 
Summary: Riemannian geometry of the twistor
space of a symplectic manifold
R. Albuquerque
rpa@uevora.pt
Departamento de Matem´atica, Universidade de ´Evora
´Evora, Portugal
September 2004
0.1 The metric
In this short communication we show some computations about the curvature of a metric
defined on the twistor space of a symplectic manifold.
Let (M, , ) be a symplectic manifold endowed with a symplectic connection (that
is = 0, T = 0). Recall that the twistor space
Z = j End TxM : x M, j2
= 1, type (1,1) for j and ( , j ) > 0
is a bundle : Z M, with obvious projection, together with an almost complex
structure J defined as follows. First, notice the connection induces a splitting
0  V  TZ = H V
d

TM  0
