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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612909
• 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，欧几里德几何只是时空的真实的几何的近似。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预先确定。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的空间（任意维），但允许长度和角度的概念（以下称为度量）变化。这个约束现在由指定里奇张量形式的方程给出，里奇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。我的建议的主要目标是找到和分类所有这些可能性，并研究它们的几何性质。 在这种普遍性中，这个目标太宽泛了许多研究人员花费了他们的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探索的方向。这涉及的情况下，空间有最大的内部对称性，这是一个通用的条件。为了使这个建议更加可行，我打算研究一类空间，在这类空间中，存在一个独特的时间方向，并且对于这类空间，允许某些剩余的空间方向在特定的时刻平滑地坍缩。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模拟。同样，所研究的约束方程是常Ricci曲率方程或梯度Ricci孤子方程。后者是前者的修正，当人们考虑类似于热扩散的自然过程时，它会出现，这允许我们逐渐修改度量的初始选择，希望Ricci曲率最终成为常数。 将采用几何学、拓扑学、微分方程和数值计算等技术。来自对称性和数学物理学研究的概念也将发挥重要作用。 寻找新的空间配备一个或多个度量满足上述约束方程将是重要的研究人员在理论物理和其他几何相关学科，无论是纯粹的和应用。对于孤子方程，新的解决方案是特别重要的，因为虽然有许多理论结果的一般孤子的性质，很少的孤子方程的一般解决方案是实际已知的。在我的建议中开发的方法可能是有用的其他情况下，其中一个必须构造具有指定曲率属性的几何对象。一个例子是在正厄米特矩阵的集合上设计度量，其区分电气工程中的各种信号类。