Metrics of special curvature

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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.