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

Musso, Monica; Pejsachowicz, Jacobo; Portaluri, Alessandro

doi:10.1051/cocv:2007037

Morse index and bifurcation of

p

-geodesics on semi riemannian manifolds

Musso, Monica ; Pejsachowicz, Jacobo ; Portaluri, Alessandro ¹

¹ Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-900, São Paulo, SP Brazil; Current address: Dipartimento di Matematica, Politecnico di Torino, Italy

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

Résumé

Given a one-parameter family ${g_{λ} : λ \in [a, b]}$ of semi riemannian metrics on an $n$ -dimensional manifold $M$ , a family of time-dependent potentials ${V_{λ} : λ \in [a, b]}$ and a family ${σ_{λ} : λ \in [a, b]}$ of trajectories connecting two points of the mechanical system defined by $(g_{λ}, V_{λ})$ , we show that there are trajectories bifurcating from the trivial branch $σ_{λ}$ if the generalized Morse indices $μ (σ_{a})$ and $μ (σ_{b})$ 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 Schöenberg type.

MR Zbl

DOI : 10.1051/cocv:2007037

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

Affiliations des auteurs :

Musso, Monica ; Pejsachowicz, Jacobo ; Portaluri, Alessandro ¹

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     title = {Morse index and bifurcation of $p$-geodesics on semi riemannian manifolds},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {598--621},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {3},
     year = {2007},
     doi = {10.1051/cocv:2007037},
     mrnumber = {2329179},
     zbl = {1127.58005},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2007037/}
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Musso, Monica; Pejsachowicz, Jacobo; Portaluri, Alessandro. Morse index and bifurcation of $p$-geodesics on semi riemannian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 3, pp. 598-621. doi : 10.1051/cocv:2007037. https://www.numdam.org/articles/10.1051/cocv:2007037/

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