Geometrization conditions for perfect fluids, scalar fields, and electromagnetic fields

The classical Rainich conditions are a system of geometric conditions, expressed purely in terms of the spacetime metric, which are necessary and sufficient for the metric to define a solution to the Einstein-Maxwell equations with a non-null electromagnetic field. We obtain analogous ``geometrization'' conditions for other matter sources. Specifically, we find geometric conditions which are necessary and sufficient for a metric to define a solution to the Einstein equations with a perfect fluid source, and to define a solution to the Einstein-scalar field equations. These conditions work in any dimension, allow for a cosmological constant, and allow for an arbitrary self-interaction potential in the scalar field case. We also generalize the classical Rainich conditions to include a cosmological constant and we obtain geometrization conditions which are applicable to the case of null electromagnetic fields.