  We introduce a geometrical framework for double field theory in which
generalized Riemann and torsion tensors are defined without reference to a
particular basis. This invariant geometry provides a unifying framework for the
frame-like and metric-like formulations developed before. We discuss the
relation to generalized geometry and give an `index-free' proof of the
algebraic Bianchi identity. Finally, we analyze to what extent the generalized
Riemann tensor encodes the curvatures of Riemannian geometry. We show that it
contains the conventional Ricci tensor and scalar curvature but not the full
Riemann tensor, suggesting the possibility of a further extension of this
framework.
