Index theorems for symplectic systems

✍ Scribed by P. Piccione; D.V. Tausk

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 839 KB
• Volume: 47
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(01)00423-0

✦ Synopsis

The Jacobi equation along a geodesic (\gamma) in a semi-Riemannian manifold (\left(M^{n}, g\right)) produces, by a parallel trivialization of the tangent bundle (T M) along (\gamma), a Morse-Sturm equation in (\mathbb{R}^{n}). More generally, the linearized Hamilton equation along a solution (\Gamma) of a Hamiltonian vector field (\vec{H}) in a symplectic manifold (\left(\mathcal{M}^{2 n}, \omega\right)) produces a first order linear differential system in (\mathbb{R}^{n} \oplus \mathbb{R}^{n *}) whose flow preserves the canonical symplectic form; such systems are called symplectic differential systems. By "index theorems" for symplectic systems we mean those results that relate two or more of the following objects: (1) the conjugate (or focal) points of the system, (2) the index or the co-index of (suitable restrictions of) the so called index form associated to the system, (3) the spectrum of the second order linear differential operator associated to the system. In this paper we present a collection of index theorems that were proven recently (References [5], [6], [8], [9], [10], [11]).