The manifold of the Lagrangean subspaces of a symplectic vector space

Abstract

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 — </?, q'y. We put ^ 0: = {(p,0)|p e R } and μ0: = {(0, q)\qe R } which are of course Lagrangean subspaces. Then the (real representation of) unitary group U(n) naturally acts on Λ(ή): = A(R) transitively, and its isotropic subgroup at Λo is given by O(n). Thus Λ(ή) is diffeomorphic to U(n)/O(n). Now M = U(n)/O(n) has a structure of a compact symmetric space whose riemannian structure comes from the Killing form of the Lie algebra of U{ή). In the present note we shall determine the cut locus and the first conjugate locus of a point of M, from which we may prove the assertion mentioned above. For compact simply connected symmetric spaces, it is known that the cut locus and the first conjugate locus of any point coincide with each other (see [2]). Note that 7Γχ(M) = Z for our manifold M = U(n)/O(n). Finally we shall determine all closed geodesies of M and calculate their intersection number with the oriented cycle U**iΛ*(n).