Einstein manifolds and related structures

本文讨论了一些类似广义相对论中爱因斯坦场方程的几何微分方程的解的存在性、模量和几何性质。这些方程适用于所有大于2维的空间，在这些空间中存在切向量之间的长度和角度的概念。所研究的方程在微分几何中具有基本的兴趣，因为它们要么表示曲率自然概念的常数，要么表示给定空间上黎曼度量空间上自然流解的自相似性。这些方程也自然地出现在超引力理论中，在超引力理论中，人们寻求广义相对论与解释自然界其他力的理论的统一。

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.