Einstein manifolds and related structures

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This proposal deals with the existence, moduli, and geometrical properties of solutions of a number of geometric differential equations analogous to Einstein's field equation in General Relativity. These equations make sense for spaces of all dimensions greater than two on which there is a notion of lengths and angles between tangent vectors. The equations under study are of fundamental interest in Differential Geometry because they express either the constancy of a natural notion of curvature or self-similarity for solutions to a natural flow on the space of Riemannian metrics on a given space. These equations also arise naturally in supergravity theories, in which one seeks a unification of General Relativity with theories explaining the other forces in nature.