Generalizations of (hyper-)Kähler geometry and geometric flows related to Ricci-flat Riemannianmanifolds

我的研究项目与所谓的伯杰列表中出现的不同类型的几何有关。这个列表对非对称单连通黎曼流形的所有可能的完整群进行了分类。这些几何中的几个自动是Ricci平坦的，并且具有平行的旋量场。这两个性质也使它们对物理学家非常有吸引力，在物理学中，它们在高维超对称理论的紧凑中以“内部空间”的形式出现。更普遍的是，物理学家使用具有几何结构的内部空间，具有所谓的“特征”联系。我的研究项目大致可以分为两个部分。鉴于第一部分是关于Kähler和Hyperkähler几何的某些推广，第二部分考察了与Berger列表中的Ricci平坦几何有关的不同几何流动方程。第一部分更准确地讨论了SKT结构和复辛结构，SKT结构是具有特征联系的Kähler结构的推广，复辛结构是Hyperkähler结构的推广。在这两种情况下，目标都是与其他数学家合作，在不同类别的幂零和可解李群上，在左不变的背景下对这些结构进行分类。请注意，在SKT的案例中，我们将使用剪切结构进行分类，这种结构是以前与我的合作者在项目的这一部分中开发的。项目的第二部分是关于旋量流、修正的拉普拉斯同流以及希钦流和群收缩之间的相互作用。此外，所有这些子项目都是与来自伦敦和欧洲其他地方的不同数学家合作的。前两个流的临界点分别是来自Berger列表的任意或七维Ricci平坦黎曼流形(带有附加数据)。我们的目的是研究这些相对较新的流在齐次环境和其他“对称”情况下的例子和性质，以便更好地了解这些流的一般情况。希钦流是六维几何流，它产生G2中具有完整的七维Ricci平坦黎曼流形(在Berger列表中是一个“例外”情况)。通过所谓的群压缩，物理学家从六维李群上的希钦流的左不变解出发，构造了六维李群上希钦流的左不变解。我们现在的目的是在上述情况下详细地理解群收缩与希钦流之间的相互作用，并系统地研究S^3×S^3和其他李群上的这种相互作用。

My research project is related to the different kinds of geometries occuring in the so-called Berger list. This list classifies all possible holonomy groups of non-symmetric simply-connected Riemannian manifolds. Several of these geometries are automatically Ricci-flat and possess a parallel spinor field. These two properties make them also very attractive for physicists and they occur in physics as "internal spaces" in compactifications of higher-dimensional supersymmetric theories. More generally, physicists use internal spaces with geometric structures which possess a so-called "characteristic" connection.My research project can be divided roughly into two parts. Whereas the first part is on certain generalizations of Kähler and hyperkähler geometry, the second part examines different geometric flow equations related to the Ricci-flat geometries from Berger's list.The first part deals more exactly with SKT-structures, which are generalizations of Kähler structures which possess a characteristic connections, and with complex-symplectic structures, which are generalizations of hyperkähler structures. In both cases, the goal is to classify, in cooperation with other mathematicians, these structures in a left-invariant context on different classes of nilpotent and solvable Lie groups. Note that in the SKT case, we will use the shear-construction for the classification, a construction which has been developped before together with my collaborator in this part of the project.The second part of the projects is on the spinor flow, the modified Laplacian coflow and the interplay between the Hitchin flow and group contractions. Also all these subprojects are collaborations with different mathematicians from London and other places in europe.The critical points of the first two flows are arbitrary or seven-dimensional Ricci-flat Riemannian manifolds (with additional data) from Berger's list respectively. Our aim is to study examples and properties of these relatively new flows in a homogeneous setting and other "symmetric" cases in order to gain a better understanding of these flows in general.The Hitchin flow is a geometric flow in six dimensions which produces seven-dimensional Ricci-flat Riemannian manifolds with holonomy in G2 (an "exceptional" case in Berger's list). Via so-called group contractions, physicists constructed left-invariant solutions of the Hitchin flow on six-dimensional Lie groups from left-invariant solutions of that flow on S^3\times S^3. We aim now at understanding this interplay between group contractions and the Hitchin flow in detail in the just mentioned cases and want to study this interplay systematically on S^3\times S^3 and other Lie groups.