Topological Methods in Nonlinear Analysis

A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds

Monica Musso, Jacobo Pejsachowicz, and Alessandro Portaluri

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Abstract

Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse index by its relative form, given by the spectral flow of an associated family of index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context, and prove that our generalized Morse index equals the total number of conjugate points. Finally we study the relation with the Maslov index of the flow induced on the Lagrangian Grassmannian.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 25, Number 1 (2005), 69-99.

Dates
First available in Project Euclid: 23 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1466705096

Mathematical Reviews number (MathSciNet)
MR2133393

Zentralblatt MATH identifier
1101.58012

Citation

Musso, Monica; Pejsachowicz, Jacobo; Portaluri, Alessandro. A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds. Topol. Methods Nonlinear Anal. 25 (2005), no. 1, 69--99. https://projecteuclid.org/euclid.tmna/1466705096


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