Metrics of special curvature

这个项目是在微分几何领域。这里的研究对象是带有度规的流形，一个数学对象，允许我们定义和研究诸如长度、体积和曲率等概念。微分几何中许多有趣的问题都可以用与曲率有关的方程或不等式来表示。利用表象理论，曲率可以分解成具有不同性质的部分：Weyl张量(共形不变部分)、标量曲率(高斯曲率的高维模拟)和Ricci曲率，后者是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Ricci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，这是很容易理解的，它等同于常曲率条件。在维度4中，存在一些拓扑障碍，即已知一些流形不允许使用爱因斯坦度量(求解爱因斯坦方程的度量)。在更高的维度中，这种情况不太清楚，因为我们不知道爱因斯坦度规存在的任何拓扑障碍，但同时我们也没有真正通用的技术来证明爱因斯坦度规的存在。这个项目的目的是通过使用对称假设来降低方程的复杂性，获得爱因斯坦度规的新例子和相关方程的解。要做到这一点，一种方法是施加所谓的上齐性一个条件，即要求一个群保持度规并以实余维一的一般轨道作用。这仍然导致了高度非平凡的方程(非线性动力系统)，但在某些情况下，分析流的轨迹并推导出具有良好整体性质的爱因斯坦度量的存在是可能的。我们还想研究更高的同齐性度量的问题，其中得到的方程实际上是偏微分方程组，尽管变量数量比一般系统少。这是一个相对知之甚少的领域。在特殊情况下，甚至可以使用方程的可积性结构来获得爱因斯坦度规的显式表达式。我们主要关注黎曼度量的情况，其中度量是正定的。然而，这些技术中的一些在洛伦兹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，如利玛窦孤子方程或准爱因斯坦方程。里奇孤子也对理解(通过采取爆发极限)里奇流的奇点感兴趣，这是一个强烈的洋流活动区。我们还想研究爱因斯坦背景下的各种方程(例如狄拉克方程)，特别是在我们对度规有相当明确的描述的情况下。这个项目属于EPSRC几何和拓扑学研究领域，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.