The manifold of the Lagrangean subspaces of a symplectic vector space

Let (E, a) be a real 2n-dimensional symplectic vector space with symplectic form a, i.e., a is a nondegenerate skew-symmetric bilinear form on E. Then an n-dimensional subspace λ of E will be called a Lagrangean subspace if alλ = 0 holds. The set Λ(E) of all Lagrangean subspaces of (E, a) has a structure of n(n + l)-dimensional compact connected regular algebraic variety. If we put A\λ): = [μ 6 Λ(E) I dim (λ Π μ) = k} for λ £ Λ(E), then Λ°U) is a cell (i.e., diίϊeomorphic to i r ( n + 1 ) / 2 ) for any λ e Λ(E). Moreover Σ tf): = |J*2>i Λ*W) is an algebraic subvariety of Λ(E), and defines an oriented cycle of codimension one, whose Poincare dual is a generator of H(Λ(E), Z) = Z and defines the Maslov-Arnold index [1], [3], [4]. This index plays an important role in the proof of Morse index theorem in the calculus of variations [4]. In the present note, we shall give a differential geometric characterization of 2 U)> i ^ ? by introducing an appropriate riemannian metric on Λ{E) we shall show that 2 (λ) is the cut locus of some μ e Λ(E) and Λ°(X) is the interior set of μ. In fact, take a basis {eu fά] (1 , and ao((p, q), (p\ q')): = (q, p'y —