Einstein Metrics and Related Geometric Structures

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Einstein’s theory of General Relativity tells us that Euclidean geometry is only an approximation to the real geometry of space-time. This geometry is constrained by Einstein’s field equation, but is otherwise not predetermined as in Newtonian physics. My research program deals with an analogous situation. It seeks to determine those spaces (of arbitrary dimension) which locally look like Euclidean space but for which the notion of lengths and angles (referred to as the metric in the following) is allowed to vary. The constraint is now given by equations which specify the form of the Ricci tensor, an object that is determined by the metric chosen via differentiation. However, different spaces and different metrics can obey the same constraint equation. The main objective of my proposal is to find and classify all these possibilities, and study their geometric properties. In this generality, this objective is too broad—many researchers spend their entire careers studying different aspects of this problem. My proposal focusses on a relatively unexplored direction. This deals with the case in which the spaces have maximal internal symmetry, which is a generic condition. To make the proposal more feasible, I plan to examine a class of spaces in which there is a distinguished time direction and for which some of the remaining spatial directions are allowed, at a specific instant in time, to collapse smoothly. This structure is a higher-dimensional Euclidean analogue of many space-times studied in General Relativity. As well, the constraint equation to be studied is either the constant Ricci curvature equation or the gradient Ricci soliton equation. The latter is a modification of the former and it arises when one considers a natural process similar to the diffusion of heat that allows us to modify an initial choice of metric gradually with the hope that the Ricci curvature eventually becomes constant. Techniques from geometry, topology, differential equations, and numerical computation will be employed. Concepts from the study of symmetry and mathematical physics will also play an important role. Finding new spaces equipped with one or more metric satisfying the above constraint equations will be important to researchers in theoretical physics and other geometry-related disciplines, both pure and applied. For the soliton equation, new solutions are especially significant because while there are many theoretical results about properties of generic solitons, very few generic solutions of the soliton equation are actually known. Methods developed in my proposal may be useful for other situations in which one has to construct geometric objects with specified curvature properties. An example is the design of metrics on the set of positive Hermitian matrices which distinguish between various signal classes in electrical engineering.