×

Some properties of the spectral flow in semi-Riemannian geometry. (English) Zbl 0928.53020

The authors study geodesics joining two points of a semi-Riemannian manifold \(M\) and study the derivative of the spectral flow for simple eigenvalues. They apply their results in the Riemannian setting to show that the spectral flow is strictly decreasing and give a proof of the Morse index theorem on the number of conjugate points along a geodesic. They also show that in semi-Riemannian geometry, the analoguous Morse index theorem cannot exist in the same form. They also apply their results to study the standard static Lorentzian \(2\)-manifold \(M=M_0\times \mathbb{R}\) where the metric takes the form \(ds^2=dx^2-\beta(x)dt^2\) (\(M_0=\mathbb{R}\) or \(M_0=S^1\)) where \(0<\beta(x)\leq N\) and \(M\) is complete. They show that (1) every spacelike geodesic does not have conjugate points, (2) if two points are causally related, then there are no spacelike geodesics joining them, (3) if two points are not causally related, then there is exactly one geodesic joining them and that geodesic is spacelike.
Reviewer: P.Gilkey (Eugene)

MSC:

53C22 Geodesics in global differential geometry
58J30 Spectral flows
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beem, J. K.; Ehrlich, B. H.; Easley, K. L., Global Lorentian Geometry (1996), Marcel Dekker: Marcel Dekker New York
[2] Benci, V.; Fortunato, D.; Giannoni, F., On the existence of multiple geodesics in static space-times, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8, 79-102 (1991) · Zbl 0716.53057
[3] Benci, V.; Masiello, A., A Morse index for geodesics in static Lorentz manifolds, Math. Ann., 293, 433-442 (1992) · Zbl 0735.58011
[4] Fortunato, D.; Giannoni, F.; Masiello, A., A Fermat principle for stationary space-times and applications to light rays, J. Geom. Phys., 15, 159-188 (1995) · Zbl 0819.53037
[5] Giannoni, F.; Masiello, A., On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Funct. Anal., 101, 340-369 (1991) · Zbl 0756.53029
[6] Hawking, S. W.; Ellis, R., The large scale structure of space-time (1973), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0265.53054
[7] Helfer, A. H., Conjugate points on spacelike geodesics or pseudo-self-adjoint Morse-Sturm-Liouville systems, Pacific J. Math., 164, 321-350 (1994) · Zbl 0799.58018
[8] Mahwin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer: Springer New York
[9] Masiello, A., Variational Methods in Lorentzian Geometry, (Pitman Research Notes in Mathematics, vol. 309 (1994), Longman: Longman London) · Zbl 0816.58001
[10] O’Neill, B., Semiriemannian Geometry with Applications to Relativity (1983), Academic Press: Academic Press New York
[11] Palais, R. S., Morse theory on Hilbert manifolds, Topology, 2, 299-340 (1963) · Zbl 0122.10702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.