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Morse index and bifurcation of \(p\)-geodesics on semi Riemannian manifolds. (English) Zbl 1127.58005
Summary: Given a one-parameter family \(\{g_\lambda: \lambda\in [a,b]\}\) of semi-Riemannian metrics on an \(n\)-dimensional manifold \(M\), a family of time-dependent potentials \(\{ V_\lambda : \lambda\in [a,b]\}\) and a family \(\{\sigma_\lambda: \lambda\in [a,b]\} \) of trajectories connecting two points of the mechanical system defined by \((g_\lambda, V_\lambda)\), we show that there are trajectories bifurcating from the trivial branch \(\sigma_\lambda\) if the generalized Morse indices \(\mu(\sigma_a)\) and \(\mu (\sigma_b)\) are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate points along a trajectory using an explicit computation of the Morse index in the case of locally symmetric spaces and a comparison principle of Morse-Schoenberg type.

MSC:
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
53C22 Geodesics in global differential geometry
58J30 Spectral flows
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References:
[1] R. Abraham and J.E. Marsden , Foundations of Mechanics , 2nd edition. Benjamin/Cummings, Ink. Massachusetts ( 1978 ). MR 515141 | Zbl 0393.70001 · Zbl 0393.70001
[2] L. Andersson and R. Howard , Comparison and rigidity theorems in Semi-Riemannian geometry . Comm. Anal. Geom. 6 ( 1998 ) 819 - 877 . Zbl 0963.53038 · Zbl 0963.53038
[3] S.B. Angenent and R. van der Vorst , A priori bounds and renormalized Morse indices of solutions of an elliptic system . Ann. Inst. H. Poincaré Anal. Non Linéaire 17 ( 2000 ) 277 - 306 . Numdam | Zbl 0964.35037 · Zbl 0964.35037
[4] V.I. Arnol’d , Sturm theorems and symplectic geometry . Funktsional. Anal. i Prilozhen. 19 ( 1985 ) 1 - 10 . Zbl 0606.58017 · Zbl 0606.58017
[5] J.K. Beem , P.E. Ehrlich and K.L. Easley , Global Lorentzian Geometry . Mercel Dekker, Inc. New York and Basel ( 1996 ). MR 1384756 | Zbl 0846.53001 · Zbl 0846.53001
[6] V. Benci , F. Giannoni and A. Masiello , Some properties of the spectral flow in semiriemannian geometry . J. Geom. Phys. 27 ( 1998 ) 267 - 280 . Zbl 0928.53020 · Zbl 0928.53020
[7] A.L. Besse , Manifolds all of whose geodesics are closed . Ergebnisse der Mathematik und ihrer Grenzgebiete 93, Springer-Verlag ( 1978 ). MR 496885 | Zbl 0387.53010 · Zbl 0387.53010
[8] O. Bolza , Lectures on Calculus of Variation . Univ. Chicago Press, Chicago ( 1904 ). JFM 35.0373.01 · JFM 35.0373.01
[9] S.E. Cappell , R. Lee and E.Y. Miller , On the Maslov index . Comm. Pure Appl. Math. 47 ( 1994 ) 121 - 186 . Zbl 0805.58022 · Zbl 0805.58022
[10] I. Chavel , Riemannian geometry: a modern introduction , in Cambridge tracts in Mathematics 108, Cambridge Univerisity Press ( 1993 ). MR 1271141 | Zbl 0810.53001 · Zbl 0810.53001
[11] P. Chossat , D. Lewis , J.P. Ortega and T.S. Ratiu , Bifurcation of relative equilibria in mechanical systems with symmetry . Adv. Appl. Math. 31 ( 2003 ) 10 - 45 . Zbl 1025.37032 · Zbl 1025.37032
[12] C. Conley and E. Zehnder , The Birhoff-Lewis fixed point theorem and a conjecture of V .I. Arnold. Invent. Math. 73 ( 1983 ) 33 - 49 . Zbl 0516.58017 · Zbl 0516.58017
[13] M. Crabb and I. James , Fibrewise Homotopy Theory . Springer-Verlag ( 1998 ). MR 1646248 | Zbl 0905.55001 · Zbl 0905.55001
[14] M. Daniel , An extension of a theorem of Nicolaescu on spectral flow and Maslov index . Proc. Amer. Math. Soc. 128 ( 1999 ) 611 - 619 . Zbl 0938.58025 · Zbl 0938.58025
[15] K. Deimling , Nonlinear Functional Analysis . Springer-Verlag ( 1985 ). MR 787404 | Zbl 0559.47040 · Zbl 0559.47040
[16] I. Ekeland , Convexity methods in Hamiltonian systems . Ergebnisse der Mathematik und ihrer Grenzgebiete 19, Springer-Verlag, Berlin ( 1990 ). MR 1051888 | Zbl 0707.70003 · Zbl 0707.70003
[17] Guihua Fei, Relative Morse index and its application to Hamiltonian systems in the presence of symmetries. J. Diff. Eq. 122 (1995) 302-315. Zbl 0840.34032 · Zbl 0840.34032
[18] P.M. Fitzpatrick and J. Pejsachowicz , Parity and generalized multiplicity . Trans. Amer. Math. Soc. 326 ( 1991 ) 281 - 305 . Zbl 0754.47009 · Zbl 0754.47009
[19] P.M. Fitzpatrick , J. Pejsachowicz and L. Recht , Spectral flow and bifurcation of critical points of strongly-indefinite functional . Part I. General theory. J. Funct. Anal. 162 ( 1999 ) 52 - 95 . Zbl 0915.58091 · Zbl 0915.58091
[20] P.M. Fitzpatrick , J. Pejsachowicz and L. Recht , Spectral flow and bifurcation of critical points of strongly-indefinite functional . Part II. Bifurcation of periodic orbits of Hamiltonian systems. J. Differ. Eq. 161 ( 2000 ) 18 - 40 . Zbl 0985.37057 · Zbl 0985.37057
[21] A. Floer , Relative Morse index for the symplectic action . Comm. Pure Appl. Math. 41 ( 1989 ) 335 - 356 . Zbl 0633.58009 · Zbl 0633.58009
[22] I.M. Gel’fand and S.V. Fomin , Calculus of Variations . Prentic-Hall Inc., Englewood Cliffs, New Jersey, USA ( 1963 ).
[23] I.M. Gel’fand and V.B. Lidskii , On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients . Amer. Math. Soc. Transl. Ser. 2 8 ( 1958 ) 143 - 181 . Zbl 0079.10905 · Zbl 0079.10905
[24] R. Giambó , P. Piccione and A. Portaluri , On the Maslov Index of Lagrangian paths that are not transversal to the Maslov cycle . Semi-Riemannian index Theorems in the degenerate case. ( 2003 ) Preprint. arXiv
[25] A.D. Helfer , Conjugate points on space like geodesics or pseudo self-adjoint Morse-Sturm-Liouville systems . Pacific J. Math. 164 ( 1994 ) 321 - 340 . Article | Zbl 0799.58018 · Zbl 0799.58018
[26] J. Jost , X. Li-Jost and X.W. Peng , Bifurcation of minimal surfaces in Riemannian manifolds . Trans. Amer. Math. Soc. 347 ( 1995 ) 51 - 62 . Zbl 0835.53010 · Zbl 0835.53010
[27] T. Kato , Perturbation Theory for linear operators . Grundlehren der Mathematischen Wissenschaften 132, Springer-Verlag ( 1980 ). Zbl 0435.47001 · Zbl 0435.47001
[28] W. Klingenberg , Closed geodesics on Riemannian manifolds . CBMS Regional Conference Series in Mathematics 53 ( 1983 ). MR 714330 | Zbl 0539.53003 · Zbl 0539.53003
[29] W. Klingenberg , Riemannian Geometry . de Gruyter, New York ( 1995 ). MR 1330918 | Zbl 0495.53036 · Zbl 0495.53036
[30] M.A. Krasnoselskii , Topological methods in the theory of nonlinear integral equations . Pergamon, New York ( 1964 ). MR 159197 | Zbl 0111.30303 · Zbl 0111.30303
[31] D.N. Kupeli , On conjugate and focal points in semi-Riemannian geometry . Math. Z. 198 ( 1988 ) 569 - 589 . Article | Zbl 0658.53059 · Zbl 0658.53059
[32] S. Lang , Differential and Riemannian Manifolds . Springer-Verlag ( 1995 ). MR 1335233 | Zbl 0824.58003 · Zbl 0824.58003
[33] E. Meinrenken , Trace formulas and Conley-Zehnder index . J. Geom. Phys. 13 ( 1994 ) 1 - 15 . Zbl 0791.53040 · Zbl 0791.53040
[34] J. Milnor , Morse theory . Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies 51, Princeton University Press, Princeton, N.J. ( 1963 ). MR 163331 | Zbl 0108.10401 · Zbl 0108.10401
[35] M. Musso , J. Pejsachowicz and A. Portaluri , A Morse Index Theorem and bifurcation for perturbed geodesics on Semi-Riemannian Manifolds . Topol. Methods Nonlinear Anal. 25 ( 2005 ) 69 - 99 . Zbl 1101.58012 · Zbl 1101.58012
[36] B. O’Neill , Semi-Riemannian geometry with applications to relativity . Academic Press, New York ( 1983 ). Zbl 0531.53051 · Zbl 0531.53051
[37] R.S. Palais , Foundations of global non-linear analysis . W.A. Benjamin, Inc., New York ( 1968 ). MR 248880 | Zbl 0164.11102 · Zbl 0164.11102
[38] G. Peano , Lezioni di Analisi infinitesimale , Volume I, pp. 120 - 121 , Volume II, pp. 187 - 195 . Tipografia editrice G. Candeletti, Torino (1893). JFM 25.0450.01 · JFM 25.0450.01
[39] P. Piccione , A. Portaluri and D.V. Tausk , Spectral flow , Maslov index and bifurcation of semi-Riemannian geodesics. Ann. Global Anal. Geometry 25 ( 2004 ) 121 - 149 . Zbl 1050.58015 · Zbl 1050.58015
[40] A. Portaluri , A formula for the Maslov index of linear autonomous Hamiltonian systems . ( 2004 ) Preprint.
[41] A. Portaluri , Morse Index Theorem and Bifurcation theory on semi-Riemannian manifolds . Ph.D. thesis ( 2004 ).
[42] P.J. Rabier , Generalized Jordan chains and two bifurcation theorems of Krasnosel’skii . Nonlinear Anal. 13 ( 1989 ) 903 - 934 . Zbl 0686.47047 · Zbl 0686.47047
[43] J. Robbin and D. Salamon , The Maslov index for paths . Topology 32 ( 1993 ) 827 - 844 . MR 1241874 | Zbl 0798.58018 · Zbl 0798.58018
[44] J. Robbin and D. Salamon , The spectral flow and the Maslov index . Bull. London Math. Soc. 27 ( 1995 ) 1 - 33 . Zbl 0859.58025 · Zbl 0859.58025
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