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.***

爱因斯坦的广义相对论告诉我们，欧几里得几何只是真实时空几何的近似。这种几何形状受到爱因斯坦场方程的约束，但不像牛顿物理学那样是预先确定的。 我的研究项目涉及类似的情况。它试图确定那些局部看起来像欧几里得空间但长度和角度的概念（在下文中称为度量）允许变化的空间（任意维度）。现在，约束由指定 Ricci 张量形式的方程给出，Ricci 张量是由通过微分选择的度量确定的对象。然而，不同的空间和不同的度量可以服从相同的约束方程。我提案的主要目标是找到所有这些可能性并对其进行分类，并研究它们的几何特性。***总的来说，这个目标过于宽泛——许多研究人员花费了整个职业生涯来研究这个问题的不同方面。我的建议集中在一个相对未经探索的方向。这涉及空间具有最大内部对称性的情况，这是一个通用条件。为了使这个提议更加可行，我计划研究一类空间，其中有一个明显的时间方向，并且允许一些剩余的空间方向在特定的时刻平滑地崩溃。这种结构是广义相对论中研究的许多时空的高维欧几里得类似物。 同样，要研究的约束方程是常Ricci曲率方程或梯度Ricci孤子方程。 后者是前者的修改，当人们考虑一种类似于热扩散的自然过程时，它就会出现，它允许我们逐渐修改度量的初始选择，并希望里奇曲率最终变得恒定。****将采用几何、拓扑、微分方程和数值计算的技术。对称性和数学物理研究中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度量的新空间对于理论物理和其他几何相关学科（无论是纯粹的还是应用的）的研究人员来说都非常重要。 对于孤子方程，新的解尤其重要，因为虽然有许多关于泛型孤子性质的理论结果，但实际上已知的孤子方程的泛型解却很少。 我的提案中开发的方法可能适用于必须构造具有指定曲率属性的几何对象的其他情况。一个例子是正厄米特矩阵集的度量设计，该矩阵区分电气工程中的各种信号类别。***