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Instituto Superior Tecnico Departamento de Matematica
 

Summary: Instituto Superior T´ecnico
Departamento de Matem´atica
SYMPLECTIC GEOMETRY - 2nd
Semester 2010/11
Problem Set 4
Due date: May 9
1. Let (M, J) be an almost complex manifold and f : M C a smooth function such
that 0 C is a regular value. Show that if (¯f)p = 0 at any point p N := f-1
(0),
then N is a complex submanifold of (M, J).
2. Let H : Rn
Sn symmetric n × n matrices, be a smooth map such that H(x) is
non-singular for all x Rn
. Consider the almost complex structure J defined on R2n
by
J(x,y) =
0 -H(x)-1
H(x) 0
a) Show that J is integrable if and only if there exists a smooth function h : Rn
R

  

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

 

Collections: Mathematics