Differential Geometry
Computation of the Maslov index and the spectral flow via partial signatures
Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 397-402.

Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of Fredholm self-adjoint operators on real separable Hilbert spaces. As applications of the theory, we obtain a semi-Riemannian version of the Morse index theorem for geodesics with possibly conjugate endpoints, and we prove a bifurcation result at conjugate points along semi-Riemannian geodesics.

Etant donné un chemin régulier de lagrangiens, nous introduisons dans le cas de la dimension finie et le cas (Fredholm) de dimension infinie la notion de signatures partielles en chaque intersection isolée d'un tel chemin avec le cycle de Maslov. En utilisant les signatures partielles, nous donnerons une formule de calcul de l'indice de Maslov. Une formule semblabe vaut pour le flux spectral de chemins réel-analytiques d'opérateurs auto-adjoints de Fredholm sur des espaces de Hilbert réels et séparables. Comme application de la théorie, nous obtenons une version semi-Riemannienne du théorème de l'indice de Morse dans le cas de géodésiques avec des points initiaux conjugués. Enfin, nous démontrons un résultat de bifurcation en ces points conjugués le long des géodésiques semi-Riemanniennes.

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DOI: 10.1016/j.crma.2004.01.004
Giambò, Roberto 1; Piccione, Paolo 2; Portaluri, Alessandro 2

1 Dipartimento di Matematica e Informatica, Università di Camerino, 62032 Camerino, MC, Italy
2 Departamento de Matemática, Instituto de Matemática e Estatı́stica, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-900, São Paulo, SP, Brazil
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Giambò, Roberto; Piccione, Paolo; Portaluri, Alessandro. Computation of the Maslov index and the spectral flow via partial signatures. Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 397-402. doi : 10.1016/j.crma.2004.01.004. http://www.numdam.org/articles/10.1016/j.crma.2004.01.004/

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