We reformulate type II supergravity and dimensional restrictions of eleven-dimensional
supergravity as generalised geometrical analogues of Einstein
gravity. The bosonic symmetries are generated by generalised vectors, while
the bosonic fields are unified into a generalised metric. The generalised tangent
space features a natural action of the relevant (continuous) duality
group. Also, the analogues of orthonormal frames for the generalised metric
are related by the well-known enhanced local symmetry groups, which
provide the analogue of the local Lorentz symmetry in general relativity.
Generalised connections and torsion feature prominently in the construction,
and we show that the analogue of the Levi-Civita connection is not
uniquely determined by metric compatibility and vanishing torsion. However,
connections of this type can be used to extract the derivative operators
which appear in the supergravity equations, and the undetermined pieces
of the connection cancel out from these, leaving the required unique expressions.
We find that the bosonic action and equations of motion can be interpreted
as generalised curvatures, while the derivative operators appearing
in the supersymmetry variations and equations of motion for the fermions
become very simple expressions in terms of the generalised connection.
In the final chapter, the construction is used to reformulate supersymmetric
flux backgrounds as torsion-free generalised G-structures. This is
the direct analogue of the special holonomy condition which arises for supersymmetric
backgrounds without
flux in ordinary Riemannian geometry.