zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] Conley, Inv. Mat. 73 pp 33– (1983)
[2] Floer, J. Diff. Geom. (1988)
[3] Floer, Pure Appl. Math. 41 (1988)
[4] and , Instantons and Four-Manifolds, Springer, New York, 1984. · doi:10.1007/978-1-4684-0258-2
[5] Gromov, Inv. Math. 82 pp 307– (1985)
[6] and , Geometric Asymptotics, Amer. Math. Soc., Providence, R.I., 1976.
[7] Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. · Zbl 0451.53038
[8] and , Elliptic operators on manifolds with cylindrical ends, preprint, 1986.
[9] Morse Theory, Ann. of Math. St. 51, Princeton University Press, Princeton, N.J., 1963.
[10] Lectures on the H-cobordism theorem, Math. Notes, Princeton University Press, Princeton, N.J., 1969.
[11] Palais, Topology 2 pp 299– (1963)
[12] Intersection de sous-varités Lagrangiennes, fonctionnalles d’action at indice des systèmes Hamiltoniens, preprint, 1986,.
[13] Witten, J. Diff. Geom. 17 pp 661– (1982)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.