zbMATH — the first resource for mathematics

The Maslov index for paths. (English) Zbl 0798.58018
In Funct. Anal. Appl. 1, No. 1, 1-13 (1967); translation from Funkts. Anal. Prilozh. 1, No. 1, 1-14 (1967; Zbl 0175.203) V. I. Arnol’d interpreted the Maslov index for a loop of Lagrangian subspaces as an interaction number with an algebraic variety known as the Maslov cycle. Arnol’d’s general position arguments apply equally well to the case of a path of Lagrangian subspaces whose endpoints lie in the complement of the Maslov cycle. In this paper the authors define a Maslov index for any continuous path of Lagrangian subspaces. This index depends on the choice of the Lagrangian subspace used to define the Maslov cycle, is invariant under homotopy with fixed endpoints and is additive for catenations. Moreover the authors use the above index to define a Maslov index for paths of symplectic matrices. This latter index is characterized axiomatically and it leads to a stratification of the symplectic group \(\text{Sp}(2n)\).

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text: DOI