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

Abstract

Given a one-parameter family {g(lambda) :lambda is an element of [ a, b]} of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials {V-lambda: lambda is an element of [a, b]} and a family {sigma(lambda): lambda is an element of [ 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 s. if the generalized Morse indices mu(sigma(a)) and mu(sigma(a)) 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 Schrodenberg type.