We reformulate eleven-dimensional supergravity, including fermions, in terms
of generalised geometry, for spacetimes that are warped products of Minkowski space with
a d-dimensional manifold M with d ≤ 7. The reformulation has an Ed(d) × R
+ structure
group and it has a local H˜
d symmetry, where H˜
d is the double cover of the maximally
compact subgroup of Ed(d)
. The bosonic degrees for freedom unify into a generalised
metric, and, defining the generalised analogue D of the Levi-Civita connection, one finds
that the corresponding equations of motion are the vanishing of the generalised Ricci
tensor. To leading order, we show that the fermionic equations of motion, action and
supersymmetry variations can all be written in terms of D. Although we will not give
the detailed decompositions, this reformulation is equally applicable to type IIA or IIB
supergravity restricted to a (d−1)-dimensional manifold. For completeness we give explicit
expressions in terms of H˜
4 = Spin(5) and H˜
7 = SU(8) representations for d = 4 and d = 7.