  Generalised diffeomorphisms in double field theory rely on an O(d,d)
structure defined on tangent space. We show that any (pseudo-)Riemannian metric
on the doubled space defines such a structure, in the sense that the
generalised diffeomorphisms defined using such a metric form an algebra,
provided a covariant section condition is fulfilled. Consistent solutions of
the section condition gives further restrictions. The case previously
considered corresponds to a flat metric. The construction makes it possible to
apply double geometry to a larger class of manifolds. Examples of curved
defining metrics are given. We also comment on the role of the defining
geometry for the symmetries of double field theory, and on the continuation of
the present construction to the U-duality setting.
