The initial value problem for the Einstein equations in Hitchin’s generalised geometry

We discuss the initial value problem for the Einstein equations in Hitchin’s generalised geometry over an $$n+1$$ n + 1 -dimensional manifold. Specifically, we establish the existence of a maximal globally hyperbolic development (MGHD). Locally, the generalised Einstein equations are a geometrisation of the equations of motion for the bosonic NS–NS sector in type-II ten-dimensional supergravity. The local dynamical fields are a spacetime metric, a scalar field known as the dilaton potential, and a two-form known as the B -field. We consider the case of closed divergence, meaning that the local dilaton potentials define a closed one-form. After a transformation depending on the choice of dilaton potential, the local system defines an Einstein-matter system. We establish well-posedness of the initial value problem for the local system by demonstrating that (in the right gauge) the PDE system is quasi-linear second-order hyperbolic. In a non-trivial patching procedure, we obtain well-posedness of the initial value problem for the generalised Einstein equations. Thus, we demonstrate the viability of the generalised Einstein equations—a geometrically and physically motivated system—as a cosmological model.