Conformal Geometry of semi-Riemannian Manifolds

几十年来，广义相对论一直在研究共形对称性，微分几何和数学物理中的许多概念都是共形不变的。在四维空间中，具有非平凡共形对称性的真空时空是由布林克曼首先引入的pp-度规。我们计划引入pp-度量的一个扩展（我们称之为广义Brinkmann空间）并在我们的项目的两个部分中研究以下问题：1.广义Brinkmann空间的共形对称性为了适当地推广pp-度量，包括具有类光Killing场的纯辐射度量，我们研究了这些度量类内的共形对称性和共形变换.广义Brinkmann空间的自旋几何我们计划分别研究pp-度量的共形紧化。广义Brinkmann空间允许扭量旋量为零。另一个研究方向是广义Killing旋量方程的解和超对称的构造。

Conformal Symmetries have been studied in General Relativity during many decades, and many concepts in differential geometry and mathematical physics are conformally invariant. In dimension four vacuum spacetimes with non-trivial conformal symmetries are pp-metrics first introduced by Brinkmann. We plan to introduce an extension of pp-metrics (which we call generalized Brinkmann spaces) and study the following questions in the two parts of our project:1. Conformal Symmetries of Generalized Brinkmann Spaces For an appropriate generalization of pp-metrics including pure radiation metrics with a lightlike Killing field we study conformal symmetries and conformal transformations inside the class of these metrics.2. Spin-Geometry of Generalized Brinkmann Spaces We plan to study conformal compactifications of pp-metrics resp. generalized Brinkmann spaces admitting twistor spinors with zeros. Another line of investigation will be concerned with solutions of generalized Killing spinor equations and the construction of supersymmetries.