  Although the introduction of generalised and extended geometry has been
motivated mainly by the appearance of dualities upon reductions on tori, it has
until now been unclear how (all) the duality transformations arise from first
principles in extended geometry. A proposal for solving this problem is given
in the framework of double field theory. It is based on a clearly defined
extension of the definition of gauge symmetry by isometries of an underlying
pseudo-Riemannian manifold. The ensuing relation between transformations of
coordinates and fields, which is now derived from first principles, differs
from earlier proposals.
