Summary: Instituto Superior T´ecnico
Departamento de Matem´atica
SYMPLECTIC GEOMETRY - 2nd
Semester 2010/11
Problem Set 5
Due date: May 23
1. Let F : Rn
Sn symmetric n × n matrices, a smooth function such that F(u) is
non-singular for all u Rn
. Consider a 2-form defined in R2n
= {(u, v) : u, v Rn
}
by the skew-symmetric matrix
(u,v) =
0 F(u)
-F(u) 0
a) Show that d = 0 if and only if there exists a smooth function f : Rn
R,
f = f(u), such that
F = Hessu(f)

Source: Abreu, Miguel - Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa

Collections: Mathematics