Morse index and bifurcation of $p$-geodesics on semi riemannian manifolds

Musso, Monica; Pejsachowicz, Jacobo; Portaluri, Alessandro

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 3, pp. 598-621.

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 $\mu \left({\sigma }_a\right)$ and $\mu \left({\sigma }_b\right)$ 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 Schöenberg type.

Classification : 58E10, 37J45, 53C22, 58J30
Mots clés : generalized Morse index, semi-riemannian manifolds, perturbed geodesic, bifurcation

[1] R. Abraham and J.E. Marsden, Foundations of Mechanics, 2nd edition. Benjamin/Cummings, Ink. Massachusetts (1978). | MR | Zbl

[2] L. Andersson and R. Howard, Comparison and rigidity theorems in Semi-Riemannian geometry. Comm. Anal. Geom. 6 (1998) 819-877. | Zbl

[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

[4] V.I. Arnol'D, Sturm theorems and symplectic geometry. Funktsional. Anal. i Prilozhen. 19 (1985) 1-10. | Zbl

[5] J.K. Beem, P.E. Ehrlich and K.L. Easley, Global Lorentzian Geometry. Mercel Dekker, Inc. New York and Basel (1996). | MR | Zbl

[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

[7] A.L. Besse, Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete 93, Springer-Verlag (1978). | MR | Zbl

[8] O. Bolza, Lectures on Calculus of Variation. Univ. Chicago Press, Chicago (1904). | JFM

[9] S.E. Cappell, R. Lee and E.Y. Miller, On the Maslov index. Comm. Pure Appl. Math. 47 (1994) 121-186. | Zbl

[10] I. Chavel, Riemannian geometry: a modern introduction, in Cambridge tracts in Mathematics 108, Cambridge Univerisity Press (1993). | MR | Zbl

[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

[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

[13] M. Crabb and I. James, Fibrewise Homotopy Theory. Springer-Verlag (1998). | MR | Zbl

[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

[15] K. Deimling, Nonlinear Functional Analysis. Springer-Verlag (1985). | MR | Zbl

[16] I. Ekeland, Convexity methods in Hamiltonian systems. Ergebnisse der Mathematik und ihrer Grenzgebiete 19, Springer-Verlag, Berlin (1990). | MR | Zbl

[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

[18] P.M. Fitzpatrick and J. Pejsachowicz, Parity and generalized multiplicity. Trans. Amer. Math. Soc. 326 (1991) 281-305. | Zbl

[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

[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

[21] A. Floer, Relative Morse index for the symplectic action. Comm. Pure Appl. Math. 41 (1989) 335-356. | Zbl

[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

[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.

[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. | Zbl

[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

[27] T. Kato, Perturbation Theory for linear operators. Grundlehren der Mathematischen Wissenschaften 132, Springer-Verlag (1980). | Zbl

[28] W. Klingenberg, Closed geodesics on Riemannian manifolds. CBMS Regional Conference Series in Mathematics 53 (1983). | MR | Zbl

[29] W. Klingenberg, Riemannian Geometry. de Gruyter, New York (1995). | MR | Zbl

[30] M.A. Krasnoselskii, Topological methods in the theory of nonlinear integral equations. Pergamon, New York (1964). | MR | Zbl

[31] D.N. Kupeli, On conjugate and focal points in semi-Riemannian geometry. Math. Z. 198 (1988) 569-589. | Zbl

[32] S. Lang, Differential and Riemannian Manifolds. Springer-Verlag (1995). | MR | Zbl

[33] E. Meinrenken, Trace formulas and Conley-Zehnder index. J. Geom. Phys. 13 (1994) 1-15. | Zbl

[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 | Zbl

[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

[36] B. O'Neill, Semi-Riemannian geometry with applications to relativity. Academic Press, New York (1983). | Zbl

[37] R.S. Palais, Foundations of global non-linear analysis. W.A. Benjamin, Inc., New York (1968). | MR | Zbl

[38] G. Peano, Lezioni di Analisi infinitesimale, Volume I, pp. 120-121, Volume II, pp. 187-195. Tipografia editrice G. Candeletti, Torino (1893). | JFM

[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

[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

[43] J. Robbin and D. Salamon, The Maslov index for paths. Topology 32 (1993) 827-844. | MR | Zbl

[44] J. Robbin and D. Salamon, The spectral flow and the Maslov index. Bull. London Math. Soc. 27 (1995) 1-33. | Zbl