A geometric formulation of exceptional field theory

We formulate the full bosonic SL(5) exceptional field theory in a coordinate invariant manner. Thereby we interpret the 10-dimensional extended space as a manifold with SL(5) x R (vertical bar) -structure. We show that the algebra of generalised diffeomorphisms closes subject to a set of closure constraints which are reminiscent of the quadratic and linear constraints of maximal seven-dimensional gauged supergravities, as well as the section condition. We construct an action for the full bosonic SL(5) exceptional field theory, even when the SL(5) x R+ -structure is not locally flat.