  The Principal Chiral Model (PCM) defined on the group manifold of SU(2) is
here investigated with the aim of getting a further deepening of its relation
with Generalized and Doubled Geometry. A one-parameter family of equivalent
Hamiltonian descriptions is introduced, and cast into the form of Born
geometries. Then O(3,3) duality transformations of the target phase space are
performed and we show that the resulting dual models are defined on the group
SB(2,C) which is the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of
the Drinfel'd double SL(2, C). Moreover, starting from the Lagrangian approach,
a new kind of duality is found between the SU(2) PCM and the natural one
defined on SB(2,C) which is not an isometry of the target phase space. A parent
action with doubled degrees of freedom and configuration space SL(2, C) is then
defined that reduces to either one of the dually related models, once suitable
constraints are implemented.
