Metrics of special curvature

这个项目属于微分几何领域。这里研究的对象是带有度规的流形，一个允许我们定义和研究诸如长度、体积和曲率等概念的数学对象。微分几何中许多有趣的问题都可以用包含曲率的方程或不等式来表示。利用表征理论，曲率可以分解为具有不同性质的块；Weyl张量（共形不变部分），标量曲率（高斯曲率的高维模拟）和里奇曲率，这是一个与度规相同类型的张量。涉及曲率的最重要的方程之一是爱因斯坦方程，它表达了里奇曲率应该与度规成比例的约束。这个方程实际上是一个非常复杂的非线性偏微分方程组。在2维或3维中，它很容易理解，等同于常曲率条件。在四维中，存在一些拓扑障碍，即已知某些流形不承认爱因斯坦度量（求解爱因斯坦方程的度量）。在更高的维度中，这种情况就不太好理解了，因为我们不知道爱因斯坦度规存在的任何拓扑障碍，但与此同时，我们也没有真正通用的技术来显示存在性。本项目的目的是通过使用对称假设来减少方程的复杂性，获得爱因斯坦度量的新例子和相关方程的解。一种方法是通过施加所谓的同质性1条件，也就是说，要求一个群保持度规并且作用于实余维为1的一般轨道。这仍然导致高度非平凡方程（非线性动力系统），但在某些情况下，有可能分析流动的轨迹并推断出具有良好全局性质的爱因斯坦度量的存在。我们还想研究高齐次度量的问题，其中得到的方程实际上是偏微分方程，尽管变量数量比一般系统少。这是一个鲜为人知的领域。在特殊情况下，甚至可以利用方程的可积性结构得到爱因斯坦度规的显式表达式。我们主要关注黎曼度规的情况，其中度规是正定的。然而，这些技术中的一些在洛伦兹的情况下也可能是有价值的。它还打算研究爱因斯坦条件的推广，如里奇孤子方程或准爱因斯坦方程。里奇孤子对理解里奇流奇点（通过取爆破极限）也很有兴趣，里奇流是一个强烈电流活动的区域。我们还想在爱因斯坦的背景下研究各种方程（例如狄拉克方程），特别是在我们对度规有相当明确描述的情况下。该项目属于EPSRC几何和拓扑研究领域，也与数学物理有很强的联系。

This project is in the field of differential geometry. The objects of study here are manifolds with a metric, a mathematical object that allows us to define and study notions such as length, volume and curvature. Many interesting problems in differential geometry can be expressed in terms of equations or inequalities involving the curvature. Using representation theory, the curvature may be decomposed into pieces with different properties; the Weyl tensor (the conformally invariant part), the scalar curvature (a higher dimensional analogue of the Gaussian curvature) and the Ricci curvature, which is a tensor of the same type as the metric.One of the most important equations involving the curvature is the Einstein equation, which expresses the constraint that the Ricci curvature should be proportional to the metric. This equation is actually a very complicated system of nonlinear partial differential equations. In dimensions 2 or 3 it is well understood, being equivalent to the constant curvature condition. In dimension 4, there are some topological obstructions, that is, some manifolds are known not to admit Einstein metrics (metrics which solve the Einstein equation). In higher dimensions the situation is much less well understood, as we do not know any topological obstructions to the existence of an Einstein metric, but at the same time we do not have really general techniques for showing existence.The aim of this project is to obtain new examples of Einstein metrics and solutions to related equations by using symmetry assumptions to reduce the complexity of the equations. One way to do this is by imposing the so-called cohomogeneity one condition, that is, requiring that a group act preserving the metric and acting with generic orbits of real codimension one. This still leads to highly nontrivial equations (nonlinear dynamical systems), but in some cases it it possible to analyse the trajectories of the flow and deduce the existence of Einstein metrics with good global properties. We would also like to investigate the question of higher cohomogeneity metrics, where the resulting equations are actually partial differential equations, though in a smaller number of variables than the general system. This is an area where comparatively little is known. In special cases it may even be possible to obtain explicit expressions for the Einstein metric using an integrability structure for the equations. Our primary focus is on the case of Riemannian metrics, where the metric is positive definite. However, some of these techniques may also be valuable in the Lorentzian case.It is also intended to study generalisations of the Einstein condition like the Ricci soliton equation or the quasi-Einstein equation. Ricci solitons are also of interest in understanding (via taking blowup limits) singularities of the Ricci flow, an area of intense current activity. We would also like to study various equations (for example the Dirac equation) in an Einstein background, particularly in cases where we have a fairly explicit description of the metric.This project falls within the EPSRC Geometry and Topology research area, and also has strong links with Mathematical Physics.