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

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.