Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1894347 | Journal of Geometry and Physics | 2009 | 15 Pages |
Abstract
Given a complex structure J on a real (finite or infinite dimensional) Hilbert space H, we study the geometry of the Lagrangian Grassmannian Î(H) of H, i.e. the set of closed linear subspaces LâH such that J(L)=Lâ¥. The complex unitary group U(HJ), consisting of the elements of the orthogonal group of H which are complex linear for the given complex structure, acts transitively on Î(H) and induces a natural linear connection in Î(H). It is shown that any pair of Lagrangian subspaces can be joined by a geodesic of this connection. A Finsler metric can also be introduced, if one regards subspaces L as projections pL (=the orthogonal projection onto L) or symmetries ϵL=2pLâI, namely measuring tangent vectors with the operator norm. We show that for this metric the Hopf-Rinow theorem is valid in Î(H): a geodesic joining a pair of Lagrangian subspaces can be chosen to be of minimal length. A similar result holds for the unitary orbit of a Lagrangian subspace under the action of the k-Schatten unitary group (2â¤kâ¤â), with the Finsler metric given by the k-norm.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematical Physics
Authors
Esteban Andruchow, Gabriel Larotonda,