  In recent development of double field theory, as for the description of the
massless sector of closed strings, the spacetime dimension is formally doubled,
i.e. from D to D+D, and the T-duality is realized manifestly as a global O(D,D)
rotation. In this paper, we conceive a differential geometry characterized by a
O(D,D) symmetric projection, as the underlying mathematical structure of double
field theory. We introduce a differential operator compatible with the
projection, which, contracted with the projection, can be covariantized and may
replace the ordinary derivatives in the generalized Lie derivative that
generates the gauge symmetry of double field theory. We construct various gauge
covariant tensors which include a scalar and a tensor carrying two O(D,D)
vector indices.
