Summary: Instituto Superior T´ecnico
Departamento de Matem´atica
SYMPLECTIC GEOMETRY - 2nd
Problem Set 1
Due date: March 10
1. Let (V, ) be a symplectic vector space. Show that any codimension 1 subspace S V
2. (a) Let E be a real vector space. Show that E E
has a canonical symplectic
structure 0 determined by 0(u , v ) = (u) - (v).
(b) Let L be a Lagrangian subspace of a symplectic vector space (V, ). Show that
there exists a symplectic linear map : (V, ) (L L
, 0) such that (u) =
u 0 , u L.
3. Let (V, ) be a symplectic vector space, J J (V, ) a complex structure compatible
with and gJ (·, ·) = (·, J·) the associated inner product. Show that a subspace
L V is Lagrangian iff J(L) = L
orthogonal complement of L with respect to gJ .
Conclude that L is Lagrangian iff J(L) is Lagrangian.