Morse index and bifurcation of p-geodesics on semi riemannian manifolds
ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 3, p. 598-621

Given a one-parameter family {g λ :λ[a,b]} of semi riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials {V λ :λ[a,b]} and a family {σ λ :λ[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.

DOI : https://doi.org/10.1051/cocv:2007037
Classification:  58E10,  37J45,  53C22,  58J30
Keywords: generalized Morse index, semi-riemannian manifolds, perturbed geodesic, bifurcation
@article{COCV_2007__13_3_598_0,
     author = {Musso, Monica and Pejsachowicz, Jacobo and Portaluri, Alessandro},
     title = {Morse index and bifurcation of $p$-geodesics on semi riemannian manifolds},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {3},
     year = {2007},
     pages = {598-621},
     doi = {10.1051/cocv:2007037},
     zbl = {1127.58005},
     mrnumber = {2329179},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2007__13_3_598_0}
}
Musso, Monica; Pejsachowicz, Jacobo; Portaluri, Alessandro. Morse index and bifurcation of $p$-geodesics on semi riemannian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 3, pp. 598-621. doi : 10.1051/cocv:2007037. http://www.numdam.org/item/COCV_2007__13_3_598_0/

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