Bi‐Hamiltonian systems and Maslov indices

TLDR: It was shown in this paper that any Lagrangian distribution on a 2n-dimensional symplectic manifold is equivalent to a principal O(n) bundle obtained as a reduction of the frame bundle.

Abstract: It is shown that any Lagrangian distribution on a 2n‐dimensional symplectic manifold Γ is equivalent to a principal O(n) bundle obtained as a reduction of the frame bundle. Bisymplectic manifolds with Nijenhuis recursion operators are studied and it is shown that the set of all bi‐Lagrangian distributions is a trivial bundle the structure group being the homogeneous space U(1)n/O(1)n. Various formulas for Maslov indices of closed curves are given including one using only data from the recursion operator.