In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is n(n+1)/2 (where the dimension of V is 2n). It may be identified with the homogeneous space
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U(n)/O(n),where U(n) is the unitary group and O(n) the orthogonal group. Following Vladimir Arnold it is denoted by Λ(n). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V.
A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space V of dimension 2n. It may be identified with the homogeneous space of complex dimension n(n+1)/2
Sp(n)/U(n),where Sp(n) is the compact symplectic group.
Topology
The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem:
In particular, the fundamental group of
For a Lagrangian submanifold M of V, in fact, there is a mapping
M → Λ(n)which classifies its tangent space at each point (cf. Gauss map). The Maslov index is the pullback via this mapping, in
H1(M, Z)of the distinguished generator of
H1(Λ(n), Z).Maslov index
A path of symplectomorphisms of a symplectic vector space may be assigned a Maslov index, named after V. P. Maslov; it will be an integer if the path is a loop, and a half-integer in general.
If this path arises from trivializing the symplectic vector bundle over a periodic orbit of a Hamiltonian vector field on a symplectic manifold or the Reeb vector field on a contact manifold, it is known as the Conley-Zehnder index. It computes the spectral flow of the Cauchy-Riemann-type operators that arise in Floer homology.
It appeared originally in the study of the WKB approximation and appears frequently in the study of quantization and in symplectic geometry and topology. It can be described as above in terms of a Maslov index for linear Lagrangian submanifolds.