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

¹ Dipartimento di Matematica, Politecnico di Torino, Torino, Italy; jacobo.pejsachowicz@polito.it
² Departamento de Matematicas, Pontificia Universidad Catolica de Chile, Avenida Vicuña MacKenna 4860, Macul, Chile; mmusso@mat.puc.cl
³ Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-900, São Paulo, SP Brazil; portalur@ime.usp.br Dipartimento di Matematica, Politecnico di Torino, Italy.

Received: 5 December 2005
Revised: 19 January 2006

Abstract

Given a one-parameter family of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials and a family of trajectories connecting two points of the mechanical system defined by , we show that there are trajectories bifurcating from the trivial branch if the generalized Morse indices and 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.