  In this paper we develop a relative version of T-duality in generalized
complex geometry which we propose as a manifestation of mirror symmetry. Let M
be an n-dimensional smooth real manifold, V a rank n real vector bundle on M,
and nabla a flat connection on V. We define the notion of a nabla-semi-flat
generalized complex structure on the total space of V. We show that there is an
explicit bijective correspondence between nabla-semi-flat generalized complex
structures on the total space of V and nabla(dual)-semi-flat generalized
complex structures on the total space of the dual of V. Similarly we define
semi-flat generalized complex structures on real n-torus bundles with section
over an n-dimensional base and establish a similar bijective correspondence
between semi-flat generalized complex structures on pair of dual torus bundles.
Along the way, we give methods of constructing generalized complex structures
on the total spaces of vector bundles and torus bundles with sections. We also
show that semi-flat generalized complex structures give rise to a pair of
transverse Dirac structures on the base manifold. We give interpretations of
these results in terms of relationships between the cohomology of torus bundles
and their duals. We also study the ways in which our results generalize some
well established aspects of mirror symmetry as well as some recent proposals
relating generalized complex geometry to string theory.
