The canonical generalised Levi-Civita connection and its curvature

Given a (semi-Riemannian) generalised metric G and a divergence operator div on an exact Courant algebroid E, we geometrically construct a canonical generalised Levi-Civita connection DG, div for these data. In this way we provide a resolution of the problem of non-uniqueness of generalised Levi-Civita connections. Since the generalised Riemann tensor of DG, div is an invariant of the pair ( G, div), we no longer need to discard curvature components which depend on the choice of the generalised connection. As a main result we decompose the generalised Riemann curvature tensor of DG, div in terms of classical (non-generalised) geometric data. Based on this set of master formulas we derive a comprehensive curvature tool-kit for applications in generalised geometry. This includes decompositions for the full generalised Ricci tensor, the generalised Ricci tensor, and three generalised scalar-valued curvature invariants, two of which are new.