Abstract
We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.
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Piccione, P., Portaluri, A. & Tausk, D.V. Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics. Annals of Global Analysis and Geometry 25, 121–149 (2004). https://doi.org/10.1023/B:AGAG.0000018558.65790.db
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DOI: https://doi.org/10.1023/B:AGAG.0000018558.65790.db