Stability of the Focal and Geometric Index in semi-Riemannian Geometry via the Maslov Index
| dc.creator | Mercuri, Francesco | |
| dc.creator | Piccione, Paolo | |
| dc.creator | Tausk, Daniel V. | |
| dc.date | 1999-05-17 | |
| dc.date | 2000-04-24 | |
| dc.date.accessioned | 2026-07-07T05:29:06Z | |
| dc.date.available | 2026-07-07T05:29:06Z | |
| dc.description | We investigate the problem of the stability of the number of conjugate or focal points (counted with multiplicity) along a semi-Riemannian geodesic $γ$. For a Riemannian or a non spacelike Lorentzian geodesic, such number is equal to the intersection number (Maslov index) of a continuous curve with a subvariety of codimension one of the Lagrangian Grassmannian of a symplectic space. Such intersection number is proven to be stable in a large variety of circumstances. In the general semi-Riemannian case, under suitable hypotheses this number is equal to an algebraic count of the multiplicities of the conjugate points, and it is related to the spectral properties of a non self-adjoint differential operator. This last relation gives a weak extension of the classical Morse Index Theorem in Riemannian and Lorentzian geometry. In this paper we reprove some results that were incorrectly stated by Helfer in a previous reference; in particular, a counterexample to one of Helfer's results, which is essential for the theory, is given. In the last part of the paper we discuss a general technique for the construction of examples and counterexamples in the index theory for semi-Riemannian metrics, in which some new phenomena appear. | |
| dc.description | 58 pages, LaTeX2e, amsart class some references updated and minor changes in the abstract and in the Introduction in the replacement of Aug. 13th, 1999 Revised version of April 24th, 2000: added Remark 6.2.2, minor modifications in the Introduction | |
| dc.identifier | https://arxiv.org/abs/math/9905096 | |
| dc.identifier | http://arxiv.org/abs/math/9905096 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78509 | |
| dc.subject | Differential Geometry | |
| dc.subject | Symplectic Geometry | |
| dc.subject | 34B24; 34C10; 34L05; 53C22; 53C50; 53C80 | |
| dc.title | Stability of the Focal and Geometric Index in semi-Riemannian Geometry via the Maslov Index | |
| dc.type | text |