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A relative Morse index for the symplectic action. (English) Zbl 0633.58009
The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function \({\mathfrak a}\) on the loop space of a manifold. In this paper, we define for any pair of critical points of \({\mathfrak a}^ a \)relative Morse index which corresponds to the difference of the two Morse indices in finite dimensions. It is based on the spectral flow of the Hessian of \({\mathfrak a}\) and can be identified with a topological invariant recently defined by Viterbo, and with the dimension of the space of trajectories between the two critical points.
Reviewer: A.Floer

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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