Lagrangian Grassmann manifold Λ(2)

Abstract

Based on the relationship between symplectic group Sp(2) and Λ(2), we provide an intuitive explanation (model) of the 3-dimensional Lagrangian Grassmann manifold Λ(2), the singular cycles of Λ(2), and the special Lagrangian Grassmann manifold SΛ(2). Under this model, we give a formula of the rotation paths defined by Arnold.

Key words： Lagrangian Grassmann manifold; Lagrangian plane; geometric representation; singular cycle

References

1 Arnold V I. Characteristic class entering in quantization conditions. Funct Anal Appl, 1967, 1(1): 1–13

2 Cappel S, Lee R, Miller E. On the Maslov index. Comm Pure Appl Math, 1994, 47: 121–186

3 Gelfand I M, V.B. Lidskii V B. On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients. Uspekhi Mat Nauk, 1955, 10: 3–40 (in Russian); Amer Math Soc Transl, 1958, 8(2): 143–181

4 Long Y. The structure of the singular symplectic matrix set. Sci China Ser A, 1991, 34: 897–907

5 Long Y.Index Theory for Symplectic Paths with Applications. Progr Math, Vol 207. Basel: Birkhäuser, 2002

6 Long Y, Zehnder E. Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In: Albeverio S, ed. Stoc Proc Phys and Geom. Singapore: World Scientific, 1990, 528–563

7 Maslov V P. Theory of Pertubations and Asymptotic Methods. Moscow: Moskov Gos Univ, 1965 (in Russian)

8 Maslov V P, Fedoriuk M V. Semi-Classical Approximation in Quantum Mechanics. Mathematical Physics and Applied Mathematics, Vol 7. Dordrecht: D Reidel Publ Company, 1981

9 Robbin J, Salamon D. The Maslov index for paths. Topology, 1993, 32: 827–844

10 Salamon D, E. Zehnder E. Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Comm Pure Appl Math, 1992, 45: 1303–1360