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

Published online by Cambridge University Press:  26 July 2007

Monica Musso ,
Jacobo Pejsachowicz  and
Alessandro Portaluri
Monica Musso
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Torino, Italy; jacobo.pejsachowicz@polito.it Departamento de Matematicas, Pontificia Universidad Catolica de Chile, Avenida Vicuña MacKenna 4860, Macul, Chile; mmusso@mat.puc.cl
Jacobo Pejsachowicz
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Torino, Italy; jacobo.pejsachowicz@polito.it
Alessandro Portaluri
Affiliation:
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.
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Abstract

Given a one-parameter family $\{g_\lambda\colon \lambda\in [a,b]\}$ of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials $\{ V_\lambda\colon \lambda\in [a,b]\}$ and a family $\{\sigma_\lambda\colon \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 $\mu(\sigma_a)$ and $\mu (\sigma_b)$ 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.

Keywords

Generalized Morse indexsemi-Riemannian manifoldsperturbed geodesicbifurcation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 3 , July 2007 , pp. 598 - 621
Copyright
© EDP Sciences, SMAI, 2007

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