Einstein Metrics and Related Geometric Structures

爱因斯坦度量和相关几何结构

• 批准号: RGPIN-2015-04346
• 负责人: Wang, Mckenzie
• 金额: $ 1.46万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677142
• 关键词: Einstein; Metrics; Related; Geometric; Structures

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.***

爱因斯坦的一般相对论告诉我们，欧几里得的几何形状只是时空几何的近似值。这种几何形状受到爱因斯坦的场方程的约束，但否则没有像牛顿物理学那样预先确定。我的研究计划处理类似的情况。它试图确定（任意维度）的那些空间，这些空间在局部看起来像欧几里得空间，但允许将长度和角度的概念（称为以下指标）变化。现在，约束是由指定RICCI张量形式的等式给出的，Ricci张量的形式是由通过分化选择的度量确定的对象。但是，不同的空间和不同的指标可以遵守相同的约束方程。我的提议的主要目的是找到和分类所有这些可能性，并研究其几何特性。我的建议集中于相对出乎意料的方向。这涉及空间具有最大内部对称性的情况，这是一种通用条件。为了使建议更加可行，我计划检查一类空间，在这些空间中有一个杰出的时间方向，并且在特定的瞬间，允许其余的某些空间方向崩溃。顺利。这种结构是许多在一般相对论中研究的空间时间的较高维度的欧几里得类似物。同样，要研究的约束方程是恒定的RICCI曲率方程或梯度RICCI Soliton方程。后者是对前者的修改，并且当人们认为自然过程类似于热量扩散时，它使我们能够逐渐修改指标的初始选择，希望RICCI曲率最终变为恒定。****从几何，拓扑，拓扑，差异方程式和数字计算中使用的技术。对称和数学物理学的研究的概念也将发挥重要作用。****寻找配备一个或多个满足上述约束方程的新空间对理论物理学和其他与地球相关的科论的研究人员都很重要，包括纯净和应用。对于孤子方程，新的解决方案尤其重要，因为尽管关于通用固体的性质有许多理论结果，但实际上很少知道孤子方程的通用溶液。在我的建议中开发的方法可能对其他情况必须构建具有指定曲率特性的几何对象的其他情况很有用。一个例子是在阳性式遗产材料的集合上设计指标，该物品区分了电气工程中的各种信号类别。***