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.

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