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We present a geometric field theory in which the action and field equation are constructed from a vector field and its covariant derivative and have full general covariance in a higher-dimensional spacetime. The field equation is the simplest possible generalisation of the Poisson equation for gravity consistent with general covariance and the equivalence principle. It contains the Ricci tensor and metric acting as operators on the vector field. If the symmetrised covariant derivative is diagonalisable across a neighbourhood under real changes of coordinate basis, spacetime coincides with a product manifold. The dimensionalities of the factor spaces are determined by its eigenvalues and hence by its algebraic invariants. Tensors decompose into multiplets which have both Lorentz and internal symmetry indices. The vector field decomposes into conformal Killing vector fields for each of the factor spaces.The field equation has a `classical vacuum' solution which is a Cartesian product of factor spaces. The factor spaces are all Einstein manifolds or two-dimensional Riemannian manifolds. All have a Ricci curvature of roughly the same order of magnitude, or are Ricci-flat. A worked example is provided in six dimensions.Away from this classical vacuum, connection components in appropriate coordinates include $SO(N)$ gauge fields. The Riemann tensor includes their field strength. Unitary gauge symmetries act indirectly on tensor fields and some or all of the unitary gauge fields are found amongst the $SO(N)$ gauge fields. Symmetry restoration occurs at the zero-curvature `decompactification limit', in which all dimensions appear on the same footing.