Mirror symmetry and generalized complex manifolds. Part I. The transform on vector bundles, spinors, and branes

In this paper we begin the development of 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 ∇ a flat connection on V. We define the notion of a ∇-semi-flat generalized almost complex structure on the total space of V. We show that there is an explicit bijective correspondence between ∇-semi-flat generalized almost complex structures on the total space of V and ∇^v-semi-flat generalized almost complex structures on the total space of ∇^v. We show that semi-flat generalized complex structures give rise to a pair of transverse Dirac structures on the base manifold. We also study the ways in which our results generalize some aspects of T-duality such as the Buscher rules. We show explicitly how spinors are transformed and discuss the induces correspondence on branes under certain conditions.