9,171～206.
[7] M. Vigue′-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, J.Differential Geom.11(1976),663～644.
[8] H.-B. Rademacher, On the average indices of closed geodesics, J. Differential Geom.29(1989),65～83.
[9] Y. Long, Index theory for symplectic paths with applications, Birkha¨user Verlag, Basel·Boston· Berlin,2002.
[10] V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler2-sphere,Math. Ann.346(2010),335～366.
[11] W. Ballmann, G. Thorbergsson and W. Ziller, Closed geodesics and the fundamental group,Duke Math. J.48(1981),585～588.
[12] V. Bangert and N. Hinston, Closed geodesics on manifolds with infinite abelian fundamentalgroup, J. Differential Geom.19(1984),277～282.
[13] M. Chas and D. Sullivan, String topology, arXiv:math-GT/9911159,(1999).
[14] R. L. Cohen and J. D. S. Jones, A homotopy theoretic realization of string topology, Math.Ann.324(2002),773～798.
[15] P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math.,31(1978),157～184.
[16] I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convexautonomous Hamiltonian system, Invent. Math.81(1985),155～188.
[17] D. Dong and Y. Long, The iteration formula of the Maslov-type index theory with applica-tions to nonlinear Hamiltonian systems. Trans. Amer. Math. Soc.349(1997),2619～2661.
[18] Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials,Ann. Inst. H. Poincare′Anal. Nonlinear,10(1993),605～626.
[19] Y. Long, The minimal period problem of periodic solutions for autonomous superquadraticsecond order Hamiltonian systems, J. Differential Equations,111(1994),147～171.
[20] Y. Long, On the minimal period for periodic solution problem of nonlinear Hamiltoniansystems, Chin. Ann. of Math. B,4(1997),481～484.
[21] J.D.S. Jones, Cyclic homology and equivariant homology, Invent. Math.87(1987),403～423.
[22] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math.(2)78(1963),267～288.
[23] Y. Fe′lix and J. C. Thomas, Rational BV-algebra in string topology. Bull. Soc. Math. France136(2008), no.2,311～327.
[24] Tian Yang, A Batalin-Vilkovisky Algebra structure on the Hochschild Cohomology ofTruncated Polynomials, arXiv:0707.4213.
[25] M. Bo¨kstedt and I. Ottosen, String cohomology groups of complex projective spaces,Algebraic&Geometric Topology7,2007,2165～2238.
[26] V. Bangert, Geoda¨tische Linien auf Riemannschen Mannigfaltigkeiten, Jahresber. Deutsch.Math.-Verein.87(1985),39～66.
[27] Y. Long, Multiplicity and stability of closed geodesics on Finsler2-spheres, J. Eur. Math.Soc.8(2006),341～353.
[28] I. A. Taimanov, The type numbers of closed geodesics, arXiv:0912.5226,2010.
[29] I. A. Taimanov, Closed geodesics on non-simply-connected manifolds,Russian MathematicalSurveys, vol.40, Issue6,(1985),143～144.
[30] W. Klingenberg, Lectures on closed geodesics. Springer-Verlag, Berlin, heidelberg, NewYork,1978.
[31] H. Duan and Y. Long, The index quasi-periodicity and multiplicity of closed geodesics,preprint.
[32] J. McCleary, A user’s guide to spectral sequences, second edition, Cambridge Univ. Press,2001.
[33] A. Hatcher, Spectral sequences in algebraic topology, to appear.
[34] M. Bo¨kstedt and I. Ottosen, Homotopy orbits of free loop spaces, Fund. Math.162(1999),251～275.
[35] N. Hingston, Equivariant Morse theory and closed geodesics, J. Differential Geom.19(1984),85～116.
[36] G. Bredon, Introduction to compact transformation groups, Academic press, New York andLondon,1972.
[37] A. Borel, Seminar on transformation groups, Princeton university press,1960.
[38] W. Y. Hsiang, Cohomology theory of topological transformation groups, Springer, NewYork,1975.
[39] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in AdvancedMathematics38, Cambridge university press,1994.
[40] Y. Long and W. Wang, Multiple closed geodesics on Riemannian3-spheres, Calc. Var.30(2007),183～214.
[41] H. Duan and Y. Long, Multiple closed geodesics on bumpy Finsler n-spheres, J. Differential.Equations233(2007),221～240.
[42] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math.(1999), Vol.187, No.1,113～149.
[43] Y. Long, Precise index formulae of the Maslove-type indexand ellipticity of closed charac-teristics, Adv. Math.154(2000),76～131.
[44] C. Liu, The relation of the Morse index of closed geodesics with the Maslov-type index ofsymplectic paths, Acta. Math. Sinica, English Series(2003), Vol.19, No.2,1～11.
[45] I. Ekeland, Convexity methods in Hamiltonian mechanics, Springer-Verlag, Berlin,1990.
[46] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamilto-nian systems, Math. Ann.255(1981),405～421.
[47] J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Applied Mathe-matical Sciences, vol.74, Springer-Verlag, New York,1989.
[48][uncited] A. Abbondandolo, Morse theory for Hamiltonian systems, Chapman&Hall/CRCPress, Boca Raton,2001.
[49][uncited] K.C. Chang, Infinite dimensional Morse theory and multiple solution problems,Birkha¨user, Boston,1993.
[50][uncited] A. Hatcher, Algebraic Topology, Cambridge university press,2002.
[51][uncited]C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications,Science in China(Series A)(2002), vol.45, No.1,9～28.
[52][uncited] J. Loday, Cyclic homology, Grundlehren301, Springer-Verlag,1992.
[53][uncited] Y. Long and H. Duan, Multiple closed geodesics on3-spheres, Adv. Math.221(2009),1757～1803.
[54][uncited]I. Ottosen, On the Borel cohomology of free loop spaces, Math. Scand.93(2003),185-220.
[2] C. Westerland, String topology of spheres and projective spaces, Algebr. Geom. Topol.7(2007),309～325.
[3] L. Menichi, String topology for spheres, Comment. Math. Helv.84(2009), no.1,135～157.
[4] M. Morse, The calculus of variations in the large, vol.18, New York, Colloquium Publ.,1934.
[5] D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J.Differential Geom.3(1969),493～510.
[6] R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure.Appl. Math.(1956), vol.9,171～206.
[7] M. Vigue′-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, J.Differential Geom.11(1976),663～644.
[8] H.-B. Rademacher, On the average indices of closed geodesics, J. Differential Geom.29(1989),65～83.
[9] Y. Long, Index theory for symplectic paths with applications, Birkha¨user Verlag, Basel·Boston· Berlin,2002.
[10] V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler2-sphere,Math. Ann.346(2010),335～366.
[11] W. Ballmann, G. Thorbergsson and W. Ziller, Closed geodesics and the fundamental group,Duke Math. J.48(1981),585～588.
[12] V. Bangert and N. Hinston, Closed geodesics on manifolds with infinite abelian fundamentalgroup, J. Differential Geom.19(1984),277～282.
[13] M. Chas and D. Sullivan, String topology, arXiv:math-GT/9911159,(1999).
[14] R. L. Cohen and J. D. S. Jones, A homotopy theoretic realization of string topology, Math.Ann.324(2002),773～798.
[15] P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math.,31(1978),157～184.
[16] I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convexautonomous Hamiltonian system, Invent. Math.81(1985),155～188.
[17] D. Dong and Y. Long, The iteration formula of the Maslov-type index theory with applica-tions to nonlinear Hamiltonian systems. Trans. Amer. Math. Soc.349(1997),2619～2661.
[18] Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials,Ann. Inst. H. Poincare′Anal. Nonlinear,10(1993),605～626.
[19] Y. Long, The minimal period problem of periodic solutions for autonomous superquadraticsecond order Hamiltonian systems, J. Differential Equations,111(1994),147～171.
[20] Y. Long, On the minimal period for periodic solution problem of nonlinear Hamiltoniansystems, Chin. Ann. of Math. B,4(1997),481～484.
[21] J.D.S. Jones, Cyclic homology and equivariant homology, Invent. Math.87(1987),403～423.
[22] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math.(2)78(1963),267～288.
[23] Y. Fe′lix and J. C. Thomas, Rational BV-algebra in string topology. Bull. Soc. Math. France136(2008), no.2,311～327.
[24] Tian Yang, A Batalin-Vilkovisky Algebra structure on the Hochschild Cohomology ofTruncated Polynomials, arXiv:0707.4213.
[25] M. Bo¨kstedt and I. Ottosen, String cohomology groups of complex projective spaces,Algebraic&Geometric Topology7,2007,2165～2238.
[26] V. Bangert, Geoda¨tische Linien auf Riemannschen Mannigfaltigkeiten, Jahresber. Deutsch.Math.-Verein.87(1985),39～66.
[27] Y. Long, Multiplicity and stability of closed geodesics on Finsler2-spheres, J. Eur. Math.Soc.8(2006),341～353.
[28] I. A. Taimanov, The type numbers of closed geodesics, arXiv:0912.5226,2010.
[29] I. A. Taimanov, Closed geodesics on non-simply-connected manifolds,Russian MathematicalSurveys, vol.40, Issue6,(1985),143～144.
[30] W. Klingenberg, Lectures on closed geodesics. Springer-Verlag, Berlin, heidelberg, NewYork,1978.
[31] H. Duan and Y. Long, The index quasi-periodicity and multiplicity of closed geodesics,preprint.
[32] J. McCleary, A user’s guide to spectral sequences, second edition, Cambridge Univ. Press,2001.
[33] A. Hatcher, Spectral sequences in algebraic topology, to appear.
[34] M. Bo¨kstedt and I. Ottosen, Homotopy orbits of free loop spaces, Fund. Math.162(1999),251～275.
[35] N. Hingston, Equivariant Morse theory and closed geodesics, J. Differential Geom.19(1984),85～116.
[36] G. Bredon, Introduction to compact transformation groups, Academic press, New York andLondon,1972.
[37] A. Borel, Seminar on transformation groups, Princeton university press,1960.
[38] W. Y. Hsiang, Cohomology theory of topological transformation groups, Springer, NewYork,1975.
[39] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in AdvancedMathematics38, Cambridge university press,1994.
[40] Y. Long and W. Wang, Multiple closed geodesics on Riemannian3-spheres, Calc. Var.30(2007),183～214.
[41] H. Duan and Y. Long, Multiple closed geodesics on bumpy Finsler n-spheres, J. Differential.Equations233(2007),221～240.
[42] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math.(1999), Vol.187, No.1,113～149.
[43] Y. Long, Precise index formulae of the Maslove-type indexand ellipticity of closed charac-teristics, Adv. Math.154(2000),76～131.
[44] C. Liu, The relation of the Morse index of closed geodesics with the Maslov-type index ofsymplectic paths, Acta. Math. Sinica, English Series(2003), Vol.19, No.2,1～11.
[45] I. Ekeland, Convexity methods in Hamiltonian mechanics, Springer-Verlag, Berlin,1990.
[46] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamilto-nian systems, Math. Ann.255(1981),405～421.
[47] J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Applied Mathe-matical Sciences, vol.74, Springer-Verlag, New York,1989.
[48][uncited] A. Abbondandolo, Morse theory for Hamiltonian systems, Chapman&Hall/CRCPress, Boca Raton,2001.
[49][uncited] K.C. Chang, Infinite dimensional Morse theory and multiple solution problems,Birkha¨user, Boston,1993.
[50][uncited] A. Hatcher, Algebraic Topology, Cambridge university press,2002.
[51][uncited]C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications,Science in China(Series A)(2002), vol.45, No.1,9～28.
[52][uncited] J. Loday, Cyclic homology, Grundlehren301, Springer-Verlag,1992.
[53][uncited] Y. Long and H. Duan, Multiple closed geodesics on3-spheres, Adv. Math.221(2009),1757～1803.
[54][uncited]I. Ottosen, On the Borel cohomology of free loop spaces, Math. Scand.93(2003),185-220.