Metrics of special curvature in differential geometry

这个项目是在微分几何领域，这是一个主要研究光滑物体(称为流形)的长度和曲率性质的数学分支，通常是在更高的维度上。微分几何是现代数学的重要组成部分，与代数、分析、拓扑和理论物理等众多领域有着直接而密切的联系。微分几何的一个中心研究领域是黎曼流形：光滑的物体具有黎曼度量，可以测量长度和角度。黎曼流形的曲率性质被编码为黎曼曲率张量。这以一种复杂和非线性的方式依赖于度规，因此曲率条件涉及到通常难以分析的非线性偏微分方程组。例如爱因斯坦方程和利玛窦孤子方程，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利玛奇流进行几何分析有潜在的联系。黎曼流形上的其他重要方程也涉及辅助数据，如流形上的连通线或向量或旋量丛的截面。这个项目将涉及使用对称性假设来简化这些曲率方程，将它们归结为常微分方程组或低阶偏微分方程组。要考虑的方程包括爱因斯坦方程和厄米流形或Kähler流形上的Strominger系统。爱因斯坦方程在黎曼几何学中具有基本的重要性，但我们对四维以上的解的理解相对较差，这是由于缺乏例子。Strominger系统起源于数学物理，通过研究扭转背景下的超弦，涉及到厄米度规和流形上的连接。尽管它的动机很好，数学家和物理学家都对它进行了研究，但斯特罗明格系统的解的大量例子的缺乏再次意味着，我们对该系统的理解存在明显的差距。该方法将涉及到使用齐次空间和李群的理论来实现对称性约化，以及使用来自非线性动力系统和椭圆型偏微分方程的技术来分析所得到的系统。特别是这种方法论还没有被用来研究Strominger系统，因此代表了一个新的研究方向。该方法在其他相关情况下被证明是非常强大的，因此人们可以希望在项目设置中使用该工具将同样卓有成效。项目应该导致对这些方程的更好的理解，包括解的新的存在性结果，在某些情况下可能的显式解，以及关于唯一性和模的结果。这个项目属于EPSRC的几何和拓扑学研究领域，但也与分析和数学物理有很强的联系。

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.