Morse index and bifurcation of p-geodesics on semi Riemannian manifolds

Given a one-parameter family {gλ : λ ∈ [a, b]} of semi Riemannian metrics on an ndimensional manifold M, a family of time-dependent potentials {Vλ : λ ∈ [a, b]} and a family {σλ : λ ∈ [a, b]} of trajectories connecting two points of the mechanical system defined by (gλ, Vλ), we show that there are trajectories bifurcating from the trivial branch σλ if the generalized Morse indices μ(σa) and μ(σb) 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.