A Generalized Index Theorem for Morse-Sturm Systems and Applications to semi-Riemannian Geometry
| dc.creator | Giannoni, F. | |
| dc.creator | Masiello, A. | |
| dc.creator | Piccione, P. | |
| dc.creator | Tausk, D. | |
| dc.date | 1999-08-13 | |
| dc.date.accessioned | 2026-07-07T05:30:17Z | |
| dc.date.available | 2026-07-07T05:30:17Z | |
| dc.description | We prove an extension of the Index Theorem for Morse-Sturm systems of the form $-V''+RV=0$, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M,g), we consider a geodesic $γ$ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, satisfied, for instance, if (M,g) is stationary, the theorem gives an equality between the index of the second variation of the action functional f at $γ$ and the sum of the {\em Maslov index} of $γ$ with the index of the metric g on P. Under generic circumstances, the Maslov index of $γ$ is given by an algebraic count of the P-focal points along $γ$. Using the Maslov index, we obtain the global Morse relations for geodesics between two fixed points in a stationary Lorentzian manifold. | |
| dc.description | 32 pages, LaTeX2e, amsart class | |
| dc.identifier | https://arxiv.org/abs/math/9908056 | |
| dc.identifier | http://arxiv.org/abs/math/9908056 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78942 | |
| dc.subject | Differential Geometry | |
| dc.subject | Functional Analysis | |
| dc.title | A Generalized Index Theorem for Morse-Sturm Systems and Applications to semi-Riemannian Geometry | |
| dc.type | text |