Metrics of special curvature in differential geometry

微分几何中特殊曲率的度量

• 批准号: 2271784
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2271784
• 关键词: Metrics; special; curvature; differential; geometry

This project is in the area of differential geometry, the branch of mathematics primarily concerned with the length and curvature properties of smooth objects (called manifolds), often in higher dimensions. Differential geometry is a key part of modern mathematics with direct and close links to numerous areas including algebra, analysis, topology and theoretical physics.A central area of study in differential geometry is Riemannian manifolds: smooth objects endowed with a Riemannian metric, which allows one to measure lengths and angles. The curvature properties of a Riemannian manifold are encoded in the Riemann curvature tensor. This depends in a complicated and nonlinear way on the metric, and so curvature conditions involve nonlinear partial differential equations which are usually difficult to analyse. Examples include the Einstein equations and the Ricci soliton equations, both of which are key equations in differential geometry because of their potential links to topology and mathematical physics, as well as to geometric analysis through the Ricci flow. Other important equations on Riemannian manifolds also involve auxiliary data such as connexions or sections of vector or spinor bundles on the manifold.This project will involve the use of symmetry assumptions to simplify such curvature equations by reducing them to systems of ordinary differential equations or partial differential equations of lower order. Equations to be considered include the Einstein equation and also the Strominger system on a Hermitian or Kähler manifold. The Einstein equation is of fundamental importance in Riemannian geometry, yet our understanding of solutions is relatively poor in dimensions above four, and this is due in poor to a lack of examples. The Strominger system originated in mathematical physics through the study of superstrings in torsion backgrounds, and involves a Hermitian metric and a connexion on the manifold. Although it is well-motivated and studied both by mathematicians and physicists, the paucity of non-trivial examples of solutions to the Strominger system again means that there is a clear gap in our understanding of the system. The methodology will involve using the theory of homogeneous spaces and Lie groups to implement the symmetry reduction, as well as using techniques from nonlinear dynamical systems and elliptic partial differential equations to analyse the resulting systems. This methodology, in particular, has not been used to study the Strominger system, and so represents a novel research direction. The methodology has proved to be very powerful in other related situations, and so one can hope that it will be similarly fruitful to utilize this tool in the setting of the project.The project should lead to a better understanding of these equations, including new existence results for solutions, possible explicit solutions in some cases, and results on uniqueness and moduli. The results will be of relevance not only to pure mathematicians working in differential geometry but also to mathematical physicists.This project falls within EPSRC research area Geometry and Topology, but also has strong links to Analysis and Mathematical Physics.

这个项目是在微分几何领域，数学的分支主要关注光滑物体（称为流形）的长度和曲率特性，通常在更高的维度。微分几何是现代数学的重要组成部分，与代数、分析、拓扑学和理论物理等众多领域有着直接而密切的联系。微分几何研究的一个中心领域是黎曼流形：具有黎曼度量的光滑物体，它允许人们测量长度和角度。黎曼流形的曲率性质被编码在黎曼曲率张量中。这取决于在一个复杂的和非线性的方式对度量，因此曲率条件涉及非线性偏微分方程，这通常是难以分析。例子包括爱因斯坦方程和里奇孤子方程，这两个方程都是微分几何中的关键方程，因为它们与拓扑学和数学物理以及通过里奇流的几何分析有潜在的联系。黎曼流形上的其他重要方程亦涉及辅助数据，例如流形上向量或旋量丛的连接或截面。本计划将利用对称性假设，把这类曲率方程简化为低阶常微分方程或偏微分方程。要考虑的方程包括爱因斯坦方程和厄米或凯勒流形上的斯特罗明格系统。爱因斯坦方程在黎曼几何中具有根本的重要性，但我们对四维以上解的理解相对较差，这是由于缺乏例子。Strominger系统起源于数学物理学，通过对挠率背景下超弦的研究，并涉及到厄米特度量和流形上的连接。虽然它是很好的动机和研究都由数学家和物理学家，缺乏非平凡的例子解决斯特罗明格系统再次意味着有一个明显的差距，在我们的理解系统。该方法将涉及使用齐次空间和李群的理论来实现对称性简化，以及使用非线性动力系统和椭圆偏微分方程的技术来分析所产生的系统。特别是，这种方法还没有被用来研究Strominger系统，因此代表了一个新的研究方向。该方法已被证明是非常强大的在其他相关的情况下，因此，人们可以希望，这将是同样富有成效的，利用这个工具在设置的project.The项目应该导致更好地理解这些方程，包括新的存在性结果的解决方案，在某些情况下可能的显式解决方案，并在唯一性和模的结果。结果将不仅是相关的纯数学家在微分几何工作，而且数学物理学家。这个项目属于EPSRC研究领域的几何和拓扑福尔斯，但也有很强的联系，分析和数学物理。